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Power Laws in the Information Production Process: Lotkaian Informetrics

Library and Information Science Series Editor: Bert R. Boyce School of Library & Information Science Louisiana State University, Baton Rouge Recent and Forthcoming Volumes Mike Thelwall An Information Science Approach Donald Case Looking for Information Matthew Locke Saxton and John V. Richardson Understanding Reference Transactions: Turning Art Into a Science Robert M. Hayes Models for Library Management, Decision-Making, and Planning Charles T. Meadow, Bert R. Boyce, and Donald H. Kraft Text Information Retrieval Systems, Second Edition Charles T. Meadow Text Information Retrieval Systems A. J. Meadows Communicating Research V. Fronts, J. Shapiro, & V. Votskunskii Automated Information Retrieval: Theory and Methods Harold Sackman Biomedical Information Technology: Global Social Responsibilities for the Democratic Age Peter Clayton Implementation of Organizational Innovation: Studies of Academic and Research Libraries Bryce L. Allen Information Tasks: Toward a User-Centered Approach to Information Systems

Power Laws in the Information Production Process: Lotkaian Informetrics

Leo Egghe

2005

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© 2005 Elsevier Ltd. All rights reserved. This work is protected under copyright by Elsevier Ltd, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier's Rights Department in Oxford, UK: phone (+44) 1865 843830, fax (+44) 1865 853333, e-mail: [email protected]. Requests may also be completed on-line via the Elsevier homepage (http://www.elsevier.com/Iocate/permissions). In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 20 7631 5555; fax: (+44) 20 7631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of the Publisher is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier's Rights Department, at the fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. First edition 2005

ISBN:

0 12 088753 3

& The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.

My wife is thanking me for the many quiet evenings at the time of the writing of this book. To Ute, because she asked for it.

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PREFACE Explaining and hence understanding is one of the key characteristics of

human

beings. Explaining is making logical, mathematical deductions based on a minimum of unexplained properties, called axioms. Indeed, without axioms, one is not able to make deductions. How these axioms are selected is the only non-explainable part of the theory. In fact, different choices are possible leading to different, in themselves consistent, theories which conflict when considered together. A typical example is the construction of the different types of geometries, Euclidean and non-Euclidean geometries which are in contradiction when considered together but which have their own applications.

In this book the object of study is a two-dimensional information production process, i.e. where one has sources (e.g. journals, authors, words, ...) which produce (or have) items (e.g. respectively articles, publications, words occurring in texts, ...) and in which one considers different functions describing quantitatively the production quantities of the different sources. All functions can be reduced to one type of function, namely the size-frequency function f: such a function gives, for every n = 1, 2, 3, ..., the number f(n), being the number of sources with n items. This is the framework of study and in this framework we want to explain as many regularities that one encounters in the literature as possible, using the size-frequency function f.

As explained above, we need at least one axiom. The only axiom used in this book is that the size-frequency function f is Lotkaian, i.e. a power function of the form f(n)=C/n a , where C> 0 and a 3 0, hence also implying that the function f decreases. The name comes from its introduction into the literature by Alfred Lotka in 1926, see Lotka (1926). Based on this one assumption, one can be surprised about the enormous amounts of regularities that can be explained. In all cases the parameter a turns out to be crucial and is capable of, dependent on the different values that a can take, explaining different shapes of one phenomenon.

In this book we encounter explanations in the following directions: other informetric functions that are equivalent with Lotka's law (e.g. Zipf s law), concentration theory

viii

Power laws in the information production process: Lotkaian informetrics

(theory of inequality), fractal theory, modelling systems in which items can have multiple sources (as is the case in the system: articles written by several authors), modelling citation distributions. These models are developed from Chapter II on. Although the law of Lotka is an axiom in this book, in Chapter I we investigate what other models are capable of "explaining" Lotka's law. The only purpose for these developments is to understand why Lotka's law is choosen in this book (and not another type of size-frequency function such as, e.g., an exponential function) and hence to make the choice acceptable (although this is not needed, strictly speaking). For the same reason we also give an overview of situations where Lotka's law is encountered, being the majority of the situations (including the "new" situations as networks, including the Internet).

The author is basing himself on the publications that he has written the past 20 years on the subject but also benifits from the work of many others. To them my sincerest thanks. The author is especially grateful to Professor Ronald Rousseau (co-winner of the 2001 Derek De Solla Price Award) with whom he co-authored several papers but with whom he also had numerous long discussions (often by phone) on the different topics described in this book.

We strongly hope this book will serve the informetrics community in the sense that it shows the logical links between many (at first sight unrelated) informetrics aspects. The informetrician, having read this book, can use it each time he/she encounters a new informetric phenomenon (often in the form of a data set) in the sense that one can investigate if the phenomenon shows regularities that are (or can be) explained using the arguments given in this book. The mathematical knowledge required is limited to elementary mathematics such as first-year calculus. Other, more advanced topics, are introduced in this book.

The author is indebted to the Limburgs Universitair Centrum (LUC) and the University of Antwerp (UA) for their support in doing informetric research: in LUC, the author is chief librarian and coordinator of the research project "bibliometrics" while in UA, he is professor in the School of Library and Information Science, where he teaches the courses on informetrics and on information retrieval. The author thanks

Preface

ix

Mr. M. Pannekoeke (LUC) for the excellent typing and organization of this manuscript.

Leo Egghe Diepenbeek, Belgium Summer 2004

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TABLE OF CONTENTS

Preface

vii

Table of contents

xi

Introduction

1

Chapter I Lotkaian Informetrics: An Introduction

7

1.1 Informetrics

7

1.2 What is Lotkaian informetrics ?

14

1.2.1

ThelawofLotka

14

1.2.2

Other laws that are valid in Lotkaian informetrics

19

1.3 Why Lotkaian informetrics ?

25

1.3.1

Elementary general observations

26

1.3.2

The scale-free property of the size-frequency function f

27

1.3.3

Power functions versus exponential functions for the size-frequency function f

1.3.4

32

Proof of Lotka's law based on exponential growth or based on exponential obsolescence

34

1.3.4.1 Proof of Lotka's law based on exponential growth: the Naranan model

34

1.3.4.2 Proof of Lotka's law based on exponential obsolescence: solution of a problem of Buckland

40

1.3.5

Derivation of Mandelbrot's law for random texts

42

1.3.6

"Success Breeds Success"

45

1.3.6.1 The urn model

46

1.3.6.2 General definition of SBS in general IPPs

49

1.3.6.3 Approximate solutions of the general SBS

52

1.3.6.4 Exact results on the general SBS and explanation of its real nature 55

xii

Power laws in the information production process: Lotkaian informetrics

1.3.7

Entropy aspects

65

1.3.7.1 Entropy: definition and properties

66

1.3.7.2 The Principle of Least Effort (PLE) and its relation with the law of Lotka

70

1.3.7.3 The Maximum Entropy Principle (MEP)

76

1.3.7.4 The exact relation between (PLE) and (MEP)

78

1.4 Practical examples of Lotkaian informetrics

85

1.4.1

Important remark

85

1.4.2

Lotka's law in the informetrics and linguistics literature

86

1.4.3

Lotka's law in networks

87

1.4.4

Lotka's law and the number of authors per paper

90

1.4.5

Time dependence and Lotka's law

92

1.4.6

Miscellaneous examples of Lotkaian informetrics

94

1.4.7

Observations of the scale-free property of the size-frequency function f

98

Chapter II Basic Theory of Lotkaian Informetrics

101

11.1 General informetrics theory

101

II. 1.1

Generalized bibliographies: Information Production Processes (IPPs)

101

II. 1.2

General informetric functions in an IPP

104

II. 1.3

General existence theory of the size-frequency function

110

11.2 Theory of Lotkaian informetrics

114

11.2.1

Lotkaian function existence theory

114

11.2.1.1 The case pm = oo

114

11.2.1.2 The general case pm < oo

116

11.2.2 The informetric functions that are equivalent with a Lotkaian size-frequency function f

121

Table of contents

xiii

11.3 Extension of the general informetrics theory: the dual size-frequency function h

144

11.4 The place of the law of Zipf in Lotkaian informetrics

150

11.4.1

150

Definition and existence

11.4.2 Functions that are equivalent with Zipf s law

152

Chapter III Three-dimensional Lotkaian Informetrics

157

111.1 Three-dimensional informetrics

157

111.1.1 The case of two source sets and one item set

158

111.1.2 The case of one source set and two item sets

159

III. 1.3 The third case: linear three-dimensional informetrics

161

III. 1.3.1 Positive reinforcement

163

III.1.3.2 Type/Token-Taken informetrics

168

III. 1.4 General notes

172

111.2 Linear three-dimensional Lotkaian informetrics

175

111.2.1 Positive reinforcement in Lotkaian informetrics

175

111.2.2 Lotkaian Type/Token-Taken informetrics

177

Chapter IV Lotkaian Concentration Theory

187

IV. 1 Introduction

187

IV.2 Discrete concentration theory

192

IV.3 Continuous concentration theory

196

IV.3.1 General theory

196

IV.3.2 Lotkaian continuous concentration theory

199

IV.3.2.1 Lorenz curves for power laws

199

IV.3.2.2 Concentration measures for power laws

205

IV.3.3 A characterization of Price's law of concentration in terms of Lotka's law and of Zipf s law

214

xiv

Power laws in the information production process: Lotkaian informetrics

IV.4 Concentration theory of linear three-dimensional informetrics

218

IV.4.1 The concentration of positively reinforced IPPs

219

IV.4.2 Concentration properties of Type/Token-Taken informetrics

226

Chapter V Lotkaian Fractal Complexity Theory

231

V.I Introduction

231

V.2 Elements of fractal theory

232

V.2.1

233

Fractal aspects of a line segment, a rectangle and a parallelepiped

V.2.2 The triadic von Koch curve and its fractal properties. Extension to general self-similar fractals V.2.3 Two general ways of expressing fractal dimensions

234 236

V.2.3.1 The Hausdorff-Besicovitch dimension

236

V.2.3.2 The box-counting dimension

239

V.3 Interpretation of Lotkaian IPPs as self-similar fractals

242

Chapter VI Lotkaian Informetrics of Systems in which Items can have Multiple Sources

VI.l Introduction

247

247

VI.2 Crediting systems and counting procedures for sources and "super sources" in IPPs where items can have multiple sources VI.2.1 Overview of crediting systems for sources

253 254

VI.2.1.1 First or senior author count

254

VI.2.1.2 Total author count

254

VI.2.1.3 Fractional author count

255

VI.2.1.4 Proportional author count

255

VI.2.1.5 Pure geometric author count

255

VI.2.1.6 Noblesse Oblige

256

VI.2.2 Crediting systems for super sources

256

Table of contents VI.2.3 Counting procedures for super sources in an IPP

xv 256

VI.2.3.1 Total counting

257

VI.2.3.2 Fractional counting

258

VI.2.3.3 Proportional counting

258

VI.2.4 Inequalities between Q T (c) and Q F (c) and consequences for the comparison of QT (c), Q F (c) and Q P (c)

261

VI.2.5 Solutions to the anomalies

266

VI.2.5.1 Partial solutions

267

VI.2.5.2 Complete solution to the encountered anomalies

269

VI.2.6 Conditional expectation results on QT (c), QF (c) and Qp (c)

270

VI.3 Construction of fractional size-frequency functions based on two dual Lotka laws

276

VI.3.1 Introduction

276

VI.3.2 A continuous attempt: z 6 R+

278

VI.3.3 A rational attempt: q G Q+

282

Chapter VII Further Applications in Lotkaian Informetrics

295

VII.l Introduction

295

VII.2 Explaining "regularities"

297

VII.2.1 The arcs at the end of a Leimkuhler curve

297

VII.2.2 A "type/token-identity" of Chen and Leimkuhler

298

V'11.3 Probabilistic explanation of the relationship between citation age and journal productivity

300

xvi

Power laws in the information production process: Lotkaian informetrics

VI 1.4 General and Lotkaian theory of the distribution of author ranks in multi-authored papers

304

VII.4.1 General theory

304

VII.4.2 Modelling the author rank distribution using seeds

308

VII.4.3 Finding a seed based on alphabetical ranking of authors

310

VI 1.5 The first-citation distribution in Lotkaian informetrics

313

VII.5.1 Introduction

313

VII.5.2 Derivation of the model

317

VII.5.3 Testing of the model

320

VII.5.3.1 First example: Motylev (1981) data

320

VII.5.3.2 Second example: JACS to JACS data of Rousseau

322

VII.5.4 Extensions of the first-citation model

323

VII.6 Zipfian theory of N-grams and of N-word phrases: the Cartesian product of IPPs

326

VII.6.1 N-grams and N-word phrases

326

VII.6.2 Extension of the argument of Mandelbrot to 2-word phrases

329

VII.6.3 The rank-frequency function of N-grams and N-word phrases based on Zipf s law for N = 1

333

VII. 6.4 The size-frequency function of N-grams and N-word phrases derived from Subsection VII.6.3

347

VII.6.5 Type/Token averages nN and Type/Token-Taken averages \i"N for N-grams and N-word phrases

352

Appendix

365

Appendix I

365

Appendix II

370

Table of contents xvii

Appendix III Statistical determination of the parameters in the lawofLotka

372

A.III. 1 Statement of the problem

372

A.III.2 The problem of incomplete data (samples) and Lotkaian informetrics

373

A.III.3 The difference between the continuous Lotka function and the discrete Lotka function

378

A.III.4 Statistical determination of the parameters K, a, nmax in the discrete Lotka function K/na, n = 1 ,...,nmax

386

A.III.4.1 Quick and Dirty methods

387

A.III.4.2 Linear Least Squares method

388

A.III.4.3 Maximum Likelihood Estimating method

390

A.III.5 General remarks

393

A.III.5.1 Fitting Zipf s function

393

A.III.5.2 The estimation of pm and nmax

394

A.III.5.3 Fitting derived functions such as Price's law

394

A.III.5.4 Goodness-of-fit tests

395

Bibliography

397

Subject Index

423

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INTRODUCTION The most facinating aspect of informetrics is the study of what we could call twodimensional informetrics. In this discipline one considers sources (e.g. journals, authors, ...) and items (being produced by a source - e.g. articles) and their interrelations. By this we mean the description of the link that exists between sources and items. Without the description of this link we would have two times a one dimensional informetrics study, one for the sources and one for the items. Essentially in two-dimensional informetrics the link between sources and items is described by two possible functions: a size-frequency function f and a rank-frequency function g. Although one function can be derived from the other, they are different descriptions of two-dimensionality. A size-frequency function f describes the number f(n) of sources with n = 1, 2, 3, ... items while a rank-frequency function g describes the number g(r) of items in the source on rank r = 1, 2, 3, ... (where the sources are ranked in decreasing order of the number of items they contain (or produce)). So, in essence, in f and g, the role of sources and items are interchanged. This is called duality, hence f and g can be considered as dual functions.

Rank-frequency functions are well-known in the literature, especially in the economics and linguistics literature, where one usually considers Pareto's law and Zipf s law, respectively, being power laws. Less encountered in the literature (except in information sciences) is the size-frequency function. If studied, one supposes in most cases also a power law for such a function, i.e. a function of the type f(n)= C / n a with a > 0. Such a function is then called the law of Lotka referring to its introduction in the informetrics literature in 1926, see Lotka (1926). The law of Lotka gives rise to a variety of derived results in informetrics, the description of them being the subject of this book. That we choose a size-frequency function as the main studyobject is explained e.g. by its simplicity in formulation (in the discrete setting simpler than a rank-frequency function since the latter uses ranks which have been derived from the "sizes" n but also in the continuous setting, where sizes and ranks are taken in an interval, the formulation of the size-frequency function is more appealing and direct). A size-frequency function also allows for a study of fractional quantities (see Chapter VI), needed e.g. in the description of two-dimensional informetrics in which

2

Power laws in the information production process: Lotkaian informetrics

items can have multiple sources (as e.g. is the case with articles that can be written by several authors - incidently the original framework in which the law of Lotka was formulated in 1926). Furthermore, in this book we restrict ourselves to size-frequency functions that are power functions, i.e. Lotkaian informetrics. In Chapter I (a complete overview of the chapters will follow in the sequel) several arguments are given why power functions have to be preferred above other functions (e.g. exponential functions): a major reason is that power functions satisfy the so-called "scale-free" property meaning that the multiplication of the size n with a constant C > 0 does not change the type of power function, a property that is characterizing power functions. This is an important property since direct comparisons of results (using the same function) can be made whether we work with small numbers n or high numbers n. Also scale-free functions are the only ones capable of describing the self-similar fractal aspects of the informetric system under study (Chapter V). Furthermore in Chapter II we show that power-type size-frequency functions comprise all power-type rank-frequency functions (such as Zipfs law) but also other rank-frequency functions such as Mandelbrot's function and even exponential rank-frequency functions. This is an important theoretical advantage of the use of Lotka's law above any rank-frequency law: the class of scale-free size-frequency functions is strictly wider than the class of scale-free rank-frequency functions. This means that Lotkaian informetrics describes results which have a wider application than e.g. the study of Zipfian informetrics which is included in the Lokaian one.

In Chapter I we first explain the ingredients of informetrics and of Lotkaian informetrics. The use of Lotka's law for the size-frequency function could be considered here as an axiom in the sense that it is not explained but assumed - hereby allowing for other informetrics theories based on other size-frequency functions (a bit comparable with the establishment of different types of Euclidean and non-Euclidean geometries). Nevertheless we give, in Chapter I, several "explanations" of Lotka's law. These explanations are based, of course, on other assumptions that are not explained but in this way links between different regularities are given. So we present a proof of Lotka's law based on exponential growth, as proved by Naranan (1970) and a proof of Lotka's law based on exponential aging solving hereby a problem of

Introduction 3 Buckland (1972). Mandelbrot's law, being equivalent with Lotka's law is derived for random texts. The relation between Lotka's law and the "Success Breeds Success" phenomenon is presented. Finally, we give a description of the Principle of Least Effort (PLE) and the Maximum Entropy Principle (MEP) and show how Lotka's law can be derived from the (PLE) but not from the (MEP). To fully understand this remarkable result a characterization of (MEP) in terms of (PLE) and the newly defined Principle of Most Effort (PME) (defined in Egghe and Lafouge (2004)) is given. The extensive introductory Chapter I is then ended with an overview of practical examples of Lotkaian informetrics (including the recently found power-type laws in networks). Chapter II presents the basic theory of Lotkaian informetrics: describing the system of sources and items as information production processes (IPPs) and the general informetric functions to be studied: size- and rank-frequency functions. Given A, the total number of items and T the total number of sources, we give necessary and sufficient conditions for the existence of a size-frequency function giving these numbers A and T. Based on this we determine the existence of the corresponding Lotkaian functions. Then we determine the functions that are equivalent with the law of Lotka, for the diverse values of a > 0. In this connection we find the laws of Mandelbrot, Leimkuhler and Bradford (the latter one being a group-free version of the classical law of Bradford) and determine the relations between the parameters of these laws. We further prove that Zipfs law is, at least for special values of C, also equivalent with the law of Lotka. Also in Chapter II we define the dual size-frequency function h and prove its properties. This function is used later (in Chapter VI) to determine the number h(s) of authors with a fractional score s = 1/j per paper, j = 1, 2, 3,..., where j is the number of authors of a paper. This shows that the dual framework of IPPs and of size-frequency functions is capable to handle IPPs where items can be produced by multiple sources. Chapter III is devoted to the relatively new and intricate problem of three-dimensional informetrics. Especially linear three-dimensional informetrics is studied in general and in the Lotkaian case. Here two types of studies are given. One on positive reinforcement, i.e. where sources produce items and where these items become

4

Power laws in the information production process: Lotkaian informetrics

sources that produce again items, and one on Type/Token-Taken (TTT) informetrics in which not only sources and items (i.e. in the linguistical terminology: type and token) are considered but also the use (i.e. taken) that is made of these items. In both cases, concrete formulae for the size- and rank-frequency functions are given, based on the ones of the composing IPPs and consequences are given supposing Lotkaian informetrics. In both cases we also study the new averages of these cases of linear three-dimensional informetrics and prove that in all cases the average increases. Chapter IV is devoted to the study of concentration theory of Lotkaian informetrics, i.e. the determination of measures for the inequality between sources (with respect to the number of items they contain). Especially in Lotkaian informetrics this inequality is apparent due to the skewness of the Lotka function. Inspired by methodologies coming from econometrics we define the continuous Lorenz curves, which have the property that, the higher the curve, the more unequal the studied situation is. We also calculate these curves in the case of Lotkaian informetrics and prove that these curves increase in a (i.e. inequality increases with a). Based on these curves we are then able to present several good measures of concentration (i.e. measures of inequality) for Lotkaian systems. Also the concentration of the linear three-dimensional systems, studied in Chapter III, is determined: in a case of positive reinforcement the concentration is ruled by a theorem of Fellman (1976) which is reproved here; in the case of TTT we generally show that the concentration decreases. In this Chapter we also consider the so-called law of Price which we prove to be equivalent with Zipf s law with exponent 1, hence with Lotka's law with a = 2 and where the constant C takes a special value, a fact that was already found to be true (experimentally) in Allison, Price, Griffith, Moravcsik and Stewart (1976).

Chapter V gives the already mentioned interpretation of a Lotkaian system as a selfsimilar fractal, after giving a general introduction to fractal theory. The formula for the fractal dimension Ds of such a system (a measure of its complexity) then leads, via the theory of Naranan (1970), given in Chapter I, to the important relation: Ds = a - 1 .

Introduction 5 Chapter VI is completely devoted to the general theory of systems in which items can have multiple sources and to the Lotkaian involvement in this. We start with a general overview of crediting systems for sources, such as the total, fractional, proportional,... counting methods. We show the robustness of Lotka's law when going from singlesource items systems to multiple-source items systems when one uses the total counting system. These different counting procedures are also defined and applied to so-called "super sources", e.g. countries adding scores of the different authors (sources) in these countries. Some paradoxes in the relative scores between these procedures are highlighted and remedies given. Chapter VI then continues with the construction of fractional size-frequency functions based on the Lotka laws f and its dual (cf. Chapter II) h. The surprising result, using convolution theory, being irregularly shaped size-frequency functions over some rational scores is obtained by applying "ordinary" decreasing Lotka functions f and h. The closeness with practical data is remarkable.

The final Chapter VII gives further applications in Lotkaian informetrics. We alert the reader that not all regularities to be explained are of the informetric type: this is the case with the arcs at the end of a Leimkuhler curve and with a "type/token-identity" of Chen and Leimkuhler (1989). Based on the Central Limit Theorem in probability theory an explanation is given of the relationship between citation age and journal productivity, experimentally established by Wallace (1986), where part of the graph is explained using Lotka's law. Further, we give the general and Lotkaian theory of the distribution of author ranks in the byline of multi-authored papers. Lotkaian informetrics is also able to model the cumulative first-citation distribution where Lotka's a is the key parameter which decides on a concave shape (l < a < 2) or an S-shape (a > 2) of this distribution. Finally the size- and rank-frequency distributions of N-grams and N-word phrases are determined based on the law of Zipf (which is part of Lotkaian informetrics) for N = 1. We could also speak about a theory for the Cartesian product of IPPs. The Appendix is mainly devoted to the statistical determination of the parameters in the Lotka function. We first determine the influence of incomplete data (samples) on the law of Lotka. We show that Lotka's law is the only one allowing for the same size-

6

Power laws in the information production process: Lotkaian informetrics

frequency distribution for the complete and incomplete data, making it the only one treatable in sampling problems. Then we describe the difference between the continuous Lotka functions and the discrete ones, when they are used to model the same data and we determine the relations between the parameters. Then we present some statistical methods for the determination of the parameters: after some "quick and dirty" methods we give the linear least square method and the maximum likelihood estimating method.

I

LOTKAIAN INFORMETRICS : AN INTRODUCTION

I.I INFORMETRICS The concept of informetrics is well-known nowadays and could be defined as the science dealing with the quantitative aspects of information. This is the widest possible definition comprising mathematical and statistical treatment of diverse forms of information: books, articles, journals, references and citations, libraries and other information centers, research output, collaboration and transport (e.g. in networks such as intranets or the internet). Although there might be different opinions on this, informetrics could be considered (as we will do here) as comprising other disciplines such as bibliometrics or scientometrics. Bibliometrics can be defined as the quantitative study of pieces of literature as they are reflected in bibliographies (White and McCain (1989)) or by the well-known definition of Pritchard (1969): the application of mathematical and statistical methods to books and other media of communication (see also Narin and Moll (1977) and Tague-Sutcliffe (1994)). Scientometrics, coined in Nalimov and Mul'cenko (1969) as "naukometrija", deals with quantitative aspects of science, including research evaluation and science policy.

The term informetrics, we believe introduced in 1979 by Blackert and Siegel (1979) and by Nacke (1979), gained popularity by the organization of the international informetrics conferences in 1987 (see Egghe and Rousseau (1988, 1990b)) and by the foundation (during the fourth international informetrics conference in 1993) of the ISSI, the International Society for Scientometrics and Informetrics, hereby also recognizing the importance of the term scientometrics, mainly because of the existence of the important journal with the same name. It is not the purpose to repeat in this book all historic facts on the science informetrics since this has been covered many times in an excellent way in publications as White and McCain

8

Power laws in the information production process: Lotkaian informetrics

(1989), Ikpaahindi (1985), Lawani (1981), Tague-Sutcliffe (1994), Brookes (1990) and the more recent, very comprehensive (almost encyclopaedic) Wilson (1999).

Since, however, we intend to provide mathematical foundations for a part of informetrics (called Lotkaian informetrics, of course explained further on) we will provide in this overview a concrete description of the concept of generalized bibliography (or information production process) as well as the standard definitions of informetric functions, both as a concept and as they appear in the literature (then called laws). They form the basis of the informetric theory and will be formalized in Chapter II.

The main object of study in informetrics is the generalized bibliography, also called (e.g. in Egghe (1990a)) an "information production process" (IPP). The most classical example is, indeed, a bibliography (on a certain subject) where one has a collection of articles dealing with this subject. Of course, articles are published in journals and this is the basic aspect of IPPs: in the example of a classical bibliography, journals can be considered as sources that "produce" items, i.e. the articles collected in the bibliography. The point is that, in informetrics, one can provide several other examples of sources, containing items. Indeed one can consider the publications (articles) of an author also as a source (author)-item (publication) relationship: an author "produces" a publication. Even an article (being an item in the previous examples) can become a source e.g. "producing" references or citations as items. In a library, books (as sources) are the "producers" of loans (each time a book is borrowed this is an item belonging to the source, being the book itself). Although this example might seem more abstract than the previous ones it is a very natural example and of the same nature as (although completely different from) the following example. In quantitative linguistics (for a basic reference, see Herdan (I960)) one considers texts as (in our terminology) IPPs where words are considered as sources and their use in the text (i.e. each time a word appears in the text) is considered as an item. There one uses the terminology "Type/Token"-relationship for what we call here the "Source/Item"-relationship (see also Chen and Leimkuhler (1989)). In this book we will use both versions of the terminology: mainly source/item, since this is classical in informetrics; type/token being interesting descriptions of the same phenomenon and sometimes used, e.g. where it is more convenient such as in Chapter III. We are convinced this will not confuse the reader (again: there is no difference: source/item = type/token) and in this way we underline the fact that our

Lotkaian informetrics: an introduction

9

framework is applicable outside informetrics (although linguistics is a neighbor discipline of information science!).

Really outside informetrics we are still able to find the same framework. We can give the example in econometrics of workers or employees (as sources) in relation with their productivity (as items) (Theil (1967)). Productivity can be expressed by the number of produced objects by these sources or in terms of profits (amount of money earned by these sources). Even in demography one can consider cities and villages (as sources) in relation to their populations (each member being an item).

So, in general, we can define an IPP as a triple (S,I,F) where S is the set of sources, I the set of items and F a function indicating which item i e l belongs to which source seS. In this sense we can talk about two-dimensional informetrics studying sources/items and their interrelation by means of F. Two-dimensional informetrics is hence more (higher) than two times a one-dimensional informetrics theory developed separately on the sources (e.g. number of sources) and on the items (e.g. number of items). The function F can be considered as a relation F: S —> I or as a function F: S —> 21 (the set of all subsets of I) where, for each s e S, F ( s ) d , the subset of I containing the items that are produced by (that belong to) source s. The classical way of thinking, as we will do in this first chapter, is limiting the sets S and I to finite discrete sets (e.g. finite subsets of N , the set of natural numbers, so that we can count them). For the basic theory, however, in Chapter II, we will express the fact that IPPs usually contain many sources and items, by using continuous sets for S and I (such as intervals [a,b] c R + , the positive real numbers).

The reader might remark that this common framework is nice but does it lead us somewhere, i.e. is there a reason to formulate these objects in a common way? The answer is yes, for several reasons. First of all there is the challenge in itself to detect and define common tools or frameworks (such as IPPs) among these different sciences. Once this is done we can then elaborate these tools in a unified way e.g. by defining common measures and functions (such as distribution functions - see further, but these are not the only examples). An intriguing benefit of this approach would be if we detect or prove that some of these measures or functions have the same form in the different "-metrics" sciences! While leaving this

10

Power laws in the information production process: Lotkaian informetrics

"speculation" for what it is (for the moment, but watch the sequel!) we will now give some common functions and measures that can be defined in these IPPs.

So we suppose that we have an IPP (S,I,F), where - for the time being - S and I are finite sets. As defined above, for each se S, F(s) is the set (subset of I) containing the items that belong to (or are produced by) s. Some general quantities can be defined (note that this also helps us in the definition of standards in informetrics (and beyond), the need of it being advocated e.g. in Glanzel (1996) - see also Rousseau (2002a)). Knowing (S,I,F) we can determine, for every n s N (i.e. for every number n = 1,2,3,...) the number f(n) of sources with n items (i.e. having or producing n items). The function f is called the size-frequency function of (S,I,F) and is the basic informetric function in this book. Both its simplicity (see next section) and its power to determine quantitative regularities in IPPs will be highlighted during the elaboration of this book. Note that this function f can be determined by, simply, counting for each source the number of items its contains and then, for each ne N, counting the number of sources with this number n of items. Note that T = ^ f (n) denotes the total number of sources in the IPP and that A = ^ n f (n) denotes the total number of items in the IPP. Hence

A

E nf («)

(L1)

H= — = H h

T

f n

£( > n

is the average number of items per source in this IPP, an important measure. Here E means n

the addition (sum) for all existing values of n, usually ne {1,2,3,...,nmax} where nmaX is the maximal production of a source in this IPP. We note here the (intuitively clear) fact that not all values in {l,2,...,nmax} are effectively encountered: usually small values (1,2,3,...) always occur and in high quantities (in other words: there are many low productive sources) but higher productions are rare: high n values do not occur or, if they occur, not many sources (usually 1) have this number of items. This intuitive description is basic for the rest of the book. We will see (and model) the fact that a high number of sources have a low number of items and that a low number of sources have a high number of items. Stated in terms of f (and in an intuitive way): f is a (mainly) decreasing function of n (and, of course, f(n)>0 for all n). This simple property is the start of a limiting procedure for possible functions f which

Lotkaian informetrics: an introduction

11

will lead us (in this chapter) to the arguments why we will study "Lotkaian" informetrics from Chapter II on: Lotkaian refers to a certain special, simple, type of function f, to be introduced in the next section. The inequality in terms of the source production (number of items per source), as described above, will be explicitely studied (by defining so-called inequality measures) in Chapter IV.

Let us now continue to define more types of functions and measures in the IPP (S,I,F). First we need to define a ranking in the set of sources. This is done as follows: the source with the highest number of items receives rank 1, the source with the second highest number of items receives rank 2 and so on. Rankings for sources with an equal number of items are given consecutively where it is of no importance which source receives which rank. Example: Suppose S = {s 1 ,s 2 ,s 3 ,s 4 ,s 5 ,s 6 } and let Sj have 3 items, s2 have 5 items, s3 and s4 have 1 item and s5 and s6 have 2 items. Then s2 receives rank 1, s, receives rank 2, s5 receives rank 3 and s6 rank 4 (or s5 and s6 interchanged), s3 receives rank 5 and s4 rank 6 (again s3 and s4 can be interchanged). Stated mathematically in the notation of our IPP (S,I,F), we have, if r(s) denotes the rank of source s e S, then for all s,s' e S for which |F(s)| * |F(s')|:

r(s)|F(s')|

(1.2)

where <=> means "if and only i f (sometimes abbreviated to "iff) and where |A| denotes the number of elements in the set A (here F(s), F(s') respectively) (|A| is also called the cardinality of the set A).

Other rankings are possible in case of ties: one can give the same rank to all tied sources (usually the smallest possible) or one gives an "average" ranking. The choice of one or another ranking is, of course, of influence on the practical graphs one can draw from data using this rank (for a discussion on this, see Heine (1998) and for examples of graphs: see further on) and hence the statistics applied to these graphs (e.g. fitting, average ranks,...) are also dependent on the choice of the ranking device. We stress, however, the fact that our book (apart from an informative Appendix III on statistical fitting) is not dealing with the statistics of informetrics but with the mathematics (including probability theory) of informetrics. The

12

Power laws in the information production process: Lotkaian informetrics

used models throughout the book will not be influenced by any rank-type choice since we will work with continuous ranks and with item densities, approximating the real discrete world (exact definitions follow in Chapter II). In addition to this, as is obvious, the basic informetric function f, as defined above, is independent of rank-type choices (and only for this function some statistical features will be described in the Appendix III).

Nevertheless, to fix the ideas, for the functions and measures to be defined using ranks r, we suppose the definition (1.2) (although any other type of ranking as indicated above can also be taken). An important function, denoted by g, is defined as follows: let r be the rank of a source se S. Then g(r) denotes the number of items belonging to this source. The function g is called the rank-frequency function. We encounter here a first example of duality in IPPs. With duality in IPPs we mean the general principle that if we can measure regularities of sources with respect to items, we can always look for the corresponding regularity of items with respect to sources. Stated more succinctly: sources and items can be interchanged. In this setting, the functions f and g play a dual role: f(n) denotes the number of sources with ne N items, while g(r) denotes the number of items in the source on rank r. Duality is the main feature of informetrics theory and it will be basic for the theory that we develop from Chapter II on. Note that such a duality principle is naturally seen in some examples, e.g. the informetrics of articles (sources) giving references (items). The dual interpretation is that these references (sources) are cited by these articles (now considered as items). Putting it briefly, "citing" and "cited" are dual notions and we can only grasp this important aspect of informetrics by building it in an informetrics theory from the beginning. This is certainly the case in this book. Note that the above example of "citing" and "cited" has been studied in the literature in the terminology "synchronous citation analysis" (citing) and "diachronous citation analysis" (cited), cf. Stinson (1981), Stinson and Lancaster (1987). Another example of duality is given by the author-publication relation. While it is classic to consider here authors as sources, producing publications (=items) one can consider, certainly in case of multi-authored papers (dealt with in Chapter VI) articles (sources) having a number of authors (items). Other examples of duality can be given here. Throughout the book, however, the notion will be present permanently. Note that duality is similar with the same term in (projective) geometry where, any theorem involving points and straight lines (in this order) can be rephrased (and reproved!) into a theorem involving straight lines and points (in this order).

Lotkaian informetrics: an introduction

13

Because of the importance of the duality concept, we will define the dual IPP of an IPP (S,I,F). The dual IPP of (S,I,F) is the IPP (I,S, FD) where FD = F" 1 , the inverse relation of F, here considered as a relation from S into I. So far we defined already two important functions: f(n) = the number of sources with n items and g(r) = the number of items in the source on rank r. It is clear that

g-(n) = r(n) = £ f ( m )

(1.3)

, where nmax is the maximal number of items in a source (if this source is unique, it receives rank r = 1) and where g~' denotes the inverse function of g (we assume its existence in this introductory chapter boiling down to the assumption that g is injective; Chapter II will show this to be true in the continuous model). From (1.3) we can also conclude that, for each n, g"1 (n) equals the total number of sources with a number of items greater than or equal to n.

From the rank-frequency function g we can define the cumulative rank-frequency function G as follows:

G

W=Eg( s )

a-4)

5=1

, the cumulative number of items in the first (most productive) r sources.

Next we induce the order of S into I as follows: if i,i'el belong to the same source, these items are ordered arbitrarily and further, if i,i'e I do not belong to the same source we put

ir(s)
(1.5)

where s = F~' (i) and s' = F"1 (iT) and where the inequality at the right hand side of (1.5) is defined by (1.2). Now divide the ordered set I into p equal parts (p = 2,3,... if possible). Then, for the i* part (i p = l,...,p, following the order (1.5)) we define o p (i p ) (note the p-

14

Power laws in the information production process: Lotkaian informetrics

dependence) to be the number of sources producing the items in this ip' part (we note that, possibly, <5p(ip) is not an entire number due to the fact that in the beginning and the end of each item-group we have only a fraction of the items in a source in which case we add only this fraction to a p (ip J - this and also the fact that |I| must be a p-multiple are drawbacks of the discrete setting, which will not bother us anymore from Chapter II on: a continuous setting will solve all these problems). Finally, define the function £ p :

EP(ip) = | X ( k )

d.6)

the cumulative number of sources, producing the items in the first ip parts.

The reader might remark that f, g and G are classical while a p and E p are not. Yet, all five functions have well-known versions in Lotkaian informetrics as we will indicate now.

1.2 WHAT IS LOTKAIAN INFORMETRICS ? 1.2.1 ThelawofLotka The celebrated function, being the basis for the mathematical developments in this book, is the so-called law of Lotka, defined in Lotka (1926). Its definition is very simple: it assigns a concrete simple formula to f(n), the size-frequency function, introduced in the previous section. The formula is:

f

(*) = £ >

(L7>

where C > 0 and a > 0 are constants. So the number of sources with n items is proportional to a decreasing power law.

The historic description in Lotka (1926) is in terms of contributors or authors (as sources) and contributions or papers (as items) (incidentally, Lotka himself also mentions the term

Lotkaian informetrics: an introduction

15

"items"!) and deals with publication counts of authors in the Chemical Abstracts (1907-1916) (only author names beginning with the letter A or B were covered!) and in Auerbach's Geschichtstafeln der Physik of 1910. Lotka determined the exponent a to be around 2.021 for Auerbach's data and to be around 1.888 for the Chemical Abstracts data. The method used was linear regression on the log-log version of the data and the function (1.7) (In = loge):

lnf(n) = l n C - a l n n

(1.8)

Although it is not the main topic of this book, Chapter II and the Appendix III are devoted to more correct calculation methods for the parameters of functions of the form (1.7). Rousseau (oral communication) arrives at an exponent a = 1.967 for Lotka's Chemical Abstracts data and at a = 2.055 for Auerbach's data, both very close to Lotka's original findings! Rousseau used his program to fit Lotka's law, which is available at Rousseau and Rousseau (2000), see also the Appendix III. It is underlined in Yablonsky (1985), Haitun (1982b), Ivancheva (2001), Huber (2001) that statistical fitting of a power law often is impossible, certainly for low a . This has lead some scientists to (completely) avoid the use of power laws and to work with decreasing exponential functions instead (i.e. functions of the form c.a" where c and a are constants and 0 < a < l ) (see Seglen (1992), Huber (1998c, 1999, 2001, 2002), Huber and Wagner-Dobler (2001a,b)). In the next section, however, we will give mathematical arguments (found in the literature as well as new ones) why an exponentially decreasing function is not the right one (and a power function is the right one) for the description of f(n), hereby also closing the debate on the use of the exponential function for f(n), also in statistical fitting and explaining the universality (of occurrence) of the power law.

The real merit of the paper Lotka (1926) is the introduction of the function (1.7) into informetrics (not called this way in 1926!) and to give an indication for the exponent a , called the Lotka exponent: a is indeed the more important one among the two constants (C and a ) appearing in (1.7): C is only used to make sure that Zlf (n) gives the total number T of sources (or, in case of normalization, i.e. of transforming f(n) into the fraction of sources with n items, to have that Z)f (n) = 1). The exponent a will be the most important parameter in this book, describing not only (1.7) and its theory (Chapter II), but also indicating e.g. the degree of concentration (inequality) (Chapter IV) or of complexity (Chapter V): parameter a will play a central role in the mathematical description of all the topics in every chapter of this book.

16

Power laws in the information production process: Lotkaian informetrics

The paper Lotka (1926) introduced the general law (1.7) but found values of a around a = 2. Formula (1.7) for a = 2 is considered as the "most classical" version of Lotka's law but, again, Lotka himself introduced (1.7) in Lotka (1926) for general a . If a = 2, Lotka also gives the following calculation for the fraction

of sources with n items: since

JIM

cg^2 T^n

£f T and since

&K

(U1)

(cf. e.g. Apostol (1957), Gradshteyn and Ryzhik (1965) or Protter and Morrey (1977)) we have that

£W = £ T

T

6 7i2

indicating that, if a = 2, the fraction of sources with 1 item is about 0.61; in other words, around 61% of the sources produce only 1 item! This shows how "unequal" the situation, expressed by (1.9) is: relatively many sources produce few items (and hence relatively few sources produce many items), a fact that will be discussed in mathematically more exact terms in Chapter IV. On the same lines such remarks can be made for (1.7) (any a > 0) and the exponent a will determine the "degree" of this inequality among the sources. To f(l) determine the fraction —— of sources with 1 item, for general a , we can proceed as follows: again we have

Lotkaian informetrics: an introduction

17

so

c

-

f ( 1 )

!

-

a 13) M

T

T

|J

I

f^J_ n=l

n

which can be determined from the tables of the so-called Riemann-zeta function

n=]

n

We again refer to the Appendix III for an approximative formula and Table for (1.14), given a , and for a statistical method to estimate a itself. In the theory, to be developed from Chapter II on, we will see that, in the continuous setting, it will be much easier to determine C and a , given T (the total number of sources) and A (the total number of items). The Riemann-zeta function is not needed there. We can say that Chapter II will present a method of "mathematical fitting" of the Lotka function, complementing the statistical fitting, given in the Appendix III.

The law of Lotka is the simplest among the informetric laws (to be formulated in the sequel of Section 1.2). Yet it is remarkable that its historical formulation in Lotka (1926) was in terms of authors (as sources) and their publications (as items). Why is this remarkable? Of the many examples of IPPs, formulated in Section I.I, it is the only one where items can have multiple sources, i.e. in terms of the historical formulation, articles can have multiple authors. In all the other examples an item is produced by a single source: an article by 1 journal, a reference is produced by 1 article or book (the same for a citation), a loan refers to 1 book, a word token in texts obviously comes from 1 word type, an inhabitant is linked to 1 city or village, a produced item (in economics) (e.g. money earned) to 1 employee (although in this case an example of a multiple source item can be given in the sense that a product can be produced by the collaboration of several employees).

18

Power laws in the information production process: Lotkaian informetrics

Of course, in a purely dual setting (section I.I) any IPP can, if necessary via its dual IPP (see section I.I), be considered as a system in which items can have multiple sources, since in the original IPP, sources usually have more than 1 item (and then interchange the names "source" and "item"). Especially the IPPs defined via references or citations can be considered in this way.

In any case, the author-article relationship is, because of the above remark, rather unique in informetrics (contrary to the statements in Fedorowicz (1982a) and White and McCain (1989)). This is the reason why we added a special Chapter (VI) solely devoted to the aspect of this multiple source possibility. One of the main issues in this setting is the question how to count the number of items per source. Indeed, suppose we have an item with 3 sources, e.g. an article co-authored by 3 authors. Will each author receive a (production) credit of 1 (total count) or only of — (fractional count, see also De Solla Price (1981))? Of course, the fractional counting method is in better agreement with the dual framework: "if there are n sources per item then there are, in a dual vision, — items per source" (n e N). n One can even consider straight counting which gives a credit of 1 to the first author and 0 to the other authors (Cole and Cole (1973)). A variant of this is to give a credit of 1 to the senior author (usually the last author, at least in those disciplines where the senior author is identified) and 0 to the other authors : noblesse oblige, see Zuckerman (1968) or see Subsection VI.2.1.6. For more on these and other counting methods we refer the reader to Egghe, Rousseau and Van Hooydonk (2000) and to section VI.2. Lotka (1926) used the latter method (senior author count), hence treating papers as single authored and hence also circumventing the problem of multiple authorship. Potter (1981) thinks to have an explanation for Lotka's choice of counting only senior authors. First, in Lotka's Chemical Abstracts data, only the first four authors are indexed. However, the second, third and fourth author only receive a "see" reference to the first author and from this it is a lot of work to compile author productivity data based on total counts. Potter then assumes that Lotka avoided this amount of work and therefore only counted the senior author. Yet it remains an interesting problem (addressed in Chapter VI) why Lotka's law is so widespread both in IPPs where items have only one source and in IPPs where items can have multiple sources (see Section 1.4). Even Lotka himself mentions the "wide range of applicability of the power law to a variety of

Lotkaian informetrics: an introduction

19

phenomena" and even refers to econometrics and biometrics and to aspects of concentration (inequality), see Chapter IV. Also interesting is: how do we go from laws in IPPs where we have single source items to laws in IPPs where we have multiple source items, both in the total and fractional counting system? We will see that Lotka's law will be the key function in both explanations (Chapter VI).

1.2.2 Other laws that are valid in Lotkaian informetrics

In a purely formal sense it is impossible, in the discrete setting, to produce formulae for functions (such as g, G, a p or S p introduced in Section I.I) that are equivalent (or at least imply) Lotka's law. The reason is that discrete sums (as occurring in (1.3), (1.4) and (1.6)) cannot be evaluated. A real powerful Lotkaian informetrics theory can only be produced in a continuous setting, as will be done from Chapter II on. Now we will limit ourselves to the definition of some historical laws which we will prove in Chapter II play a role in (continuous) Lotkaian informetrics; otherwise stated: laws which are compatible with Lotkaian informetrics.

Related to the function g (1.3) we have two versions. First there is the law of Zipf:

g(0 = 7'

(L15)

where D and P are positive constants. In this setting, the number of items in the source on rank r is a decreasing power function of r. In this sense, this law is equal (as a mathematical function) to the one of Lotka (1.7) but it will be seen in Chapter II that the exponents are not the same and, furthermore, that Lotka's law embraces a wider class of rank-frequency functions, named after B.B. Mandelbrot, the inventor of fractal theory (complexity theory see also Chapter V or Mandelbrot (1967, 1977a,b)): the law of Mandelbrot:

g(r)=

(T7kr

(L16)

20

Power laws in the information production process: Lotkaian informetrics

where E > 0, F and p are constants (as will be seen in Chapter II, [3 can be negative but then F < 0, keeping g decreasing).

Of course, dividing by F p in the nominator and the denominator of (1.16) leads to the following trivial equivalent variant of (1.16)

8

<"=(Ffr

E 1 where E' = —r, FT = — , see e.g. Arapov (1982), Tague-Sutcliffe (1994). Fp F While the origin of Lotka's law is very clear (the article Lotka (1926) itself!) this is not at all the case for the law of Zipf. It is sure that the law originates from linguistics and is named after G.K. Zipf because he promoted this regularity, mainly in his popular book Zipf (1949). But it is sure that the mathematical regularity (1.15), for P = l , appears already in Condon (1928). Zipf s law is sometimes called the law of Estoup because of an implicite reference to this regularity in Estoup (1916). For more on the history of this law we refer the reader to Petruszewycz (1973), Rousseau (2002b) or to Wilson (1999). The origin of the law of Mandelbrot is clearer and goes back to Mandelbrot (1954) - see also Mandelbrot (1977b).

Zipf s law is a special case of Mandelbrot's: take F = 1 in (1.16) and make the transformation r' = 1 + r . We then obtain the function (1.15) in r ' . Of course there is the different starting point of the ranks r and r' but we could take r > 0 in (1.16) (as will be done in the continuous model in Chapter II), yielding r = 1 as the starting rank in (1.15) which is natural for (1.15): starting from r = 0 in (1.15) does not make much sense, since g(r) then diverges. Using r > 1 for (1.15) in the continuous setting will also yield an application of Zipf s law in Chapters IV and VII. Taking F = 1 as above, and r = 1,2,...,9, we have that (1.16) is the so-called "law of anomalous numbers", being the rank distribution of the digits r as they appear as first digits in numbers in real life - see Brookes (1984b), Feller (1948), Brookes and Griffith (1978). The law of anomalous numbers is also known as Benford's law or the "significant-digit" law, see e.g. Benford (1938), Varian (1972) and Fairthorne (1969).

Lotkaian informetrics: an introduction

21

In Chapter II we will prove that, if a * 1, Mandelbrot's law is mathematically equivalent with Lotka's law yielding also a relation between the exponents a and p . In other words, Mandelbrot's informetrics is the same as Lotkaian informetrics; that is why we mention the law here. Zipfian informetrics, hence, is a part of Lotkaian informetrics but this implies that the law of Zipf is compatible with the law of Lotka, meaning that results proved using the law of Zipf are results that are part of Lotkaian informetrics, provided we take r > 0 in (1.16) and r > 1 in (1.15). For the exact relation between the law of Lotka and the one of Zipf we refer to Section II.4. The prominent role of Zipf s law in Lotkaian informetrics is the reason why we will encounter this law throughout this book.

It is now the moment to mention the law of Pareto. In most econometric publications one finds (1.15) as the definition of the law of Pareto, usually rewritten as follows:

D

i \ n = g(r) = -r

, hence

r = r(n) = -

r

nB which is of the form (equivalent with Zipf s law)

r(n) = ^

(1.18)

and in words: the number of sources (employees) with a number of items (income) larger than or equal to n is given by (1.18). Pareto's law is much older than Zipf s and this shows how the same regularities (albeit in different subjects) are found in an independent way. The formulation of Pareto's law goes back to 1895, see Pareto (1895). So unlike Zipf, Pareto is really the inventor of "his" law! For more on Pareto's law in econometrics, see also Theil (1967).

22

Power laws in the information production process: Lotkaian informetrics

Related to the function G (1.4) we can define the law of Leimkuhler

G(r) = alog(l + br),

(1.19)

where a and b are positive constants. In (1.19) we can take any log, e.g. loge = In. In Chapter II we will prove in the continuous setting that the law of Leimkuhler is equivalent with the law of Mandelbrot for P = 1. Variants for G, equivalent to all rank-frequency laws, discussed above, will be given in Chapter II. The name of law (1.19) comes from its introduction in Leimkuhler (1967).

We finally discuss the historic informetric law of Bradford. Introduced in Bradford (1934) for classical bibliographies, with the preparation given in section I.I (definition of the function c p ) we can formulate Bradford's law easily as

a p (i p ) = rok'>-\

(1.20)

ip = l,...,p, where r0 and k are positive constants and k > 1. Hence r0 is the number of sources in the first part of I (see the description in section I.I), called the nucleus, rok > r0 is the number of sources in the second part of I and so on: each time the previous number of sources is enlarged via the multiplication with k > 1. For this reason, k is called the Bradford multiplicator. There is much to discuss about this law. First of all, it is a regularity that we only see defined in informetrics but its formulation is so "surprising" that, in a way, it became the most "popular" informetrics law. Secondly, as formulated here, formula (1.20) is not a fixed law but changes every time we change the number p of groups: indeed, r0 and k and hence a p are p-dependent. Furthermore, in the discrete setting, as remarked in Section I.I, strictly speaking it is not possible (in most cases) to even "formulate" Bradford's law since, for a chosen number p of groups, we need that |I|, the number of items, is a p-multiple and even if this is the case, by dividing I into p parts, we will split up sources in the sense that part of their items will be in one group and the rest in the previous our next group, an extra complication for a .

Lotkaian informetrics: an introduction

23

To remedy this last problem it is clear that, if we change I and S into, say, continuous intervals, the problems of p-multiples and of splitting sources is solved since sources and items are points in a continuous interval. Of course, even in this setting, the second problem of p-dependence of crp remains. In Chapter II a unique function a, independent on p will be introduced. We can describe it here (intuitively), based on our introduction of the function Op. Let the item set I be the interval [0,A] and let i g [0,A] arbitrarily. Define

ip=Y+l.

(1-21)

where [x] denotes the entire number smaller than or equal to x, x 6 R + . Then ip denotes the number of the group to which i belongs, when there are p groups. Define (cf. also (1.6))

E ( 0 = ] s | > p M = £2EP(ip)

(L22)

and then a can be taken as £ ' , the derivative of J2 w.r.t. i (if all these operations exist!). In Chapter II we will not follow this complex way of definingCT: the function a will be defined from the very beginning of the informetrics theory, being an essential ingredient of a continuous IPP: a discrete IPP (S,I,F) (as used here) will be modified so that S and I are intervals and F is replaced by (a variant of) a . From this set-up the other laws and their properties will follow. There we will see that, in Lotkaian informetrics, equivalent with Lotka's law with exponent a = 2, the "universal" (i.e. group-free) version of Bradford's law has the (in view (1.20) expected) form

a(i) = M.K'

(1.23)

i e l = [0,A], where M and K are positive constants (K being the universal Bradford multiplicator). Since we do not see much application of the group-dependent Bradford formulation (the notions of nucleus, and important or less important sources according to the number of the group to which they belong, do not make much sense since p is not determined and even if we determine p (p = 3 is sometimes choosen for historic reasons) we do not need

24

Power laws in the information production process: Lotkaian informetrics

Bradford's law to divide the items and subsequent sources into p parts), we will not go back to it from Chapter II on. We hope that this introduction fully explains

the importance of Bradford's historical formulation the fact that it forms the basis for a new informetric function a which in turn is the basis for the informetric theory to be developed from Chapter II on. It is stressed that this function a - as far as we know - does not occur in any other "-metrics" science as discussed before although, as we will show, the other laws (e.g. Zipf, Pareto, Mandelbrot,...) are a consequence of it.

Of course, also in Chapter II, we will determine the exact form of a equivalent with Lotka's law in the case that Lotka's a ^ 2 . We can say that the introduction of a (as first done in Egghe (1989,1990a)) gives a functional "completion" of the "strange" "incomplete" law of Bradford: the only historical law that did not present a simple function as regularity. The function £ will be related to the so-called graphical formulation of Bradford's law, see Wilkinson (1973), Narin and Moll (1977) or Vickery (1948) (the law (1.20) is, in this context, sometimes called the verbal formulation of Bradford's law) in which an exponential function (as in (1.20)) is used, not to model the number of sources a p (ip J in the i* group but to model the cumulative number of sources ^

(ip) = ^ a p (k) (cf. (1.6)) in the first ip groups and we P

k=l

will make clear which forms of J2 are part of Lotkaian informetrics (and this part will be characterized). For more on the classical formulation of the graphical law of Bradford, we refer to Egghe (1988,1991). For more general or historic texts on (Lotkaian) informetrics we refer the reader to Broadus (1987), Narin and Moll (1977), White and Me Cain (1989), Ikpaahindi (1985), Lawani (1981), Bonitz (1982), Tague-Sutcliffe (1994), Bookstein (1976,1979), Soyibo and Aiyepeku (1988), Wilson (1999), Egghe and Rousseau (1990a), O'Conner and Voos (1981), Hertzel (1985), Buckland and Hindle (1969), Fairthorne (1969).

Lotkaian informetrics: an introduction

25

1.3 WHY LOTKAIAN INFORMETRICS ? In this section we will search for the reasons why the power function (1.7), being the one defined by Lotka, is important to study and why it is more important than possible other functional forms for f(n), being the number of sources with n items in a gneral IPP. At the end of this section it will be clear that the power law

f(n) = — n

(1.24)

for f is the most important type of function to be used for the size-frequency function f, certainly from a mathematical modelling point of view. In other words, in this section, we will search for explanations of Lotka's power law for the size-frequency function. Of course, as in every mathematical theory, one can only prove regularities when one makes assumptions, e.g. axioms, being unexplained in themselves. Let us make the comparison with Euclidean geometry. Here, geometry is developed starting from the Euclidean axiom that for every straight line and every point not on this line there exists exactly one straight line, through this point, that is parallel with the first straight line. There exist also non-Euclidean geometries where there are two (or more) parallel straight lines (hyperbolic geometry or geometry of Gauss, Bolyai, Lobachevsky) or where there is no parallel straight line through this point (elliptic geometry or Riemannian geometry). All three geometries have important applications and hence it depends on the problem one wants to study, which of these geometrical theories one can use - see also Greenberg (1974). So, the starting axiom is not an expression that is right or wrong in the absolute sense but an axiom is an expression that is accepted without further explanation, within the specific theory. Of course, "later on" the real (practical) value of one or another theory will be determined by the number of applications each theory gives.

In the above sense, the most elementary starting point for Lotkaian informetrics could be by simply putting formula (1.24) as an "axiom" hereby also opening possibilities for nonLotkaian informetrics. In a way this is what we will do in this book from Chapter II on. In this section, however, we will indicate that Lotka's formula (1.24) can be proved, of course then using other assumptions (as said, one has to suppose something!). The value of this is that we learn about situations (or aspects) of informetrics that belong to Lotkaian informetrics. The

26

Power laws in the information production process: Lotkaian informetrics

more "links" we can create the more we show the importance of Lotkaian informetrics. It is in this sense (only) that the word "explanations" of Lotka's power law must be interpreted.

1.3.1 Elementary general observations

When looking for (or explaining) a function f(n) with a mathematical expression (such as (1.24) but not only this one) one necessarily allows that f is a function that can also have positive non-integer arguments n and values f(n). Indeed, e.g. (1.24) can be interpreted for n e N but also for n e E + (the positive real numbers). In the latter case, f(n) represents a density of the sources with a density n of items (see Chapter II for exact definitions). In this context we can, undisputedly, state (as axioms, if you wish) that f > 0 and that f decreases in n. These elementary requirements belong - at least to the author's perception - to any informetrics theory (and not only to Lotkaian informetrics). We can then already state and prove the following elementary result, valid in any informetrics theory.

Proposition 1.3.1.1: Let f be a real valued function on 1R+ such that f > 0 and that f decreases. Then there exists a constant L > 0 such that

limf(n) = L

(1.25)

n—»cc

Proof: Since f decreases and has a lower bound (0 in this case) we have that limf (n) exists (see Apostol (1957) or Protter and Morrey (1977)). Let L be the limit. If L < 0 then, necessarily, there exists n such that f (n) < 0, contradicting f > 0.

D

Finally and evidently, the function f (defined on a subset of R + ) should be continuous. We therefore formulate the following three axioms of general mformetric theories.

Axioms 1.3.1.2: Let f be the size-frequency function on any IPP. Then

(ii)

f decreases

(iii) f is continuous

Lotkaian informetrics: an introduction

27

Note that this implies (by Proposition 1.3.1.1) that (1.25) is valid. We do not require that L = 0 but it will be a consequence of our further elaboration of the proporties of f in the direction of Lotkaian informetrics.

Of course, there is an infinity of types of functions f satisfying the axioms 1.3.1.2. One would expect that the next step towards Lotkaian informetrics is by putting extra requirements on f by means of its derivatives e.g. requiring f to be differentiable and requiring f' and/or f" to have certain signs (e.g. f " > 0 i.e.: f convex). This is, however, not necessary. The next subsection will lead us directly and naturally to Lotkaian informetrics (incorporating the convexity of the size-frequency function f).

This will then automatically exclude the exponential function as a model for the sizefrequency function f. Note that, at the present stage, both the decreasing power function (1.24) and the decreasing exponential function f(n) = can, where 0 < a < 1, satisfy the Axioms 1.3.1.2 (and also (1.25) with L = 0) and they are the most important examples of functions satisfying these axioms. Note also that, with this approach, we follow completely the lines of reasoning of Griffith (1988) pleading for a development "of the simplest bibliometric models utilizing few and explicit parameters, and then, attempts to determine how much structure in very large data sets can be readily and simply explained".

1.3.2 The scale-free property of the size-frequency function f

A natural requirement for a function that measures frequencies is its scale-free property. It is defined as follows.

Definition 1.3.2.1: A function f is called scale-free if, for every positive constant C, there is a positive constant D such that

f(Cx) = Df(x)

(1.26)

for all x in the domain of f (i.e. the set on which f is defined) such that Cx also belongs to the domain off. Note that the constant D depends on C but not on x.

28

Power laws in the information production process: Lotkaian informetrics

This is a very important property of f: it guarantees that if there is a change of scale (e.g. change of currency, change of scientific discipline, change of books, change of species, or any transformation e.g. x —> Cx, i.e. where ratios are constant and the value zero is the same - this is called a ratio scale - see Roberts (1979)) we still measure the size-frequency by the same function f. Hence, derived properties such as concentration, complexity - see further, or more simply, the informetrics theory as a whole, remain(s) the same. Hence this also covers evolution in time where the same IPP is studied but over different time intervals (e.g. a bibliography over 10 or over 30 years). Another way to express (1.26) is that f transforms ratio scales into ratio scales (cf. also Luce (1959)).

We have the following important result (see Luce (1959), Roberts (1979)). Theorem 1.3.2.2: The following assertions are equivalent:

(i)

f is a function on R + to K + , is continuous and scale-free

(ii)

f is a power function, i.e. there exist constants a € R + , b e R such that

f(x) = axb

(1.27)

for all x in the domain of f. Proof: We refer to the Appendix I or to Roberts (1979) (Section 4.2) for the complete proof of this known result. Here we content ourselves by checking the easy part (ii)=>(i). Indeed, a power function f(x) = axb with a > 0 satisfies f: K+ —> K+ , f continuous and furthermore, for each C > 0 we have

f(Cx) = a(Cx) b

= aCbxb

= Df (x) ,

Lotkaian informetrics: an introduction

29

where D = Cb > 0 is dependent on C but not on x.

The above theorem has the following important consequence for Lotkaian informetrics.

Corollary 1.3.2.3: The following assertions are equivalent:

(i)

f is a function on K+ to R + , is continuous, decreasing and scale-free (i.e. Axiom 1.3.1.2 + the scale-free property)

(ii)

f is a decreasing power function, i.e. there exist constants C, o e l

f(x) = - ^

+

such that

(1.28)

i.e. a decreasing power law, i.e. Lotka's law.

Proof:

(i) => (ii)

It follows from Theorem 1.3.2.2 that

f(x) = axb

with a e R + and b e R . Since f decreases, we have that b < 0. Hence the corollary is proved by putting a = C and a = -b.

(ii) = (i) It is clear that function (1.28) satisfies all the requirements of (i) (see e.g. the proof of Theorem 1.3.2.2).

D

Next we note the equivalency of the scale free property with the so-called product property.

30

Power laws in the information production process: Lotkaian informetrics

Definition 1.3.2.4 (Egghe (2004d)): A function f: R + -> R + is said to have the product property if there exists a positive constant E such that

f(xy) = Ef(x)f(y)

(1.29)

for all x,y in the domain of f.

Proposition 1.3.2.5 (Egghe (2004d)): The following assertions are equivalent for a function J? . TTn+

t . K.

TTJ+ .

—> IK

.

(i)

f is scale-free

(ii)

f has the product property.

Proof:

(ii) => (i)

Since for all x,y in the domain of f: f(xy) = Ef(x)f(y) by (1.29) we can take C = y yielding f(Cx) = Ef(C)f(x) = Df(x) with D = Ef(C) which is independent of x.

(i) =

(ii)

The scale-free property of f implies the existence of a function D(y), only dependent on y, such that, for all x

f(xy) = D(y)f(x)

(1.30)

(namely take C = y and D = D(y) in Definition 1.3.2.1). For x = 1, (1.30) yields

f(y) = D(y)f(l)

Now (1.30) and (1.31) together yield

(1.31)

Lotkaian informetrics: an introduction

31

f(xy) = Ef(x)f( y )

where E = —-r— > 0 is an absolute constant.

Note that, in the above proof, we did not need the continuity off. Of course, in order to have a power function, we need the continuity of f (Theorem 1.3.2.2).

The product property is, essentially, the same as the property repeatedly discussed by Bookstein in Bookstein (1977, 1990b, 2001) but there the constant E is taken as 1 due to the requirement that f (l) = 1. There is, however, no reason to require this since in case f is a distribution f (l) =A 1 and also if f denotes the actual number of sources, then also f (l) Furthermore Proposition 1.3.2.5, together with the previous results show that (1.29) is the more natural property, characterising power functions of the type (1.27) or (1.28).

In view of Corollary 1.3.2.3 we can say that Lotkaian informetrics is the same as scale-free informetrics and this explains its importance. Scale-free informetrics guarantees that different IPPs can be compared and also that evolutions in time of IPPs can be compared since the same function f - necessarily a power function - applies. As a consequence, derived measures remain the same (such as concentration measures - Chapter IV), all results derived from Lotka's law remain applicable with the same parameters and, finally, complexity theory (Chapter V) takes the same form for all scale-free (i.e. Lotkaian) informetrics theories. As a consequence of this we will show that Lotkaian IPPs (i.e. IPPs for which Lotkaian informetrics applies) are self-similar fractals, yielding a big advantage for the calculation of the fractal dimension.

We refer the reader to Katz (1999), Balasubrahmanyan and Naranan (2002), Bilke and Peterson (2001), Jeong, Tombor, Albert, Ottval and Barabasi (2000) and Barabasi and Albert (1999) for further discussions on scale-free systems (incl. scale-free aspects of networks, which is very important with respect to size changes) - see also Subsection 1.4.3.

1.

32

Power laws in the information production process: Lotkaian informetrics

Due to the given characterization of power functions, it is immediately clear that exponential functions are not scale-free. This willl be further elaborated in the next section, where additional advantages of power functions over exponential functions will be given.

1.3.3 Power functions versus exponential functions for the size-frequency function f

In the previous subsection we explained the scale free property of the power function (1.28). Suppose now, temporarily, that the size-frequency function f is of the exponential type:

f(x) = cax

(1.32)

for all x 6 M+ . Since f decreases (Axiom 1.3.1.2 (ii)), we have that 0 < a < 1 necessarily. The unimportant case a = 1 corresponds to the constant function (or a = 0 in (1.28)) and is the only function in the intersection of the class of power functions (1.28) and the class of exponential functions (1.32). Let us now take 0 < a < 1. The replacement of x by Cx (C > 0, C ^ 1) yields in (1.32):

f(Cx) = ca c *=c(a c ) x =cb*

(1.33)

where b = a c > 0, yielding a different exponential function, as remarked in the previous subsection. As will be clear from the rest of the book this is a dramatic difference between power and exponential functions, the disadvantage being on the side of the latter one. Let us now note some other pitfalls of the exponential function as applied to the size-frequency function: it must, indeed, be underlined that the exponential function is very important for measuring evolutions in time (e.g. growth or aging of literature) - see this chapter and the last one, and that the disadvantage of the exponential function is only discussed in reference to the size-frequency function of IPPs.

In Anderson and Tweney (1997) another elementary observation is made which also is in the advantage of power functions. For the power function f (formula (1.28) - now called f,) we have that

Lotkaian informetrics: an introduction f]'(x) = -aCx" ( ' + a ) = Kfi(x)/x,

33 (1.34)

where K < 0 is a constant, while for the exponential function f (formula (1.32) - now called f 2 ), we have

f2(x) = (clna)a x =Qf 2 (x),

(1.35)

where Q < 0 is a constant. In terms of memories f, the function f' measures the "forgetting" process and the function f 7f measures the rate of this change. It is generally accepted that this rate declines in x (e.g. x = t = time in this application) (as in the case of f,: the rate f,(x)/f,(x)

equals K/x) and is not constant in x (as in the case of f2: the rate

S(x)/f2(x) = Q). The above arguments do not exclude that practical data can be statistically fitted using exponential functions: this has nothing to do with the theory behind it. We refer to Huber (2001), Huber and Wagner-Dobler (2001 a,b) for pleas (based on fitting) to use the exponential function for f (in Laherrere and Sornette (1998) one even uses "stretched" exponentials, i.e. functions of the form ab ). In view of the above we are inclined to reject this way of mformetric progress. In Jeong, Tombor, Albert, Ottval and Barabasi (2000) an interesting (practical) comparison between exponential and power functions for the sizefrequency function is given: the Erdos-Renyi model of random networks (where each pair of nodes in the network is connected randomly with probability p) shows an exponential form for f, expressing the number of connections per node, while social (natural) networks, WWW and the internet show a power function for this function f.

We repeat here that our arguments against the exponential function refer to its use for the size-frequency function f. It is well-known (see e.g. Egghe and Rousseau (1990a)) that exponential functions are basic in the description of growth and aging of literature (although refinements have important applications: S-shaped growth (e.g. logistic growth) (see Egghe and Rao (1992b)) and lognormal aging (see Egghe and Rao (1992a)). One could even think of developing an informetrics theory based on these exponential laws for growth and/or ageing.

34

Power laws in the information production process: Lotkaian informetrics

In any case, the following important argument of Naranan (1970) shows that exponential growth implies Lotkaian informetrics.

1.3.4

Proof of Lotka's law based on exponential growth or based on exponential obsolescence

1.3.4.1 Proof of Lotka's law based on exponential growth: the Naranan model

The paper Naranan (1970) has until now - as far as the author can see this - not really been understood for its high informetric value. Naranan supposes exponential growth of sources (journals) as well as of items in sources (articles in journals). He furthermore assumes that the growth rate of items in a source is the same for each source. Although we cannot explain this now, these 3 assumptions are the basis for the description of an IPP as a self-similar fractal (see Chapter V). Its link with the fractal dimension of such a self-similar fractal can only be proved after the study of IPPs in Chapter II and V. Even Naranan himself seems to be unaware of this aspect of his theory. For the moment, we can only give Naranan's argument leading to a power law for the size-frequency function f. This is what we will do now and we will deduce a formula for Lotka's exponent a in function of both growth rates. This is also done in Naranan (1970) but the fractal argument, based on this formula, is missing there; we will give it in Chapter V, then showing the full importance of Naranan's work.

Of course, one remark, as given in Egghe and Rousseau (1990a) can be made: "explaining" Lotka's power law by "assuming" an exponential growth is replacing the problem of explaining Lotka's power law to explaining exponential growth. This is the negative way of looking at such arguments. One can argue in a positive way that such an argument makes a link between an informetric theory based on exponential growth and Lotkaian informetrics, which is important and not fully understood in Egghe and Rousseau (1990a), let alone its link with fractals. This is the reason why we give Naranan's argument here, hereby also indicating that the cumbersome criticism of Hubert (1976) is not in order.

Naranan's paper deals with a "classical" IPP being a bibliography consisting of journals (as sources) and articles (as items) in these journals. We will follow the general source-item terminology, since the argument is universal. He considers an IPP that grows in time. Naranan's assumptions are 3-fold:

Lotkaian informetrics: an introduction

(i)

The number of sources grows exponentially in time. Let us denote this as

N(t) = Cla;

(ii)

35

(1.36)

The number of items p(t) in each source grows exponentially in time.

(iii) The growth rate and the number of items in a source at t = 0 in (ii) is the same for every source. Because of (ii) and (iii), we have

p(t) = c2a2

(1-37)

where p(t) is the number of items in each source at time t (c 2 and a2 are the same for all sources).

We need to derive, from (1.36), the age distribution of the sources. Via the transformation (1.37) we will then obtain the production distribution of these sources, i.e. the size-frequency distribution. At any time t0 and 0 < t < t0

N(t)-N(0) K> , ,y ' N(t 0 )

(1.38)

is the fraction of the sources (at t 0 ) that started (were "born") in the time interval ]0,t]. Hence

^ 4 N(t 0 ) is the density (at t 0 ) of the sources that started at t itself: indeed

r.N£h = N(t)-N(0) °N(t0) N(t0)

J

(L39)

36

Power laws in the information production process: Lotkaian informetrics

being equal to (1.38). So, taking t = t0 — i,

N

V/)

(1.41)

N(t0) is the density of sources that are (at t0) x time units old. (1.41) equals

c,(lna,)ar c,a;»

= (lna,)ar'

(1.42)

Since this is independent of t 0 , (1.42) represents the overall age density. To go from the variable x (age of source) to the variable p (number of items in a source) we use the cumulative distribution of sources that are less than or equal to x time units old:

f(lna,)ar T 'dx'

We now use (1.37) twice:

P(X) =

C a

2 2

implies

-j2. = c2(lna2)a^

and

In -2- Uxlna,

(1.43)

Lotkaian informetrics: an introduction 37

which yields

dT

= W = T^ c 2 (lna 2 ja 2

(L44)

plna 2

and In P-' x=- ^ ~ lna 2

(1.45)

(1.44) and (1.45) in (1.43) yields (replacing the dummy variable x' by the dummy variable p ' )

1

'

F(p)= K

'

J o ina 2 p '

'

r^Avi^^ Jo I n a 2 [ c 2

p'

F(p) = / ; ^ - c ^ p ^ i d p '

(L46)

being the cumulative distribution function of sources with less than or equal to p items. Hence, the size-frequency distribution f(p) is nothing else than F'(p). Hence

f( p ) = p . c 2 t a - p l ma,

b

"

(1.47)

, hence Lotka's law with exponent a = 1+ ]na i

lna2

(I4g)

38

Power laws in the information production process: Lotkaian informetrics

So we have shown that Lotka's law can be derived from exponential growth and furthermore, the exponent a (Lotka's exponent - see formula (1.28)) is related to this (double) exponential growth via formula (1.48) which will play a crucial role in Lotkaian informetrics: (1.48) will be basic for the fractal description of Lotkaian IPPs as self-similar fractals (see Chapter V). This fact was not mentioned in Naranan and we think it is revealed here.

We now present the same argument but treating time as a discrete variable (t = 0,1,2,3,-

) and

we will show that the same result is valid, hence also showing that the differential calculus of Naranan (giving rise to the dispute in Hubert (1976)) is not needed: an elementary argument of counting provides the same result, showing again that the criticism of Hubert (1976) is not correct. So we suppose we have (1.36) and (1.37) for discrete t. Let t e N be fixed but arbitrary. At time t there is a fraction of — N(t)

sources that exist i or more time units (e.g. a

.»w year), hence with p or more items, where p = c2a2, hence i = —s-2-l. Hence the fraction lna2 N(t-i)

^Teq

, 4) N(t-i) N(t)

=

=

c,a,

'

1

41 lna.

One can write that

Lotkaian informetrics: an introduction

G

( p ) = (-) ta * 2

39

d- 49 )

which is the cumulative fraction of sources with p or more items, i.e. G = 1-F with F as in the previous (continuous) argument. Hence the size-frequency function f(p) relates to G as

G(p)= P f ( p ' ) d p ' .

Hence we have

f(p) = -G l (p)

c 2 lna 2 ( p

, hence Lotka's law C/p" (identical with (1.47)) with a as in (1.48). This shows the validity of Naranan's argument. We rephrase this important result as a theorem.

Theorem 1.3.4.1.1: In any IPP, let the sources grow exponentially in time with rate a, and let the items in the sources grow exponentially in time with fixed rate a 2 , then this implies that the size-frequency function of this IPP is of power type (hence Lotkaian) f(p) = C/p a where

a

_1

=

]naL lna2

the quotient of the logarithms of the growth rate of the sources and the one of the items in the sources. The depth of this result will be explored further on in this book.

We have the following trivial but interesting corollary.

40

Power laws in the information production process: Lotkaian informetrics

Corollary 1.3.4.1.2: If the exponential growth rate of the sources equals the exponential growth rate of the items in the sources, then we have Lotkaian informetrics with Lotka exponent a = 2.

1.3.4.2 Proof of Lotka's law based on exponential obsolescence: solution of a problem of Buckland The problem described here was originally formulated in Buckland (1972) and re-formulated and solved in Egghe (2004a).

Obsolescence is usually expressed by the decline in time of the use of a document and use is, in most cases, expressed as "cited". Here time goes to the past and one talks about synchronous obsolescence as opposed to diachronous obsolescence where one studies the citation of a document (or a group of documents) after its publication (cf. Stinson (1981), Stinson and Lancaster (1987)). Here we will limit ourselves to synchronous obsolescence and simply talk about obsolescence.

There are at least two ways of considering obsolescence: the decline of the use of an article the older it is but also, as a consequence of it, the decline of the use of a journal (in which this article is published). So, in general, we can study obsolescence of items and obsolescence of sources. Buckland (1972) uses the terminology "obsolescence" for the former and "scattering" for the latter and poses the question:

Problem of Buckland: What is the relation between obsolescence and scattering? In our terminology: what is the relation between obsolescence of the items and obsolescence of the sources?

We think this formulation is incomplete, for the following reasons. Obsolescence of sources is an incomplete way of describing scattering, the latter usually being described as the distribution of items over sources, e.g. the size-frequency distribution in two-dimensional informetrics. Secondly, the obsolescence distribution of sources does not follow from the obsolescence distribution of the items since the disappearance of an item in time (in the past when the item is not used anymore) does not necessarily lead to a disappearance of the source

Lotkaian informetrics: an introduction

41

which contains this item (since the source can still contain other items that are still used) [This is related to the success-breeds-success mechanism to be described in Subsection 1.3.6]. So, describing obsolescence essentially requires two distribution functions: the one of the itemaging and the one of the source-aging and hence we can talk about two-dimensional obsolescence. In fact this is in line with the graphs produced in Buekland (1972), where this type of two-dimensional obsolescence is depicted. With two-dimensional obsolescence given, we are in a logical position to ask for the relation with two-dimensional informetrics (scattering) as e.g. expressed by the size-frequency distribution.

Therefore we think that the problem of Buekland can be reformulated as follows (and inspired by Buekland (1972)).

Reformulation of the problem of Buekland: What is the relation between two-dimensional obsolescence and two-dimensional scattering (as e.g. expressed by the size-frequency function)?

In Egghe (2004a) this problem is solved as follows

1.

We suppose we have given an exponentially decreasing obsolescence distribution for the age of the items (citations) (to the past) and an exponentially decreasing obsolescence distribution for the age of the sources (that contain these items, e.g. the journals that contain these cited articles); also here, time is going to the past. Note that (see e.g. Egghe and Rao (1992a)) exponential aging distributions are the basic functions to describe obsolescence (i.e. aging).

2.

We reverse the time by applying the transformation t' = —t, so we transform the "past" into the "future". Looking at time this way, exponentially decreasing obsolescence distributions become exponentially increasing growth distributions, hence distributions as given by Naranan and described in Subsection 1.3.4.1: exponential growth of both the sources and the items.

3.

It then follows from Naranan's argument (Subsection 1.3.4.1) that we have a sizefrequency distribution of power-law-type, hence Lotka's law.

Hence we have the following Theorem, being an answer to the problem of Buekland.

42

Power laws in the information production process: Lotkaian informetrics

Theorem 1.3.4.2.1 (Egghe (2004a)): Let us have a two-dimensional obsolescence situation in which the number of sources decreases exponentially in time and where the number of items also decreases exponentially in time (with the same aging rate in each source), then we have an IPP where the sizefrequency function is Lotkaian.

We refer to Egghe (2004a) for more details. 1.3.5 Derivation of Mandelbrot's law for random texts

We will now consider the special type of IPP where sources are word types and items are word tokens as occurring in linguistics (cf. the examples of IPPs in Section I.I). We will derive the rank-frequency function g, as introduced in Section I.I and show that the law of Mandelbrot (1.16), introduced in Section 1.2, applies. There we indicated (but its proof will be given in Chapter II) that the law of Mandelbrot is equivalent with Lotka's law, hence, in this way, a new explanation of Lotka's law is given (albeit for random texts). The argument will result in a formula for the exponent (3 in Mandelbrot's law, similar to the result obtained in the previous subsection. In fact, Chapter II will fully explain the relation between both exponents ( a of Lotka's law and [3 of Mandelbrot's law) and will show that both results are the same.

The argument given by Mandelbrot is a support for the Naranan argument and, furthermore, Mandelbrot's reasoning will be extended, in Chapter VII, for the derivation of the rankfrequency distribution of N-word phrases. We give Mandelbrot's argument as indicated in Mandelbrot (1977b) (cf. also Nicolis, Nicolis and Nicolis (1989) or Li (1992)) and clarified in Egghe and Rousseau (1990a).

Let our alphabet (including all necessary numbers or signs) consist of N "letters". Consider a text as consisting of letters and blanks. Every letter or blank fills up the spaces in the text. We assume (greatly simplified situation) that every letter has an equal chance of being used. Let p be this probability. Therefore, since there are also blanks in the text:

Lotkaian informetrics: an introduction

43

p = P(letter)<—. V ; N The probability of having a word consisting of k letters is then (this is also an approximation of reality)

P(k) = ( l - N p ) p k

for every k = 0,1,2,... . Hence, we also have

P(k) = Popk.

(1.51)

Let r be the rank of such a word if we arrange the words in decreasing order of use. This results in

N + N2 +... + Nk~' < r < N + N2 +... + N k .

Thus, with N +... + Ns =

Ns+1 - N , N —1 N k - ' < r ^—!- + l < N k . N

(1.52)

P k= — ^ . In p

(1.53)

Now (1.51) implies that

Hence (1.52) becomes

/ N _.

w , Nlnp

< r

_k n P

£L_L + l < N ' { N

.

44

Power laws in the information production process: Lotkaian informetrics

Consequently,

e>"P

W

< r

2^1

+1<e-p

iJ.

(L54)

We denote (for reasons to be explained later on)

D , = - ^ . In p

(L55)

Then (1.54) becomes:

^

The last approximation makes r

fN —1

.

(1.56)

+ 1 equal to the average of the left and the right part

I N in (1.56) (in the light of continuous IPPs this approximation is acceptable). Hence:

r

fN —1 (N

. N + l P ~°s hi = 2 N P0

or

(N+ii P = P(r) =

^ ^

p.

(1.57)

h(¥)F Multiplying P(r) by A, the total number of word tokens (i.e. items in the general IPP terminology) yields the rank-frequency function g as introduced in Section I.I, which is of the form (1.16) given in Section 1.2, hence Mandelbrot's law. Note that

Lotkaian informetrics: an introduction

45

which will be shown in Chapter V to be the same as formula (1.50), based on equivalency results between the law of Lotka and the one of Mandelbrot, to be given in Chapter II.

Note that, if N is high (valid in all languages), we see that formula (1.57) takes the special form of the law of Zipf in linguistics:

1.3.6 "Success Breeds Success"

Success Breeds Success (SBS) is a well-known expression (a variant is "Cumulative Advantage", see e.g. De Solla Price (1976)) which describes the evolution of an IPP in time. Studying time dependence adds an important dimension to the problem (and its solution), hereby transforming it from a static to a dynamic problem capable of following evolution in time.

SBS describes this evolution as follows (intuitively): the more items a source has (say now) the higher its probability (chance) that it will produce a new item. Intuitively this leads to skew distributions of items over sources, just as in econometrics its equivalent "the richer one is, the higher chance to become even more rich" leads to an unequal distribution of wealth in a community, say a country. This fact attracted researchers as De Solla Price, Simon (1955) and others and they noted the possible similarity of application in both fields. They also searched for deriving Lotka's law (and other laws) from SBS-principles, hence providing a new explanation of Lotkaian informetrics based on SBS. There is, indeed, a possibility that this can be achieved since Lotkaian IPPs indeed show an unequal distribution of items over

46

Power laws in the information production process: Lotkaian informetrics

sources. Indeed, take the case of the most classical Lotka function, i.e. the one with exponent cc = 2:

f(n) = 4 -

(L6°)

n We derived already, in Subsection 1.2.1 that, in this model, about 61% of the sources have only one item. In addition, as is clear from (1.60), f(2) = f(l)/4, i.e. the number of sources with 2 items is only 25% of those with one item. Further f(3) = f(l)/9 meaning that the number of sources with 3 items is only l/9th of those with one item, and so on.

Before going to SBS itself let us illustrate this principle in a simple case of balls drawing from and adding to an urn and see what comes out of it, with respect to the size-frequency function f.

/. 3.6.1 The urn model

We will present here the "simplest" urn model of SBS leading to a size-frequency function "close" to Lotka. We describe De Solla Price's argument (De Solla Price (1976)), slightly adapted by Koenig and Harrell (1995) who call it the "publication game".

Players ("authors") enter the game with one publication and draw a ball from an urn that initially (first round or first level) has only two balls: an S ball (S = success) and an F ball (F = failure). If a player draws an F ball he/she puts back the F ball in the urn and has to leave the game and remains in the present state. If a player draws an S ball his/her score of publications is increased by 1, the S ball is returned to the urn and a new S ball is added in the urn. So in the first round, 1/2 of the players draw an F ball and remain with one publication and 1/2 of the players draw an S ball and have, at this moment, 2 publications and have the right to make another draw, now in an urn with 2 S balls and 1 F ball hence where, due to the previous success, the chance for a new success has increased (i.e. SBS). Hence of these 1/2 of the players, 1/3 (i.e. 1/6 of the total number of players) of them draw an F ball, leave the game and remain with 2 publications. On the other hand, 2/3 (i.e. 1/3 of the total number of players) of them draw an S ball and have now 3 publications and the right to make another draw, now in an urn with 3 S balls and 1 F ball. Of these 1/3 of the players with 3

Lotkaian informetrics: an introduction

Al

publications, 1/4 (i.e. 1/12 of the total number of players) of them draw an F ball, leave the game and remain with 3 publications. On the other hand, 3/4 (i.e. 1/4 of the total number of players) of them draw an S ball and have now 4 publications and the right to make another draw, now in an urn with 4 S balls and 1 F ball. We continue this way by complete induction in n. We start with 1/n of the total number of players with n publications that have the right to make another draw in an urn with 1 F ball and n S balls. Hence —; of the total number n(n + l) of players draw an F ball, leave the game and remain with n publications. However — n n+1

=

of the total number of players draw an S ball, have now n + 1 publications n+1

and have the right to make another draw, now in an urn with 1 F ball and n + 1 S balls. This ends the complete induction. We have proved that a fraction

^n) = ^ T )

(L61)

of the total number of players have (definitively) n publications. In terms of IPPs, sources and items and where there are T sources in total, we have that the size-frequency function is given by

f(n) = — I — = T ( V ; n(n + l) U

—| n + lj

(1.62)

, which is very close to the Lotka law (1.60). In fact, if we interpret (1.60) in continuous variables (as did Brookes (1984a)), we have

I C^V = C — U n + lj Jn' 2

(1.63)

V

J

which has the same form as (1.62). Distribution (1.61) is called Booth's law (based on Booth (1967)) and has the "telescoping" property

S9(n)= S^H)

48

Power laws in the information production process: Lotkaian informetrics

£<,(»)-1 n=l

Booth's law is rather popular in informetrics - see De Solla Price (1963, 1976), Brookes (1981, 1983, 1984a,b), Tague (1981), Haitun (1982a), Pao (1978), Fedorowicz (1982b), Chen (1989), Egghe and Rousseau (1995) and Chen and Leimkuhler (1990) but the law is also known in other fields, see Simon (1955), Baayen (2001), Feller (1968) where the telescoping parts in (1.64) (or rather their complements to 1:

, n

,...) are defined as the law of n+1

succession of Laplace of 1812:

L(n) = ^

(1.65)

(see also Brookes (1981, 1983, 1984a,b)). This elementary introduction shows that SBS is capable to explain unequal situations, albeit we do not explain SBS itself! Indeed, comparable with what is also done in other sciences and with differential equations, a type of SBS is "given" (as in the urn model: if an F ball is drawn the urn remains the same; if an S ball is drawn we add one S ball) from which the (rank- or size-) frequency function can be deduced, but the specific type of SBS itself is not explained. We could as well use another type of SBS e.g. where, more than one S balls are added when an S ball is drawn or where we start with other urn compositions than one F and one S ball. We could even implement FBF ("failure breeds failure"), together with or apart from SBS, where FBF means that a certain number of F balls are added when an F ball is drawn, certainly leading to many different size-frequency functions f.

For this reason we cannot say that a certain form of SBS gives explanations of certain laws but rather reveals a mechanism within which certain laws are valid. So SBS belongs to the informetrics theory in general but not so much to Lotkaian informetrics. We can compare it

Lotkaian informetrics: an introduction

49

with differential equations, capable of generating many functions, indicating the mechanism behind these functions rather than explaining these functions themselves - see Kawamura and Hatano (2002), Rousseau (1994a) and many references, e.g. in physics. The difference between differential equations and SBS is that in SBS we use probability theory (and discrete time mathematics) to generate functions rather than continuous time and differential calculus. We leave open the problem of determining informetric regularities using stochastic differential equations, where all aspects of the above mentioned theories are included.

Back to SBS. Since SBS is a general mechanism, possibly implying many informetric theory types and since the results on the size-frequency function f are approximations of the exact values, due to the complexity of the equations we will, in this book, proceed as follows. In the next Subsection 1.3.6.2 we will give the general definition of SBS and show the special form used by De Solla Price (1976), following Simon (1955). We will also present exact numerical calculations of the Price-Simon model, showing that even this "classical" type of SBS does not always produce decreasing size-frequency functions f (contrary to Axioms 1.3.1.2) although the approximations do deliver decreasing f (and approximately Booth's law again). In Subsection 1.3.6.3 we will show how Lotka's law can be derived from an SBS principle (different from the one of De Solla Price-Simon). Finally in Subsection 1.3.6.4 the exact link of SBS with stochastic processes and conditional expectations is given, yielding also the link with gambling theory (monitoring the gambler's fortune instead of - in informetrics monitoring the number of items in sources). We hope that this approach sheds the right light on SBS, its explaining power, but also its complexity and its generalizing capacity.

1.3.6.2 General definition of SBS in general IPPs

General SBS can be defined as follows in terms of sources and items, where we limit ourselves to the case that items can only have one source (for the theory of multiple source items, we refer to Chapter VI).

SBS describes the growth of an IPP by indicating, at each item addition, to which source it will belong. Therefore, we can consider the discrete time t £ N to be the same as the number of items in the IPP. SBS is defined inductively. Let us be at time t (i.e. with t items). The introduction of a new item t + 1 leads to the following alternatives:

50

Power laws in the information production process: Lotkaian informetrics (i)

(Source creation): With a probability a(t) e ]0,l[ this item is produced by a new source;

(ii)

(pure SBS): If the new item is produced by an already existing source (by (i), this has a probability 1 - a(t)), there is a probability x(t,n) that this item is produced by a source that has already n items (n < t, necessarily). Note that, for the moment, a(t) and the x(t,n), n = 1,.. ,,t are arbitrary upon the requirement, of course, that

X>(t,n)=i

(L66)

n=l

This requirement guarantees that (i) together with (ii) covers all cases.

Of course, if we do not specify a(t) and x(t,n), the above statements have no relation with SBS and the above formulation can include even opposite effects such as FBF. The reader can already understand that, by adopting different explicite forms for a(t) and x(t,n), SBS can lead to different frequency functions. As explained above, SBS is a mechanism, not really an explanation.

In the work of Simon (1955) and De Solla Price (1976), there are the (unexplained) assumptions that a(t) = a, i.e. constant in t and that x(t,n) is proportional to n P(t,n), where P(t,n) denotes the fraction of sources with n items in a system with t items in total. Hence, denoting our size-frequency function f by f(t,n) (to show the t-dependence) we have

/ N f(t,n) P(t,n) = A_J

(1.67)

where T(t) denotes the total number of sources at t.

We can, of course, accept a(t) = a, constant in t, as a first simplification: this does not harm any explanatory power of SBS. This is not the case for the assumption that x(t,n) is proportional to n P(t,n). There is not a rationale for it except the fact that one seems to be expressing that x(t,n) increases with n (expressing that, the higher n, the higher chance that the next ((t + l) th ) item be produced by a source that has (at t) already n items). However this

Lotkaian informetrics: an introduction

51

argument is wrong since x(t,n) also depends on the unknown P(t,n). It is shown in De Solla Price (1976) (but see also the clarifications of the argument given in Egghe and Rousseau (1990a)) that it follows from the SBS of De Solla Price, that P(t,n) approximates Booth's law (1.61). Going back to this SBS principle we then see that x(t,n) is proportional to

, hence n+1

decreasing in n. Another interpretation of the requirement that x(t,n) must be proportional to nP(t,n) is that the latter equals the fraction (at t) of the items that belong to a source with n items. Again, it is not clear whether this is a natural requirement or not. Finally, the approximations to the De Solla Price SBS principle (i.e. the size-frequency function is approximately Booth's law), imply that P(t,n) is decreasing in n, a natural requirement in informetrics, following the Axioms 1.3.1.2. In Egghe and Rousseau (1995) we showed that this is not always the case: we calculated exactly, up to t = 8, all possibilities of item-source links offered by the De Solla Price SBS. This has lead to Fig. 1.3.1 and Table 1.3.1. Fig. 1.3.1 gives the situation up to t = 4 and Table 1.3.1 gives the situation up to t = 8 for E(P(t,n)), the expected values of the P(t,n), n = 1,.. .,8.

Fig. 1.3.1

Exact SBS situation in the case of De Solla Price: a(t) = a constant, b = 1-a and x(t,n) proportional to n P(t,n), in the cases t = 1,2,3,4.

Reprinted from Egghe and Rousseau (1995), Fig. 1, p. 428. Copyright John Wiley & Sons Limited. Reproduced with permission.

52

Power laws in the information production process: Lotkaian informetrics

Table 1.3.1 Exact results for the De Solla Price SBS, t = 8, n = 1,. ..,8, in the cases a = 0.1, a = 0.2, a = 0.5 and a = 0.9. E refers to expected value of the distributions P(8,n). Reprinted from Egghe and Rousseau (1995), Table 1, p. 428. Copyright John Wiley & Sons Limited. Reproduced with permission.

a = 0.1

a = 0.2

a =0.5

a = 0.9

0.1775

0.3099

0.5808

0.9044

E(P(8,2))

0.0644

0.1102

0.1756

0.0822

E(P(8,3))

0.0396

0.0673

0.0910

0.0115

E(P(8,4))

0.0342

0.0570

0.0594

0.0017

E(P(8,5))

0.0389

0.0609

0.0415

0.0002

E(P(8,6))

0.0569

0.0765

0.0277

0.0000

E(P(8,7))

0.1101

0.1086

0.0162

0.0000

E(P(8,8))

0.4783

0.2097

0.0078

0.0000

E(P(8,1))

From this table we immediately see that P(8,n) decreases in n, only in the last two columns; P(8,n) does not decrease for a = 0.1 and a = 0.2. This jeopardises the De Solla Price SBS principle at least as an explanatory mechanism for Lotka's law and even any size-frequency function since we expect such a function to be decreasing (Axioms 1.3.1.2).

In the next subsection we will outline how we can obtain approximations for the expected values E(P(t,n)) and what type of functions (incl. Lotka's law) can be obtained.

1.3.6.3 Approximate solutions of the general SBS Since no exact solutions can be given for any SBS principle we will provide the approximative formulae calculated in Egghe and Rousseau (1995). SBS says how the (t + 1)situation can be derived from the t-situation (this is called in mathematics: recursion formulae). Hence, after a long but easy calculation, we obtained the following recursion relations for E(P(t + l,n)), n = l,...,t + 1 in function of E(P(t,n)), n = l,...,t (for notational reasons, we dropped all E(.) notations):

Lotkaian informetrics: an introduction

P(t + U) =

i + (l-a(t)) P(U)-4?

53

(1-68)

and forn = 2,...,t

P(t+ l,n) = a ( t ) 2 m ^ + (l-a(t))[p(t,n) + X ( t ' n - T 1 ) t -^ t - n )].

(1.69)

and for n = t+1

P(t + l,t + l) = ( l - a ( t ) ) ^

(1.70)

From these, applying a further approximation, we obtained the following general approximations, being direct formulae for E(P(t,n)) for all teNand n = l,...,t (dropping again the E(.) notation):

[

a(t)T(t)-(l-a(t))x(t,l)(l + T(t)) a(t)T(t)-(l-a(t))x(t,t)(l + T(t))

'}

(

'

j

and forn = 2,...,t

Thn) P

(l-»(O)('(t.°-l)-x(t,n))(l + T(t))

^'n)-

a(t)T(t)-(l-a(t))x(t,t)(l + T(t)) '

l

^

being essentially of the form

P(t,l) = C(t)x(t,l)

(1.73)

P(t,n) = D(t)(x(t,n-l)-x(t,n)),

(1.74)

54

Power laws in the information production process: Lotkaian informetrics

where C(t) and D(t) are constants, depending only on t. Formulae (1.72) and (1.73) are able to provide several different size-frequency functions: for this it suffices to give a formula for x(t,n), n = l,...,t. Of course, this is an unexplained fact as well and, as said before, we are only capable of showing the SBS mechanism rather than giving explanations.

If we take x(t,n) proportional to n P(t,n) then, of course, as follows from the definition of SBS, we refind the Simon-De Solla Price size-frequency function as proved in Simon (1955), De Solla Price (1976), see also Egghe and Rousseau (1990a): f(t,n) being proportional to a p function B(n,m), where m is a parameter; again, as mentioned above, this can be approximated by Booth's law. This result is used in Bornholdt and Ebel (2001) to "explain" the fact that the size-frequency distribution of in-links in WWW is a power law with exponent around 2. As discussed in Subsection 1.3.6.1, this is not an explanation but only a link to an SBS mechanism (unexplained in itself).

If we take x(t,n) to be proportional to

1-gi

(..75,

then, since x(t,n-l)-x(t,n) is proportional to — , we find that P(t,n), hence the size-frequency n° function f(t,n) is proportional to — , hence f(t,n) is the law of Lotka. Of course, again, na formula (1.75) itself is not explained this way. In Egghe and Rousseau (1995) other sizefrequency functions (such as the truncated geometric and the truncated Poisson) can be obtained. We now leave these approximative results and we will indicate in the next subsection some exact results concerning the general SBS principle, incl. the real nature of this principle, being a stochastic process.

Lotkaian informetrics: an introduction

55

/. 3.6.4 Exact results on the general SBS and explanation of its real nature

This subsection is dealing with SBS as a stochastic process, i.e. a sequence of functions with index t e N being time as well as the number of items as explained in the definition of SBS. This subsection is rather theoretical but is not needed to follow the rest of the book and, hence, can be skipped if the reader wishes to do so.

SBS explains, given any t, what are the possibilities at t + 1 and with what probability they will occur. The totality of all possibilities hence can be depicted as a tree with nodes and branches leading each node at, say, level t, to the set of possible nodes at level t + 1. At each node we know the exact IPP and the "growth" history of its construction. This implies that, within the same level, say t, the same IPPs but with a different history, can occur. Fig. 1.3.2 depicts the first levels of a general SBS, indicating also the probabilities with which these nodes occur (b(t) is defined to be 1 - a(t), being the probability that the (t + 1)* item goes to an already existing source).

Fig. 1.3.2

General SBS in the form of a tree-structure up to four levels t = 1,2,3,4.

To show how this figure is constructed, consider the IPP (211) in level t = 4. This IPP appears 3 times as a possible outcome. Let us describe the one at the right side. This is constructed as follows: at level t = 1 we always have 1 source with 1 item, at level t = 2 we have 2 sources with 1 item (probability a(l)), at level t = 3 we have 3 sources with 1 item (probability a(2)),

56

Power laws in the information production process: Lotkaian informetrics

at level t = 4 we have 3 sources: 1 source with 2 items and 2 sources with 1 item (probability b(3)x(3,l), being the probability that the fourth item goes to an already existing source with 1 item at level t = 3). Fig. 1.3.2 can be compared with the simplified Fig. 1.3.1 where, at each level, identical IPPs are put together, where the "growth" process to reach an IPP is (partially) forgotten and where the probabilities are the one of the De Solla Price SBS. Fig. 1.3.2 shows the first 4 levels of the general SBS and with "growth" history.

Since a tree as in Fig. 1.3.2 contains all the SBS information, we can also ask the question: can the evolution (in levels t) be followed for functions such as T(t), the total number of sources at level t, \i (t), the average number of items per source at level t (i.e. —— ) or f(t,n) (for each n = 1,2,... ,t fixed) being the number of sources with n items at level t. Let us denote by X(t) any of the above functions (T(t), n(t), f(t,l), f(t,2),..., f(t,t)) in order to explain the method for all these functions in one story. Suppose we know X(t) up to level t and we want to estimate what X(t + 1 ) can be. In gambling theory, X(t) could be the gambler's fortune and, having done t bets (hence knowing X(t)), the gambler is most interested in X(t + 1), his/her fortune after one more bet but, obviously, this is not known. The only thing that can be calculated from general SBS (say Fig. 1.3.2), since we know all transition probabilities a(t) and (l-a(t))x(t,n) = b(t)x(t,n) (n = 1,.. .,t), is the average of all possible X(t + l)-values, given X(t) (as said, we know X(t) at t). This is called the conditional expectation of X(t + 1), knowing level t and hence this is denoted as Et(X(t + 1)). Let us give a simple example. Suppose X(t) = 5 (e.g. 5 sources if X(t) = T(t) or €5 if X(t) is the gambler's fortune). Suppose that, when going from level t to level t + 1, with probability - , X(t + 1) will be X(t) + 1 (i.e. 5 will become 6), with probability —, X(t + 1) will be X(t) - 1 (i.e. 5 will become 4) and with 4 probability - , X(t + 1) will be X(t) itself. Then 4

Et(x(t + l)) = i(X(t) + l) + i(x(t)-l) + ix(t) Hence

Lotkaian informetrics: an introduction

SI

E,(X(t + l)) = X(t) + i

In terms of gambling, the gambler can expect, when he/she makes a new bet (the (t + l)th), to win, although he/she can also loose. So we do not know X(t + 1) but we know Et(X(t + 1)). In our examples, T(t) can never decrease, but (j.(t) and f(t,n) (any n = l,...,t) can decrease (e.g. for n(t) > 1, when, from t + 1 on, many new sources arrive (probability a(t + 1),...), |i(t) will start diminishing since these new sources (temporarily), have only 1 item).

Note that, for all t, E, (X(t)) = E(X(t)), the expectation of X(t). In terms of Fig. 1.3.2: E, (X(t)) = E(X(t)) gives the (conditional) expectation of X(t) only based on the knowledge at t = 1. Notice how fine the conditional expectations Et(X(t + 1)) are: here we only average points on level t + 1 which have a common splitting point at level t, hence giving finer estimates for X(t + 1) than the rough E(X(t +1)).

The sequence X(t), t = 1,2,... together with the conditional expectations Et(X(t + 1)), t = 1,2,... form a stochastic process (as defined in probability theory - see e.g. Neveu (1975), Egghe (1984)): typical for such a process is that it is not possible (although tried in Subsection 1.3.6.2 with approximations), for any t = 1,2,... to determine X(t + 1) from X(t) (this would mean that the gambler can predict the outcome!) but that from X(t) one can determine Et(X(t + 1)). This is what SBS is all about, in an exact sense. It is now clear how to proceed: the theory of stochastic processes is available (again, see Neveu (1975), Egghe (1984)) and the results are hence applicable to SBS of IPPs. But first, we must determine the exact form of Et (X(t +1)) for the different informetric processes:

1.

T(t), the total number of sources at t

2.

\i (t), the average number of items per source, being

*

( 1 )

^

as we have indentified discrete time t with the total number of items. 3.

for every n = 1,2

t fixed, f(t,n), the number of sources with n items.

58

Power laws in the information production process: Lotkaian informetrics

Recall that we work with the general SBS principle as defined in (i), (ii) in Subsection 1.3.6.2.

Theorem 1.3.6.4.1 (Egghe and Rousseau (1996a)): For every t = 1,2,...

1.

3.

E,(T(t + l)) = T(t) + a(t)

(1.76)

For n= 1.

E t (f(t + l,l)) = f(t,l) + a(t)-x(t,l)(l-a(t))

(L79)

E t (f(t + l,n)) = f(t,n) + (l-a(t))(x(t,n-l)-x(t,n))

(1.80)

E t (f(t + l,t + l)) = (l-a(t))x(t,t)

(1.81)

Forn = 2,...,t

For n = t+1

Proof:

1.

Et(T(t + 1)) = a(t)(T(t) + 1) + (1 - a(t))T(t) = T(t) + a(t)

by the SBS definition. Note that the values x(t,n) in the second part of the SBS

Lotkaian informetrics: an introduction

59

definition are not needed: indeed T(t) is determined by the fact that the new item (the (t + l)th) will go to a new or an already existing source (respectively with probabilities a(t) and 1 - a(t)) and not by which already existing source (i.e. not by the number of items that are already in this existing source at t). The same remark goes for [i since H(t) = t/T(t).

2.

E, W . + .))-E I

,

J5 '

>

a(t)

l-a(t)

The other formula follows from \i (t) = t/T(t).

3.

For the f(t + l,n), n = 1,2,...,t + 1, obviously, the x(t,n) are involved. For n = 1, we have

Et(f(t + 1,1)) = a(t)(f(t,l) + 1) + (1 - a(t))[(f(t,l) - l)x(t,l) + (1 - x(t,l))f(t,l)]

= f(t,l) + a(t)-x(t,l)(l-a(t)) For n = 2,...,t, we have

Et(f(t + l,n)) = a(t)f(t,n) + (1 - a(t))[x(t,n)(f(t,n) - 1) + x(t,n - l)(f(t,n) + 1) + (1 - x(t,n) -x(t,n-l))f(t,n)]

= f(t,n) + (1 - a(t))(x(t,n - 1) - x(t,n))

Finally, for n = t + 1, the formula E(f(t + l,t + 1)) = (1 - a(t))x(t,t) is clear.

D

60

Power laws in the information production process: Lotkaian informetrics

These are the basic SB S equations for the stochastic processes T(t), u (t) and f(t,n), n = 1,... ,t. Let, again, X(t) be any of such processes. We can apply the following definitions and results of stochastic processes to our SBS formulae.

Definitions 1.3.6.4.2:

1.

A stochastic process X(t) is called a martingale if

E,(X(t + l)) = X(t)

(1.82)

for all t e N .

2.

A stochastic process X(t) is called a submartingale if

E t (x(t + l))>X(t)

(1.83)

for all t e N

3.

A stochastic process X(t) is called a supermartingale if

E,(x(t + l))<X(t)

(1.84)

for all t 6 N. In terms of the gambler's fortune, we have a martingale if the game is fair in the sense that the gambler can expect to keep his/her capital after gambling. In case of a submartingale, the gambler is expected to win. Casinos will not allow this: at least to cover their expenses, games are usually supermartingales in which a gambler is expected to loose. We have the following results on (sub-) (super-) martingales, cf. Neveu (1975).

Theorem 1.3.6.4.3:

Let X(t) be a submartingale such that

Lotkaian informetrics: an introduction

supE(x(t))
61

(1.85)

teN

(recall that E(X(t)) denotes the expectation of X(t), i.e. the average of X(t)), then the process converges to an integrable function. This result is also true for all positive supermartingales.

The meaning of these results is that in these cases we have stable processes: in terms of IPP growth we have growth processes that are limited, finite. These facts are completely controlled by the SBS basic "ingredients" a(t) and x(t,n), n = l,...,t. We have the following conditions for convergence of the functions T(t) and u(t).

Theorem 1.3.6.4.4 (Egghe and Rousseau (1996a)): 1.

The process T(t) is always a submartingale. Condition (1.85) is satisfied if and only if

f>(a(t))
(1.86)

t=i

(note that in the notation E(a(i)) we have added another generalization of SBS: not only can we choose a in function of the level t but we can also let a(t) be dependent on the particular node at which we are in the SBS-tree at level t, cf. Fig. 1.3.2; if a(t) is independent of this, then, simply, write a(t) instead of E(a(t))).

2.

The process \i (t) is a (necessarily positive) supermartingale if and only if

i

(L87)

for all t (and a submartingale if > is replaced by < ) .

In the cases (1.86) and (1.87), the processes T(t) and p. (t) tend to finite limits.

62

Power laws in the information production process: Lotkaian informetrics

Proof: Formula (1.76) contains all the information:

E,(T(t + l)) = T(t) + a(t)

>T(t)

hence a submartingale. Taking expectations of both sides of (1.76) (and noting that E(E t ) = E, (E t ) = E since E is the overall average) we get

E(T(t + l)) = E(T(t)) + E(a(t)).

for all t. Hence

E(T(t))=E(T(t-l)) + E(a(t-l))

= E(T(t-2)) + E(a(t-2)) + E(a(t-l))

= E(T(l)) + f>(a(i))

= l + gE(a(i)). i=i

Hence, condition (1.85) is satisfied if and only if

£E(a(t))
For \x (t), formula (1.77) implies

Lotkaian informetrics: an introduction

63

E,(n(t + l))<ji(t) if and only if

This is equivalent with

, W

~

t+ 1

hence (1.87). In all these cases the stability result, Theorem 1.3.6.4.3, can be invoked, leading to finite limits for T and (i.

Similar results, based on formulae (I.79)-(L81) can be obtained for f(t,n), n = l,...,t, see Egghe and Rousseau (1996a). Finally (on SBS) we note the following interesting result on E((j.(t)), the expected average number of items per source, in case a(t) = a, a constant (i.e. in SBS: there is a constant probability for a new source).

Theorem 1.3.6.4.5 (Egghe and Rousseau (1996a)):

If a(t) = a, a constant, then

E(n(t)) = H

^ 3.

Proof:

We use again that \x (t) = t/T(t), hence

(1.88)

64

Power laws in the information production process: Lotkaian informetrics

E(^(t)) = t E ^ .

(1.89)

But

Since a(t) = a is constant and denoting b = 1 - a, we have

P(T(t) = i) = (|l 1 1 )a i -V- i

(since T(l) = 1). Hence, putting (1.91) and (1.90) in (1.89) yields

(since l(S- i)=M

Hence

E(Kt))=iE(!)^ a i_i \ /

= I((a + b)'-b')

using the well-known binomial formula

Edja'b-^a + b)'.

(1.91)

Lotkaian informetrics: an introduction

65

But a + b = 1 and hence (1.88) follows.

More on SBS can be read in Kot, Silverman and Berg (2003), Brookes (1983), Kantor (1978), De Solla Price (1978), Schapiro (1994), Chen, Chong and Tong (1994), Chen (1989), Fedorowicz (1982b), Mandelbrot (1959), Oluic-Vukovic (1997, 1998), Huber (1998b), Tague (1981), Rao (1980), Bookstein (1991), Onodera (1988), Chen and Leimkuhler (1989). In the latter publication one indicates that the approximate formula

IW+^IW.1 t

(1.92)

lnt

is valid and can be proved from the classical SBS principle. We refer the reader to Subsection VII.2.2 for a simple, purely mathematical explanation of (1.92) since (1.92) has nothing to do with SBS.

This closes our treatment of SBS in which we hope to have explained that SBS is nothing else than a formulation of how conditional expectations change over time and hence that, for exact results, the theory of stochastic processes can be used. These results are more general than, but include, Lotkaian informetrics: in the author's opinion, the important notion of SBS deserves a correct description, as given above.

1.3.7 Entropy aspects

The notion of entropy implies - as we will prove in the sequel - the laws of Lotka, Zipf and Mandelbrot; as indicated in Subsection 1.2.2 (and to be proved in Chapter II) the last two laws also belong to Lotkaian informetrics. So entropy plays a fundamental role in the topic of this book. Of course, one needs to fully understand the important notion of entropy as a measure of information content. Therefore, in the next subsection, we will first recall the definition and some basic properties of entropy. After this, the link with Lotkaian informetrics will be shown.

66

Power laws in the information production process: Lotkaian informetrics

1.3.7.1 Entropy: definition and properties

Suppose we have a machine producing, at regular times, symbols, where each symbol has a probability p, 0 < p < 1 to occur. Usually symbols are letters and numbers or any other symbol that can occur in a text. But symbols can also be words (here the probability of words is taken into account) or any set of objects in which each object has a probability p to occur. The rationale behind entropy, as given in Shannon and Weaver (1975), is as follows.

Before the machine produces a symbol (with probability p that a particular symbol will be given), we have uncertainty (p) which is removed after the machine gives the symbol: now we have an information content per symbol, clearly dependent on p and denoted by H(p). Necessarily H(p) > 0. The information content is higher (say the "surprise" or the "sensation") the lower p and hence H(p) is a decreasing function of p. The probability p, obviously, is also the chance to predict (before the outcome) correctly which symbol (with this probability) will be produced; H(p) is the information content of a symbol after the symbol is produced. Suppose now that we want to predict two consecutive, independent outcomes (independent means that the second outcome is not influenced by the first one). Suppose the first outcome has a probability p, and the second one has a probability p 2 . Because of independence, the probability to predict both outcomes (in that order) is p,p 2 . However, the information content is the sum of both information contents H(p,) and H(p 2 ) separately, hence H (p,) + H (p 2 ). Hence we have the functional relation

H( P l p 2 ) = H(p,) + H(p 2 ).

(1.93)

This is the key relation for entropy, comparable with (but different from) the product property (1.29) in Subsection 1.3.2 (leading to a power law). We now have that, for continuous functions with the property (1.93) (i.e. transforms products into sums), we necessarily have that H is a logarithmic function. For a proof, we refer the reader to the Appendix I (or to Roberts (1979)). Since H must be decreasing and positive (as indicated above), we necessarily have that

H(p) = - l o g p.

(1.94)

Lotkaian informetrics: an introduction

67

Here any logarithm can be used: it is obvious that, for any logarithm loga to the basis a > 1, (1.93) is satisfied and that H is positive (since 0 < p < 1) and decreasing in p. Note that (1.94) implies that H(l) = 0. This is natural: a sure event does not provide (when it happens) any information content. In binary systems (including computer sciences) one uses log2 for clear reasons: there binary digits (bits) are used, i.e. every text consists of only 0s and Is. Then formula (1.94) also expresses the number of bits necessary to binary coding m symbols. Indeed: if there are m symbols we need x 0s and Is to binary code these m symbols where 2X = m, hence x = log, m = - log, — = - log2 p where p = — is indeed the prediction chance of (m)

m

each of these m symbols. A classical example is the ASCII coding, having 256 = 28 symbols and hence H = 8 being indeed the number of bits necessary to code these 256 symbols in a binary way (see e.g. Heaps (1978) for a partial list). So, the function H in (1.94) has bits as a unity if log2 is used. Note that log2 is not only used in computer science but that it also occurs in any binary searching technique such as e.g., searching for a word in a dictionary. Indeed, suppose a dictionary contains m words and we look up a specific word. Usually, searching is executed binarily where at each step we decide in which part of the dictionary the word is to be found. So we start with one operation: we decide if the word we are looking for is situated in the first or the second half of the dictionary. Once this is done we neglect the deselected part and treat the selected part as our new "dictionary" in which we again apply the same technique. At each step we divide the number of possible words into two parts: —, —r, —-,... and we stop when the requested word is found, i.e. at the point where we 2 22 23 arrive at one word. We hence need H steps such that

m

-1

hence H = log2 m

(1.95)

Note again that (1.95) equals (1.94) since p = —, the probability to pick the requested word at m random.

68

Power laws in the information production process: Lotkaian informetrics

Note that also non-binary logarithms are usuable. Take the example of the decimal numbers, i.e. numbers composed of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Now we need x digits to form m decimal numbers, where 10" = m, hence x =log10 m (e.g. m = 1,000, x = 3 to form the numbers 0 = 000, 1 = 001,..., 998, 999). In this context, one could call log10 m the number of "dits" (decimal digits) necessary to represent m symbols in a decimal way! In general we can say that (1.94) represents the number of a-ary digits necessary to code m = — symbols, where P log = loga is used and it represents the (a-ary) information content per symbol.

Note that transforming loga into logb is easy and can be done using the trivial formula: for each x:

1 l°8a X lOg b X = — 2 2 -

(1.96)

log a b which also means that any log can be calculated with a pocket calculator using the log10 or the logc = In (Neperian logarithm) button. Example

log2 x = 3.322log10x = 1.443Inx

for every x, as is readily verified.

The graph of function H in formula (1.94) is given in Fig. 1.3.3 and clearly shows that for small p, H(p) is very high while for p close to 1, H(p) is small. In natural language, if the chance for an event p is very small (e.g. to win the lottery) the information content ("sensation"), when it happens, will be very high (and vice-versa). This is linked with the sensation law of Weber-Fechner expressing that the sensation is equal to the logarithm of the stimulus (see e.g. Roberts (1979), Gleitman (1981), Rescher (1978) or the historical book Fechner (I860)). This means, for example, if a person carries a weight of 1 kg and if you add one more kg, this person will feel the difference more than in case the person carries a weight of, say, 20kg and where you add one kg. The sensation difference is, as expressed by the

Lotkaian informetrics: an introduction

69

Weber-Fechner law: first case (and using log10 - by (1.96), there is only a constant involved and hence comparisons are possible) the sensation difference is log10 2 — log101 = log10 2 = 0.301 while in the second case, the sensation difference is log10 21 — log10 20 = log10 — = 0.021 , a clear difference with the first case.

Fig. 1.3.3 Graph of H(p) = -log p as a function of p e ]0,l].

Of course, if there are m symbols with unequal probability of occurrence (as e.g. the letters in an alphabet - see e.g. Heaps (1978) for a table, produced by Dewey, of the occurrence distribution of letters in the English language), the average information content per symbol, H, is given as follows: suppose that the m symbols have a probability of occurrence equal to p,,p 2 ,...,p m . Then their average information content per symbol equals (any log can be used)

m

H = -^PilogPi

'

(1.97)

i=l

hence the weighted average of the numbers log p- (weighted by p ( ). H is called the entropy of the system. Of course, note that the p((i = l,...,m) are limited to p; > 0 (if p( = 0 the symbol (or more general, the object) is considered as non-existing, hence deleted from the index set {l,...,m}) and to

70

Power laws in the information production process: Lotkaian informetrics

i=l

since we limit ourselves to all actually occurring symbols. It is well-known that H is maximal if and only if all p{ are equal: p, = p 2 = ... = p m = p = — in which case (1.97) m reduces to (1.94) - see e.g. Jones (1979) or Mansuripur (1987) for a reference of this wellknown result. From now on we will not limit ourselves to letters or words but we will treat the probabilities p,,...,p m in a general way being probabilities of occurrence of m objects r = l,...,m.

1.3.7.2 The Principle of Least Effort (PLE) and its relation with the law ofLotka

Besides an average information content, expressed by H in (1.97), we have also an average effort (also called cost or sometimes (in the area of physics) energy) expressed by

E=J>rPr

(1.99)

r=l

where E7 is the effort of using object r. In the example of a text consisting of words, Er is the cost of using the word on rank r, where words are arranged according to increasing length. In this setting it is clear that the E,,...,Em are increasing. Our application is more general, however, but it will be convenient to assume that the E ]; ...,E m are increasing. This is no loss of generality: simply arrange the objects r = l,...,m such that the E,,..., Em are increasing.

Definition 1.3.7.2.1: The Principle of Least Effort (PLE) The principle of least effort requires E in (1.99) to be minimal, subject to the restriction (1.97) (a fixed given value of H ) and the restriction (1.98). Intuitively this means that, within a fixed information content, we seek this situation which requires the least effort.

Lotkaian informetrics: an introduction

71

The (PLE) is attributed to Zipf (1949) who promoted (PLE) but Zipf did not present a mathematical formulation as above. In its historical formulation, (PLE) is to be understood as a sociological behavior of persons and groups of persons, see also Petruszewycz (1973).

We have the following proposition.

Proposition 1.3.7.2.2 (Egghe and Lafouge (2004)): (PLE) implies

pr=cp-E'

(1.100)

r = l,...,m, 3 c > 0 , 3 p > l . Proof: Since (PLE) is a constraint extremum problem a necessary (but not always sufficient) condition for (PLE) to be satisfied is given by the method of the multiplicators of Lagrange (see, e.g. Apostol (1957)): consider the function

G = £ E r P r + X , H + £p 7 logp r \ + X2 1 - ^ P r r=l

I

r=l

)

\

r=l

a-101) I

(i.e. minimise (1.99), subject to the constraints (1.97) and (1.98)). The methodology requires = 0 for all r = l,...,m. Hence

| ^ = E r +X 1 (log P r + l ) - ^ 2 = 0

hence (taking log = loge = In but any other base will yield the same result (1.102) below).

pr=e1'

e k'

72

Power laws in the information production process: Lotkaian informetrics

which is of the form (r = 1,... ,m)

pr=cp"E'

where c = e 1 '

(1.102)

> 0 and p = e1' > 0 . Note that ^ , ^ 0 since Xl = 0 implies p — oo and

hence p r = 0 for all r = l,...,m, by (1.102), contradicting (1.98) (and the fact that all probabilities are strictly positive). So we proved (1.100) but only for p > 0. That p > 1 will follow from (PLE) as we will show now. By (1.101), p > 1 will be proved if we can show that the p,,...,p m decrease (since we assumed the E,,...,Em to be positive and to increase). Let p vt 1. Suppose that the p,,...,p m satisfy (PLE), hence are of the form (1.102) and that they do not decrease. Let then i < j , i,je{l,...,m} be such that P; < Pj. Define % to be the elementary permutation of {l,...,m} defined as

(n(l),...,7t(m)) = (l,...,i-l,j,i + l,...,j-l,i,j + l,...,m)

(1.103)

- X > r l n P r = H = - EP,WtaP»(r)

( L104 )

Then we have that

r=l

r=l

and

EPr=1

=

EP* W

r=l

r=l

But

E E 'P«r) = EiP*(.) + EjP^j) + E E r P , W r=l

r=l

( L105 )

Lotkaian informetrics: an introduction

73

= E iPl + E JPJ + g E rP
= gErPr

+

(E J -E i )(p i(j) - Pii(i) )

since TC(i) = j and n(j) = i and 7t(r) = r , V r e { l , . . . , m } \ { i , j } . But i < j im P lies E ; < Ej (since the E r are increasing and since E; = Ej im P lies (1.102), contradiciting

Pi

<

Pj)

Pi

=

Pj

by the already proved formula

and p ,., ~PM = PJ ~ P J < 0 . Hence

(EJ-E,)(PI(J)-P<1))
Consequently

m

m

E E rP, W <E E rP r r-1

( L106 )

r-1

Now (1.104), (1.105) and (1.106) contradict the fact that the plt...,pm

satisfy (PLE). So the

p,,...,p m decrease, hence p > 1 (if p ^ 1). Consequently, p > 1.

Note: The use of the method of the multiplicators of Lagrange is not new: it can also be found in Rapoport (1982), Nicolis, Nicolis and Nicolis (1989). The part that Proves that p > l was P roved

recently in Egghe and Lafouge (2004).

That Lotka's law and Zi P f s law is implied by Proposition 1.3.7.2.2 will be proved now. Theorem 1.3.7.2.3: (PLE) with

Er = logr

(1.107)

74

Power laws in the information production process: Lotkaian informetrics

(any log can be used) implies the laws of Zipf and Lotka.

Proof: For any log we have that there exists a constant E such that

Er = Elnr

Hence (1.100) yields

P r = c e Onp)(-B.nr)

p r =cr- E '" p

v, = jr

(U08)

where c > 0 and b > 0 because p > 1 as follows from Proposition 1.3.7.2.2, hence Zipf s law since the r denote ranks.

Since Zipf s law implies Lotka's law (as we will prove in Chapter II), the latter also follows. But Lotka's law also follows directly from (PLE). Indeed, if we change the index from r into n (the number of items in a source) and if we use p n =

, the fraction of the sources with

n items (f(n) is the size-frequency function of Section I.I and T is the total number of sources) then, in exactly the same way, (PLE) yields a power law but now for pn , hence also for f(n), i.e. Lotka's law (1.7) of Subsection 1.2.1. [This approach was followed in Yablonsky (1980) (see also Haitun (1982c)) where the same logarithmic cost function En =logn was used].

Note: (PLE) was heavily promoted by Zipf himself in Zipf (1949) although a rigorous derivation of Zipf s law from (PLE) is not provided in Zipf (1949) (see also the remarks in Wyllys (1981)).

Lotkaian informetrics: an introduction

75

So it seems (cf. also Subsection 1,2,2) that neither the formulation of Zipf s law (1.15) nor its explanation (e.g. via (PLE)) were provided by Zipf!

A logarithmic effort (or cost) function is logical as the following explanation shows.

1.3.7.2.4 Explanation of the logarithmic effort function Let us take the example of words with i letters (i = 1,2,3,.-.) as they occur in a text. Expressing the use of i-letter words (i = 1,2,3,...) as a cost i, we arrive at Er being a logarithmic function of r. Indeed (cf. also the arguments in Subsection 1.3.5 - see also Mandelbrot (1977a), Nicolis, Nicolis and Nicolis (1989), Rapoport (1982) and Li (1992)) if there are N letters in our alphabet then the words consisting of 1 letter occupy the ranks 1,2,. ..,N, the words consisting of 2 letters occupy the ranks N + 1,N + 2,...,N + N2 and so on: the words consisting of i letters occupy the ranks

N w + N " 2 + . . . + l to N' + N M + . . . + N

(cf. also the argument given in Subsection 1.3.5) hence these ranks are in the order r ~ N', Er = i Ri logN r. Note that a continuous interpretation of r and i would yield

Er = logN r

exactly, hence we will use the latter equation. Also note that this relation is the same as the Weber-Fechner law mentioned in Subsection 1.3.7.1. Also the above argument is the same as the one leading to the definition of entropy as given in Subsection 1.3.7.1 - now for N-ary digits.

We refer the reader to an extension of the above argument, where one drops the assumption that all the letters in our alphabet cost the same. This argument then leads to Mandelbrot's law, as can be read in Rapoport (1982). Note, however, that Rapoport discusses (PLE) but applies the maximum entropy principle (defined below) (see formula (9) on page 9 in Rapoport (1982)). There are problems with this argument as the next subsection will show.

In Lafouge (1998) and Lafouge and Michel (2001) there is a similar confusion concerning the maximum entropy principle and (PLE) in the sense that one suggests that (PLE) is nothing else than the maximum entropy principle (see below) and hence (PLE) as formulated in Subsection 1.3.7.2 is not given.

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Power laws in the information production process: Lotkaian informetrics

For more on (PLE) we refer the reader to Fairthorne (1969), Naranan (1991) or Buckland and Hindle (1969) and also the sequel for its relation to the maximum entropy principle, to be introduced now.

1.3.7.3 The Maximum Entropy Principle (MEP)

This principle "reverses" the requirements in the sense that now the average effort E (see (1.99)) must remain constant but that one tries to maximise the average information content H (formula (1.97)). Of course also the formula (1.98) must be valid. Intuitively we could say that, within a fixed average effort, we seek this situation which yields the maximum information content.

Definition 1.3.7.3.1: The Maximum Entropy Principle (MEP) The maximum entropy principle requires H in (1.97) to be maximal subject to the restriction (1.99) (a fixed given value of E) and the restriction (1.98).

The (MEP), introduced in 1957 by Jaynes (Jaynes (1957)) in the area of physics is wellknown in the exact sciences such as, apart from physics, in information science, chemistry, biology, artificial intelligence and, of course, also in probability theory and analysis. For these applications see e.g. Kapur and Kesavan (1992) and Kapur (1992).

We have the following proposition.

Proposition 1.3.7.3.2: (MEP) implies

p r = cp~E'

r = l,...,m, 3 c > 0 , 3 p > 0 .

(1.109)

Lotkaian informetrics: an introduction

11

Proof: Using again the method of the multiplicators of Lagrange for the constraint extremum problem (MEP) (see Apostol (1957)) we consider the function

m

I

m

\

(

m

\

G = - g p r l o g p I + X I E - £ ) E r P r +A.2 l - £ ) p r

The requirements

(1.110)

= 0 for all r = 1,.. ,,m now yield (again we take log = In) c>Pr

|^- = -l-lnpr-X,Er-X2=0 "Pr hence

P r =CP^ E '

where c = e~'~Xl > 0 and p = eX| > 0 for all r = l,...,m, hence (1.110).

U

It is not possible now to prove sharper limits for p (as could be done in case of (PLE)). This is because E contains the parameters E,,...,Em while H does not contain any parameters. In fact in the next subsection we will show that the necessary condition for MEP expressed in Proposition 1.3.7.3.2, is also sufficient for (MEP). There, also the sufficiency will be proved for the necessary condition for (PLE), expressed in Proposition 1.3.7.2.2, so that we arrive at the conclusion that (PLE) and (MEP) are not equivalent. However, in the next subsection, the exact relation between (PLE) and (MEP) will be established.

Note that (MEP), via Proposition 1.3.7.3.2, using Er = logr as in Theorem 1,3.7.2.3 leads, as in this Theorem, to a power law but one that is not necessarily decreasing: if 0 < p < 1, then (1.109) increases which is not conform to Zipf s or Lotka's law.

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Power laws in the information production process: Lotkaian informetrics

For more on (MEP) we refer the reader to Bashkirov and Vityazev (2000) and to Lafouge and Michel (2001) (see also Yablonsky (1980), Haitun (1982c) and Nicolis, Nicolis and Nicolis (1989) for a definition of (MEP) using continuous variables). In Lafouge and Michel (2001) other functions for the parameters Er (other than the logarithmic functions (1.107)) are tried (e.g. the linear one) yielding non-power type functions for p r (or p n ) such as the geometric distribution. As noted above, however, the use of a logarithmic functionality for the E r (r = l,...,m) is the most important one and this leads to power type laws for p r and pn (i.e. the law of Zipf, respectively the law of Lotka).

1.3.7.4 The exact relation between (PLE) and (MEP)

All the results in this subsection are recently proved in Egghe and Lafouge (2004).

In the previous subsection we proved already that (PLE) implies pr = cp~E', 3c> 0, 3p > 1, r = l,...,m and that (MEP) implies p r =cp~ E ', 3c> 0, 3 p > 0 , r = l,...,m . Since (MEP) refers to p > 0 and (PLE) refers to p > 1 we conjecture that, in the determination of the relation between (PLE) and (MEP) another principle is needed. We will define it now and then show that indeed this third optimization principle is the "missing link" in the establishment of the mathematical relation between the principles (PLE) anf (MEP).

Definition 1.3.7.4.1: The Principle of Most Effort (PME) (Egghe and Lafouge (2004)): The principle of most effort requires E in (1.99) to be maximal, subject to the restriction (1.97) (a fixed given value of H) and the restriction (1.98).

This principle (PME) is introduced in Egghe and Lafouge (2003) for the first time. We have the following necessary condition for (PME).

Proposition 1.3.7.4.2 (Egghe and Lafouge (2004)): (PME) implies

P r =cp" E <

(I.I 11)

Lotkaian informetrics: an introduction 19

r = l,...,m, 3c>0, 30
Proof: Using exactly the same function G as in Proposition 1.3.7.2.2 we again find that

pr=cp-E'

(1.112)

r = l,...,m, 3 c > 0, 3p > 0 (in fact at this stage we only used the constrained extremality of E which also contains (PME)). We now have to show that 0 < p < 1. In other words, by (1.112) and the fact that the E p ...,E m increase, we have to prove that the p,,...,p m increase. This proof goes along the lines of the similar proof in Proposition 1.3.7.2.2. Let p ^ 1. Supposing the p,,...,p m not increasing, we have i < j , i, j e {l,...,m} such that p( > pj (hence also E, * Ej by (I.I 12)). Taking the same permutation rc as in (1.103) we have now (1.104), (1.105) and

m

m

E E r P,W>E E rPr r-1

contradicting that the P[,...,pm

(I-H3)

r=l

satisfy (PME). So the

„

increase and hence

0
We will now show, in one proof, that all the necessary conditions, proved in Propositions 1.3.7.2.2,1.3.7.3.2 and 1.3.7.4.2, are also sufficient.

Proposition 1.3.7.4.3 (Egghe and Lafouge (2004)): (i) p r = cp- E '

r = l,...,m, 3c>0, 3p > 0, implies (MEP)

(1.114)

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Power laws in the information production process: Lotkaian informetrics

(ii) pr=cp-E'

(1.115)

r = l,...,m, 3c > 0 , 3p > 1, implies (PLE) (iii) P r =cp- E <

(1.116)

r = l,...,m, 3 c > 0 , 3 0 < p < 1, implies (PME).

Proof: Given one of the situations (1.114), (1.115) or (1.116), define the following function fr of the variable x r > 0, r = l,...m:

f r (x r ) = x r l n x r - ( l + lnc)x r + (lnp)E r x r

(1.117)

Then we have that

f'r (x r ) = 1 + lnx r - 1 - l n c + (lnp)E r = 0

implies that

_ lnc-(lnp)Er v A —C

xr =cp" E l

r = l,...,m, hence the given functions (1.114), (1.115) or (1.116) respectively. Furthermore

fr"(xr) = — > 0 X

Lotkaian informetrics: an introduction

81

for all x r . Hence the given functions (1.114), (1.115) and (1.116) satisfy f^(pr) = 0 and f r ( p r ) > 0 , hence fr has a minimum in xr = pr with p r as in (1.114), (1.115) or (1.116) respectively, r = l,...,m. So for all x r > 0, r = l,...,m:

f,(x r )>f r ( P r )

i.e.

x r l n x r - ( l + lnc)x r +(lnp)E r x r > p r l n p r - ( l + lnc)p r +(lnp)E r p r

(1.118)

Hence

Jx r lnx r -(l + lnc)Jxr+(lnp)jErxI r=l

r=l

T=1

m

m m

>I>rtop r -(l + lnc)£>r + (lnp)£;Erpr r=l

r=l

(I.H9)

r=l

The proof is now split into 4 parts: (I) represents the proof of (i), (II) represents the proof of (ii) for p =* 1, (III) represents the proof of (iii) for p * 1 and (IV) represents the proof of (ii) and (iii) for p = 1.

(I)

Let now (I.I 14) be given: p > 0 and require

r=l

r=l

jE r x r =jE r P r =E r=l

r=l

be constants. Then (I.I 19) implies, for any p > 0:

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Power laws in the information production process: Lotkaian informetrics

-^x r lnx r <-£;p r lnp r . r-1

r=l

Hence the p,,.--,Pm of the form (1.114) (for any p > 0 ) satisfy the (MEP). This completes the proof of (i). (II)

Let now (1.115) be given and let p ^ 1: so p > 1 and

m

m

r-l

r=l

lllX

-LX

r = "I]Pr lnPr =

H

be constants. Then (1.119) implies, for p > 1 (hence lnp > 0):

jE r x r >J E r P r . r=l

r=l

Hence the p,,...,p m of the form (1.115) ( p > l ) satisfy the (PLE). (III)

Let now (I.I 16) be given and let p ^ 1: so 0 < p < 1 and

r=l

m

r=l

m

~J2Xr

lllX

r=l

r = -£Pr

ln

P, =

H

r=l

be constants. Then (LI 19) implies, for 0 < p < l (hence lnp<0):

EEA<EE,P,. r-l

r-l

Lotkaian informetrics: an introduction

83

Hence the p,,...,p m of the form (1.116) ( 0 < p < l ) satisfy the (PME). (IV)

Let now p = 1. Hence p, = p 2 =... = p m = —. Now (1.119) cannot be used anymore m since lnp = 0. We will show directly that (PLE) and (PME) are valid. Let

m

m

r=l

r=l

—^x r lnx r =— £ p r l n p r = H(=lnm). r=l

r=l

Then x r = p r for all r = l,...,m. This is well-known since H attains its free (apart from

the requirement

^xr=l)

maximum

in only one point,

namely

r=l

(x,,...,x m )= —,...,— = (p,,...,p m ), as was already remarked in Subsection 1.3.7.1. (m m) Hence

m

m

£E r x r =£E r=l

r=l

-i

m

in

l£E r r=l

hence (PLE) as well as (PME). Note that this case is degenerate since only one point —,...,— is involved here. This completes the proof of (ii) and (iii) and hence of [m m) the Proposition.

We hence have proved the following Theorem: Theorem 1.3.7.4.4 (Egghe and Lafouge (2004)): (I)

(MEP) is equivalent with

P, = cp~Er

84

Power laws in the information production process: Lotkaian informetrics r = l,...,m, 3 c > 0 , 3p > 0

(II)

(PLE) is equivalent with

p r = cp- E '

r = l,...,m, 3 c > 0 , 3p > 1 (III)

(PME) is equivalent with P, = cp~E'

r = l,...,m, 3 c > 0 , 3 0 < p < l The case p = 1 is the only one in the intersection of (PLE) and (PME) and corresponds to the (degenerate) case of the (MEP) where an unconstrained (w.r.t. E) maximum is obtained for H.

Finally, from Theorem 1.3.7.4.4, we obtain the exact relation announced in the heading of Subsection 1.3.7.4.

Theorem 1.3.7.4.5 (Egghe and Lafouge (2004)): The following assertions are equivalent:

(i)

(MEP)

(ii)

(PLE) or (PME).

Note 1.3.7.4.6: As follows from Theorem 1.3.7.4.4 and from the argument developed in the proof of Theorem 1.3.7.2.3, the (MEP) does not always produce a decreasing power law if (1.107) is applied (only (PLE) does). In this sense, the papers Yablonsky (1980), Haitun (1982c) and Nicolis, Nicolis and Nicolis (1989), which use (MEP) (but with continuous variables), do not provide a complete explanation of Zipf s or Lotka's law as a decreasing function.

Lotkaian informetrics: an introduction

1.4

85

PRACTICAL EXAMPLES OF LOTKAIAN INFORMETRICS

1.4.1 Important remark

In this section we will not be involved with the statistical fitting of Lotkaian laws (power functions). First of all, because of the non-Gaussian nature of these laws (i.e. the Central Limit Theorem is not valid here due to infinite moments - see e.g. Yablonsky (1980, 1985), Haitun (1982b) or Huber (2001)) it is not possible to perform accurate statistical tests on power laws (of course, as will be explained in the Appendix III, it is possible to make some "optimal" calculations of the values of C and a in formula (1.7), which has practical value). Secondly, and more importantly, a statistical fitting of such laws does not provide evidence for its mathematical acceptance. For the latter, one needs arguments as the ones given in section 1.3.

It is even so that - in statistical fitting - it is very well possible that more than one function can be accepted. The author can refer here to the "old joke" that one provides data points of the form l x , v x ) , say for x = 1,2,...,20 (hence following perfectly the function y = Vx) which can be fitted e.g. by a straight line, hereby "proving" the linear relationship! This is not showing the weakness of statistics but merely the fact that it serves another goal (than the mathematical-probabilistic goal of this book), namely to provide a variety of frameworks in which one can possibly work. Thirdly, even in cases where Lotka's law evidently is not valid (e.g. in cases where f(n) is not overall decreasing but increases and then decreases as in some examples of number of authors per paper - see further in this section) we can use Lotka's law as a first approximation or as a simplification in an otherwise too intricate mathematical theory. This will be done e.g. in the modelling of fractional author scores (Chapter VI) where the proved model is so stable that a simplification as mentioned above does not jeopardise the quality of the final model.

So in this section, we mainly give an overview of areas (also outside informetrics) where power functions for the size-frequency function f are possible or have been studied. In essence, the basic requirements are given in the Axioms 1.3.1.2, together with the scale-free property which then (via Corollary 1.3.2.3) guarantees a decreasing power law. It is the

86

Power laws in the information production process: Lotkaian informetrics

author's opinion that, instead of each time performing statistical tests (on which, as said, there is criticism) one better checks if, in a certain field or application, the scale-free property applies, in which case one has even mathematical evidence of the validity of a power law. This is also a suggestion made in Roberts (1979) where one gives numerous "psychophysical" examples of power laws but where one does not try to fit a power law but merely checks the scale-free property, by experimentation (we come back to this in this section: Subsection 1.4.7).

1.4.2 Lotka's law in the informetrics and linguistics literature

We start this overview by giving some examples in the field of informetrics itself and in its "neighboring" field of linguistics.

The well-known bibliography collected by Hjerppe (1980), covering 80 years of bibliometrics and citation analysis (say in modern terms: informetrics) was analysed in Alvarado (1999) and found to be conform with Lotka's law. As in Lotka (1926), they used straight counts (giving only a credit of 1 to the senior author and nothing to the other authors) to the solution of the multiple author problem. The found exponent a of the power function was around a = 2.3.

In library and information science, there is the study of Sen, bin Taib and bin Hassan (1996), using the LISA (Library and Information Science Abstracts) database in the years 1992 and 1993, resulting in a confirmation of Lotka's law with rather high values of the exponent around a = 3.2. Schorr (1974) collects data in library sciences (in the journals Library Quarterly and College and Research Libraries, 1963-1972) but his method to deny Lotka's law is far from scientific. Yet, in another paper Schorr (1975a), using the same intuitive approach, he accepts Lotka's law in map librarianship (but see the criticism in Coile (1977)).

In Voos (1974), H. Voos studied Lotka's law in information science, using the ISA (Information Science Abstracts) database in the years 1966-1970. Again high exponents around a = 3.5 were found. See also Coile (1975) for comments on these high values.

In Sharada (1993) the field of linguistics was examined using some abstracting journals in this field (1977-1987). Only Lotka's law with exponent a = 2 was examined and found to be acceptable. We will not report on the many studies in linguistics of Zipf s law, which is basic

Lotkaian informetrics: an introduction

87

in this field. For this, see Baayen (2001), Herdan (1960), Kanter and Kessler (1995). The same goes for the econometric equivalent of the law of Zipf, being Pareto's law. For this, see the vast econometric literature. Lotka's law ( a = 2) also seems to be confirmed for the use of Chinese words. For this see Rousseau and Zhang (1992). The relativity of these results is clear from the following: in the same publication (Rousseau and Zhang (1992)) Zipf s law (although part of Lotkaian informetrics) is "rejected" for the rank-frequency distribution of Chinese words but "accepted" in Shtrikman (1994). In Rousseau and Rousseau (1993), Lotka's law is confirmed in an example of song texts.

In Newby, Greenberg and Jones (2003) Lotka's law is confirmed for the distribution of "authors" (programmers, open source developers) of open source software.

A recent study, Pulgarin and Gil-Leiva (2004), shows the validity of Lotka's law in the literature on automatic indexing in the period 1956-2000. The found exponent a was 2.75

1.4.3 Lotka's law in networks

It is clear that this is an important example: networks (having their mathematical basis in graph theory) gain increasing interest and importance: intranets, the internet and WWW but also citation networks and collaboration networks (see e.g. Egghe and Rousseau (2002, 2003a)). A network (more generally a graph) consists of points (called vertices or nodes) and connections between points (called edges). Graphs can be unidirectional in case, if (a,b) is an edge (between the points a and b), (b,a) is not an edge; if this restriction is not applicable we call the graph bidirectional. An example of a unidirectional graph is an article-citation network, where the vertices (nodes) are papers and the (one-way) links are citations (although counter-examples (due to "invisible colleges") exist we can, quite logically, assume that if paper a cites paper b, the reverse is not true because of the time difference). An example of a bidirectional graph is the so-called collaboration graph: here vertices (nodes) are authors and there is a link between them if they collaborated, i.e. if they wrote an article (or so) together. Note that this is an example of a weighted graph in which the weight of the edge is equal to the number of documents in which both authors collaborate. Other examples of graphs can be given e.g. the well-known WWW where vertices are web pages and edges the links between them (e.g. via the clickable buttons or hyperlinks).

88

Power laws in the information production process: Lotkaian informetrics

All these networks are examples of so-called "social networks" in which the size-frequency law, being P(k) = the fraction of nodes with k edges (i.e. the degree of connectivity of a node) follows a power law. Their exponents a vary between 2 and 4 (see Bilke and Peterson (2001), Jeong, Tombor, Albert, Ottval and Barabasi (2000), Barabasi, Jeong, Neda, Ravasz, Schubert and Vicsek (2002), Adamic, Lukose, Puniyani and Huberman (2001), Barabasi and Albert (1999) and the excellent recent book Pastor-Satorras and Vespignani (2004)). For reasons explained in Subsection 1.3.2 such networks are called scale-free networks and this property is very important e.g. in the determination of their fractal dimension (i.e. their complexity - see Chapter V). These social networks are in contrast with random networks in which links (edges) between the nodes are given in a random way, given a fixed probability p for a link. These so-called Erdos-Renyi networks (see Erdos and Renyi (1960) or Bollobas (1985)) show an exponential decay for P(k) (see Jeong, Tombor, Albert, Ottval and Barabasi (2000) or Barabasi, Jeong, Neda, Ravasz, Schubert and Vicsek (2002)). The latter result is refined in Krapivsky, Redner and Leyvraz (2000) where one shows the following result for random networks. If the probability to add a new link to an edge that has already k links is dependent on k, say Ak (cf. the SBS-principle of Subsection 1.3.6!) then, if Ak is proportional to k', P(k) is an exponential distribution only if y < 1. For y > 1, nearly all nodes connect to nearly all other nodes and if y = 1 , P(k) is found to be of power type with exponent a variable in the interval ]2,+°°[.

The rank-frequency version of the function P was found to be Zipfian (hence P is Lotkaian) in Adamic and Huberman (2002). There, some consequences of this fact are given. The powertype of the function k —> P(k) implies the large degree of skewness meaning that many nodes have few (say just one or two) connections while a few nodes have many connections. This is judged in Adamic and Huberman (2002) as a two-edged sword as far as resilience of the network is concerned: if a node fails at random it is most likely one with very few connections, hence its failure will not affect the performance of the network overall. However, if one targets a few (or even one) of the high degree nodes, their removal would require rerouting through longer and less optimal paths. If a sufficient number of high degree nodes are removed, the network itself can become fragmented (i.e. disconnected) without a way to communicate from one location to another.

Lotkaian informetrics: an introduction

89

In WWW it is reported (Aida, Takahashi and Abe (1998) and references therein) that the number of URLs which are accessed n times (in a certain time period) is a classical Lotka law with exponent a = 2. Note the clear scale-free property of this example. The latter publication shows the need for a standardization of the terminology (cf. also the pleas of Glanzel (1996) and Rousseau (2002a), already mentioned in Section I.I): they do not use the name of Lotka but call its law, together with Zipf s law the "dual Zipfian model". Note that the term "dual" conforms with our dual notion between sources and items and with the duality between the size-frequency function f and the rank-frequency function g, as explained in the same section I.I.

In Kot, Silverman and Berg (2003) one finds that the rank-frequency function, describing the relation between the rank of contributors to a newsgroup and the number of their submissions to that newsgroup, clearly is Zipfian.

In Egghe (2000d), the author gives an overview of regularities in the internet as found in several (electronic) texts. Nielsen (1997) gives evidence that websites, ranked according to their page views per year, follow Zipf s law. Huberman, Pirolli, Pitkow and Lukose (1998) report on the distribution of the number of web pages that a user visits within a given website, to be (approximately) of power law type with exponent around 1.5. One talks in this connection about "the law of surfing" (see also Huberman (2001)).

Adamic and Huberman (2001, 2002) (see also Huberman (2001)) give evidence for Zipf s power law for the rank of websites versus 4 attributes: number of pages, number of users, number of out-links (i.e. hyperlinks given by the sites), number of in-links (i.e. hyperlinks pointing to the sites). The latter (on in-links), in size-frequency version, is also studied in Bornholdt and Ebel (2001): the found Lotka function has an exponent around 2.1. This is confirmed in Albert, Jeong and Barabasi (1999) and Barabasi, Albert and Jeong (2000) where also the size-frequency distribution of the out-links in WWW is studied: here a power law (Lotka function) is found with exponent equalling 2.45. Size-frequency functions of in- and out-links are also studied in Thelwall and Wilkinson (2003) and found to be Lotkaian although no indication of the exponents' values is given. In Faloutsos, Faloutsos and Faloutsos (1999) one studies both size- and rank-frequency functions for out-links and one finds in both cases values of the exponents. Of course, as we will see in Chapter II, one type of exponent implies the other which is not noticed in the mentioned article. This again

90

Power laws in the information production process: Lotkaian informetrics

underlines the need for a concise theory on size- and rank-frequency functions, to be developed from Chapter II onwards. Their values for the Lotka exponent a are around 2.2. In Adamic and Huberman (2002) one finds Zipf s law also in the relation of website ranks versus the number of visitors (via the provider America Online) on December 1, 1997.

Rousseau (1997) gives statistical arguments for the fact that the distribution of hyperlinks between websites follows Lotka's law with exponent in the order of 2.3 (similar to the findings on social networks described above). The same goes for the distribution (occurrence) of domain names (such as .edu, .com, .uk and so on); here an exponent of 1.5 is found. Rousseau calls hyperlinks "sitations", an interesting name referring to "sites" and the more classical "citations". On the latter, we can report on the following results.

Redner (1998) reports on the number of papers with n citations (from ISI data on papers published in 1981 and data of the journal Physical Review D, vols. 11-50) being a power law with exponent a ss3. The far more limited study of Momcilovic and Simeon (1981) reports on a Lotka law with exponent a = 2. The paper of Katz (1999), rightly stressing the importance of a scale-free theory (hence of Lotkaian informetrics) studies the connection between the number of papers that one publishes and the number of citations (recognition) that one receives. Note the ratio scales in this type of problem. Katz finds a power law with exponent around 1.3. Several Zipf-type laws (rank-frequency functions), which are also scalefree, are also mentioned in Katz (1999).

1.4.4 Lotka's law and the number of authors per paper

The classical formulation of Lotka's law is in terms of authors (as sources) and the number of articles (items) they have published. In a dual way (cf. Section I.I), interchanging sources and items, one can consider articles as sources of authors (= items), in other words one can consider the size-frequency function f describing the number (or fraction) of articles with n authors. So, this is a very important topic since it deals with the intricate problem of multiple authorship (see also Chapter VI).

One constatation is very clear: in some cases the number of articles with n authors is decreasing in n = 1,2,3,..., and in other cases there is a short increase in n followed by a long decrease in n. In Gupta, Kumar and Rousseau (1998), 11 of the 12 data sets (in biology) are

Lotkaian informetrics: an introduction

91

decreasing; the other one has a small increase until n = 3 and then decreases. In Rousseau (1994c), 7 data sets in information science were examined and all data sets are decreasing in n. In Ajiferuke (1991), 16 data sets were analysed but it is not indicated whether they are decreasing or not. In the topic of information systems, Cunningham and Dillon (1997) gives a data set consisting of 5 journals showing a small increase up to n = 2 from which the decrease starts. Finally, in sciences (physiology), data in Kuch (1978) show an increase up to n = 2 from which the decrease starts.

In general one can conclude that the humanities and social sciences often show a pure decrease, due to the fact that collaboration is lower than e.g. in "hard" sciences, such as physical and life sciences (cf. also the conclusions in Cunningham and Dillon (1997)).

Next there is the issue of which size-frequency function f = f(n) can be applied to the number of articles with n authors. Even in the decreasing case, most authors seem to indicate, based on their statistical fittings, that Lotka's law does not apply. Yet a power law is the simplest function to work with (among the other possible size-frequency functions tested, e.g. forms of Poisson, geometric, Waring, negative binomial,...). Secondly we indicated already the problem of fitting a power law. So, it is, for mathematical purposes, not clear that Lotka's law, as a basic simple expression, should be rejected. To formulate it otherwise: to make a mathematical theory of size-frequency functions and derived properties for IPPs, we want to make as few assumptions as possible, e.g. restricting ourselves (basically) to Axioms 1.3.1.2. But for a mathematical theory we need a workable simple analytical form for the sizefrequency function f. We explained already that a power law is the simplest form (hereby excluding decreasing exponential functions - see Subsection 1.3.3) and hence that the Lotkatype law is the first to be used in models although real data sometimes seem to deviate from it. This is the more true if the distribution of the number of authors per article is not decreasing but shows a fast increase (up to a small value of n = the number of authors per article, usually n = 2 o r n = 3 - see the examples above) and then a decrease from this n onwards. Lotka's law can be taken here as a first approximation of this regularity (a bit comparable with lognormal ageing curves - see Mattricciani (1991) or Egghe and Rao (1992a), being approximated by decreasing exponential ageing (obsolescence) functions).

In conclusion: most of the co-authorship data show a decreasing frequency function f; the others show a quick small increase (usually up to number of authors n = 2 or 3) after which

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Power laws in the information production process: Lotkaian informetrics

the decrease is clear. In both cases we will use - as a first approximation - the power law function as an analytical description. This will enable us to derive mathematical results as e.g. in the case of the fractional frequency distribution; i.e. the size-frequency distribution if authorship in multiple-authored papers is credited fractionally (see Chapter VI), where both size-frequency functions: the number of papers per author and the number of authors per paper, are used and taken as decreasing power laws. The obtained results are then assumed (due to an assumed robustness of the model) also to be valid for the general case of Axioms 1.3.1.2, including - of course - other size-frequency functions f. Note (again as in Subsection 1.3.1) that this approach is in line with the strong request of Griffith (1988) for simple explanations.

In Gupta and Karisiddippa (1999) a variant of collaboration is studied, namely the number of collaborators (co-authors) per author. One found a very classical power law with exponent a = 2.1 for the number of authors with a certain average number of collaborators. This variant was also studied in Qin (1995) and further studied in Chapter III.

1.4.5 Time dependence and Lotka's law

As in any informetrics (or even other types of -metrics) problem, time is an important issue. Adding a time dimension to the study is transforming the problem (and its solution) from a static to a dynamic one. We encountered already an example in the study of SBS (Subsection 1.3.6), but also Naranan's model (Subsection 1.3.4) deals with time as a (main) factor in explaining power laws. In all these models a source, after creating a first item, can, from time to time, produce another item. Hence, at any time t, in our IPP, we have a mixture of "living" sources, some very young, others older. Some authors as Huber, Wagner-Dobler and Berg (references are given in the sequel) (see also Huberman (2001)) have posed some important and interesting questions around this mixture. Expressed in terms of authors (as sources) and their productivity (i.e. publications as items) they rightly state that (in our terminology) in general IPPs there are authors (or inventors) who just start their careers and older inventors, where sometimes a whole career is covered. The article Wagner-Dobler and Berg (1995) agrees with the validity of Lotka's law for such general ("mixed") IPPs (they present many examples of authors in mathematical logic, where their productivity (number of publications) follows Lotka's law (with exponents in the range 1.4-1.7)). On the other hand they pose the interesting question of how is the size-frequency function changed if we restrict our IPP to

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authors with a fixed number of years of activity (in their examples: 6 respectively 10 years). Then their data show a deviation from Lotka's law, especially for the production quantities n = 2,3,4,5 (n=l was not taken in consideration) which are "more equal" than in Lotka's case which came out in a "mixed" IPP of career duration but where a time period of publications was also fixed at 6 respectively 10 years. This is remarkable and needs a mathematical explanation, which is not available at this moment, at least to the best of the author's knowledge.

In the same direction goes the work of Huber: Huber (1998c, 1999, 2001, 2002) and the articles Huber and Wagner-Dobler (2001a,b). The intention is to "breakdown" an IPP, hereby only analyzing "homogene" parts e.g. of equal career duration and then studying also the distribution of career duration. The methodology applied in Huber (1998c, 1999, 2001, 2002) and Huber and Wagner-Dobler (2001a,b) is statistical and by simulation and the following ingredients are used: most authors tend to produce at a constant rate over their careers, rankfrequency distributions of career duration are exponential, time pattern of outputs within a career is random (hence Poisson distributed) and rank-frequency distributions of pure productivity i.e. rate of publication, i.e. number of publications in a unit time interval (e.g. a year) also is exponential. The double use of the exponential rank-frequency distribution is certainly in view of Subsection 1.3.3 - not very clear to the present author, but the division of a mixed IPP of author (or more general any source) production as above is certainly interesting and needs to be studied from a mathematical modelling point of view.

Indeed, while there is nothing wrong with the study of general "mixed" IPPs, i.e. where sources of any "lifetime" are featuring, such a kind of study is (as Wagner-Dobler and Berg (1995) state) a description of the state of a scientific discipline and not of a distribution of psychological characteristics of individuals. We can add to this that one type of study can add new findings to the other type of study and hence leads to a better understanding of informetrics as a whole.

We can close this subsection by concluding that, most probably, Lotkaian informetrics describes the totality of a subject in which sources of different ages can occur, as e.g. also expressed by SBS orNaranan's model (Subsections 1.3.6 and 1.3.4 respectively).

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Power laws in the information production process: Lotkaian informetrics

1.4.6 Miscellaneous examples of Lotkaian informetrics

Of course there is the original article Lotka (1926) itself which is also discussed in De Solla Price (1963). De Solla Price tries to understand the universally applicability of Lotka's law but is not very successful. It is in these discussions that De Solla Price is formulating (implicitely, see p. 46) his well-known Price law of concentration. This will be discussed in a rigorous way in Chapter IV. There we will shed light on the "feeling" of De Solla Price (p. 49-50) that Lotka's law is related with the so-called law of Weber-Fechner (from psychology) stating that sensation is proportional to the logarithm of the stimulus (as was discussed in more detail in Subsection 1.3.7.1).

Lotka's law has been confirmed in physics (1800-1900) in Wagner-Dobler and Berg (1999), both for the number of authors versus number of papers (exponent a = 1.9) as for the number of authors versus the number of areas in which they published (exponent a = 2.1). So we can say that the classical Lotka function f (n) = — was found. In semiconductor literature (1978n 1997) also a = 2 is found but only when using truncated data, see Tsay, Jou and Ma (2000). In Indian physics, Gupta, Sharma and Kumar (1998) report on weak applicability of Lotka's function but the results are not checkable since the data are not available. In Gupta (1989) (not the same Gupta as in the previous reference) Lotka's law was confirmed in biochemical literature of Nigeria (1970-1984), where exponents in the range a = 2.0-2.6 are found. A general criticism is needed here (as well as for the previous paper and several other ones): after checking Lotka's law, they continue to check derived properties such as the 80/20 rule (see Chapter IV). There is no reason for this since inequality measures (such as the (generalized) 80/20 rule) are a mathematical consequence of laws like the one of Lotka (see Chapter IV) and hence should not be checked in addition to checks for Lotka's power law.

In Gupta and Kumar (1998), Lotka's law was confirmed in 11 core journals in theoretical population genetics (1971-1980) with exponents in the range a = 1.6-2.4 (so, roughly around a = 2). The same was done in Gupta and Karisiddippa (1996) for theoretical population genetics in the period 1900-1980 with exponent values close to a = 2 (again the 80/20 rule was, unnecessarily, checked).

Lotkaian informetrics: an introduction

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In drug literature there is the paper Windsor (1975) reporting on the applicability of Lotka's classical law ( a = 2) but one did not check for better fitting a s . In legal medecine, Schorr (1975b) rejects Lotka's law but on non-scientific grounds (and only checking a = 2). In the Spanish medical literature Lotka's law, with exponent a close to 2, is confirmed in Terrada andNavarro (1977).

In computer science there is the often cited paper Radhakrishnan and Kernizan (1979). Papers in the period 1968-1972 in the journals Communications of the Association for Computing Machinery (ACM) and the Journal of the ACM where checked for Lotka's law and accepted for a around 3 (per journal). But they recognise the limitation of their study and suggest further experimentation. The high value of a = 3 was explained in Subramanyam (1979) by the fact that Radhakrishnan and Kernizan used total counting of authorship (see also Chapter VI for the treatment of multiple authorship) with the consequence of a high number of authors with 1 publication, pushing up the value of a (this is correct as will follow from Chapter IV); they indicate that, if only senior authors are counted (as did Lotka in his historic paper Lotka (1926)), values of a more close to 2 are found. The bibliography of microcomputer software of Nieuwenhuysen (1988) was checked for Lotka's law and found to be acceptable in Rousseau (1990b) with a around 2.2. Bogaert, Rousseau and Van Hecke (2000) even found Lotka's law in percolation models (where, in a 2-dimensional square lattice with mxm cells there is a probability p for a cell to be occupied). Then one can check the sizes of the soformed clusters of occupied cells and the corresponding size-frequency distribution. In Bogaert, Rousseau and Van Hecke (2000) one confirms Lotka's law with an exponent (evidently) dependent on p. They (experimentally) found the remarkable relation (a power law itself!)

«=

6 18

-

l,

(100p)0311

(1.120)

showing that only Lotka's exponent a is enough to model this phenomenon.

A special area of scientific information is given by the size-frequency function of patents. That it is of Lotkaian type was shown (experimentally) in Huber (1998a) via the Pareto distribution, which belongs to Lotkaian informetrics, as mentioned in Subsection 1.2.2 and as

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Power laws in the information production process: Lotkaian informetrics

will be proved in Chapter II (via Zipf s law, which is mathematically the same as Pareto's law, see Subsection 1.2.2).

The publication activity of an institute is studied in Vinkler (1990) and confirmed to be Lotkaian (although no indication of its exponent is given). Of course here, the activity of an institute is measured according to the distribution of papers of the researchers in this institute. In this context, the paper Coleman (1992) is interesting. There, author productivity (authors from 101 laboratories) is compared with lab productivity. In both cases Lotka's law was acceptable but with different exponents: for authors the exponent was around 1.9 while for labs it was only around 1.1. A rationale (at least for the difference of both IPPs) will be given in Chapter III where positive reinforcement will be studied in the connection with threedimensional informetrics. Another study was undertaken in Kumar, Sharma and Garg (1998) where Lotka's law (with a = 2) was confirmed for industrial firms but denied for institutional productivity in engineering sciences.

The rank- and size-frequency distributions of misprints in references to a paper (that had acquired 4,300 citations) are found in Simkin and Roychowdhury (2002) to be Zipfian/Lotkaian respectively, with Lotka's exponent a ~ 2 . The fact that many mistakes are repeated frequently is explained in Simkin and Roychowdhury (2002) by the fact that many references are given by reproducing earlier references (hence "copying" the same mistakes) without reading the papers themselves. Based on a model given in Simkin and Roychowdhury (2002) the authors conclude that only about 20% of citations have actually been read by the citers!

For firm sizes, Zipf s law with exponent pS = 1 (corresponding with Lotka's a = 2 - see Chapter II) was found in Axtell (2001) while the same was done for the distribution of city populations in Marsili and Zhang (1998) (but in a more theoretical way and allowing for more general exponents) and in Ioannides and Overman (2003). The distribution of city sizes even goes back to Zipf (1949) with exponent p = 1 (see also Batty (2003)).

Lotka's law with exponent around 2 is found for the number of hits by artists, see Cook (1989). Journal articles on the American Revolution were analyzed in Pao (1982a) and found to be of Lotka type with exponent around 2.6.

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97

In Lemoine (1992), Lotka's law (for a in the range 1.5-2) was found for the productivity of Venezuelan scientists. Distinction was made between male and female scientists and between articles in national or international journals. The found lower productivity of women versus men (except in national journals) is interpreted in terms of different Lotka exponents (higher exponents are found for female production). Note that higher a s only indicate a higher inequality among women scientists (as will be proved in Chapter IV) and not a lower productivity, as argued in Lemoine (1992). A similar study (on the difference between male and female productivity) was executed in Gupta, Kumar and Aggarwal (1999), this time in an Indian research institute. Here, no difference between the productivity between male and female scientists is found (reflecting the high degree of emancipation of women in the Indian scientific community, which is also apparent when one visits scientific institutes in India). Lotka's law was found to deviate from the different data sets.

In Nath and Jackson (1991), Lotka's law was examined for management information systems articles in the period 1975-1987 and a good "fit" with exponent a around 2.7 was concluded.

Potter (1981) reports on monograph productivity based on data of Me Callum and Godwin (unpublished) (Library of Congress Study on author headings on MARC tapes) and of himself in Potter (1980) on personal authors in the University of Illinois library catalogue. Here all authors are counted (Potter explains why Lotka only counted senior authors - see Subsection 1.2.1). Only the exponent a = 2 was checked and reasonable fits were found. Potter (1981) also reviews work of Leavens on papers in the journal Econometrica (1933-1952). Again only the exponent a = 2 for Lotka's law is considered and the latter was confirmed.

The review article White and Me Cain (1989) discusses work of Nicholls (1988), where Lotka's law is found in the humanities and natural sciences ( a = 2.5 on average), and in the social sciences ( a = 3.0 on average). Also here one then checks De Solla Price's law on the unequal division of author productivity (see further) but this is unnecessary since, in Chapter IV, we will see that the exact form of De Solla Price's law can be derived from the one of Lotka; hence separate tests are not necessary. Lotka's law is confirmed (for a = 2) in Murphy (1973) (but denied for this value in Coile (1977), where, however, no further tests using other exponential values are presented) in the humanities (in Technology and Culture, 1960-1969).

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Power laws in the information production process: Lotkaian informetrics

Let us also refer to Pao (1986) for a review of many of the above mentioned applications but where the Pao testing procedure (see the Appendix III) is applied.

1.4.7 Observations of the scale-free property of the size-frequency function f

If we want to establish the size-frequency function of an IPP it is interesting to consider the data on a double logarithmic scale (for those who work manually: on double-logarithmic paper) and check for linearity, i.e. if the data follow a straight line. The reader might argue that this is exactly what most tests do and that we criticized this methodology - see Subsection 1.2.1 (formula (1.8)) or the Appendix III. The criticism is still there on the fitting of Q

the right power law f (j) = —, i.e. to determine the best C and a . But to decide that we have a power law it is one of the best ways to do. In terms of the size-frequency function f, what we do is to transform f into

F(x) = log,f(a x )

(1.121)

where a usually is taken to be 10 or e (but any a > 1 can be used).

An example of this procedure is shown in Roberts (1979) where one reports on work in Stevens (1960) where tests were done for several possible scale-free (hence power) functions of the type: express (by putting force in a handgrip - the ordinate) the value of a given stimulus (the abscissa). Examples of stimuli were: electric shock, warmth, lifted weights, pressure on palm, cold, vibration, white noise, tones, white light. The results are shown in Fig. 1.4.1.

Note that in these examples no fitting of the power law is done. In the same way one finds examples in Katz (1999) on recognition (citations) versus publishing size or on publishing size versus rank of publishing size (Zipf type functions, which are also scale-free) or recognition (citations) versus recognition rank or impact versus impact rank. We also refer to Brown and West (2000), a book that is completely devoted to scaling and power laws in biology.

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99

We encourage the execution of more of these type of studies leading to exact conclusions as to whether or not we have a power function, i.e. Lotkaian informetrics.

Fig. 1.4.1

Testing for scale-free properties (i.e. ratio scales).

Reprinted from Roberts (1979), Fig. 4.4, p. 184. Reproduced with permission from Cambridge University Press.

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II BASIC THEORY OF LOTKAIAN INFORMETRICS

II.l GENERAL INFORMETRICS THEORY Before developing the Lotkaian informetrics theory we will describe, formally, the functions that are needed in general informetrics - e.g. a formal description of the functions introduced in Section I.I and we will prove their properties. The basis of this general theory is duality between sources and items, i.e. their interchangeability or the possibility of replacing "produces" by "is produced by". Exact definitions will be given, followed by a heuristic interpretation. We hope, this way, to give also an intuitively clear description of the necessary formalism which, however, limits itself to aspects of real analysis (mainly derivatives and integrals). The reader who wants an update on these techniques is referred to the vast literature on real analysis, e.g. Apostol (1957) or Protter and Morrey (1977). That we use derivatives and integrals is evident from the arguments given in the first chapter: the simplicity of the functional formalism and its calculations (as compared to the evaluation of sums) and also the simplicity of the relations between the different functions and their parameters and further, its applicability to the practical cases of generalized bibliographies since, the larger the datasets (as they are more and more common because of literature growth and the automation of its collection), the better they are represented by continuous models (see also the review article Wilson (1999)). In addition, they allow for fractional counting (rational, non-entire numbers) as will become clear in Chapter VI.

II.l.l Generalized bibliographies: Information Production Processes (IPPs)

An information production process (IPP) is a triple of the form (S,I,V) where S = [0,T] is the set of sources, I = [0,A] is the set of items and where

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Power laws in the information production process: Lotkaian informetrics

V:S^I

(II.l)

is a strictly increasing differentiable function such that V(0) = 0 and V(T) = A.

Intuition; S and I stand for the discrete sets of sources {1,2,...,T} and of items {1,2,...,A} respectively, hence the interval lengths (T respectively A) stand for the total number of sources, respectively items. The function V represents the cumulative number of items in the least productive sources. Since we will identify S with the source rankings (in decreasing order of number of items), V(r) (for re[0,T]) denotes the cumulative number of items in the sources belonging to the interval [T — r,T] (i.e. the sources in the interval of length r, containing the least productive sources).

We next introduce the notion of duality. Let (S,I,V) = ([0,T], [0,A], V) be an IPP as defined above. The dual IPP of the IPP (S,I,V) is defined to be the IPP

(I,S,U) = ([O,A],[O,T],U),

(II.2)

U(i) = T - V - ' ( A - i )

(II.3)

where U : I —> S is the function

for all ie[0,A]. Note that V"1 (the inverse of V) exists since V is strictly increasing.

Note also that it follows from (II.3) that i = V(r) « T - r = U(A - i).

Intuition: When replacing the words "source" and "item" in the IPP (S,I,V) we obtain the IPP (I,S,U). Now, for each ie[0,A], U(i) denotes the cumulative number of sources that produce the items belonging to the interval [0,i], for each ie[0,A].

Note that, since V is differentiable and strictly increasing, the same is true for U. Denote, for every iel and reS

Basic theory ofLotkaian informetrics

103

a(i) = U'(i)

(II.4)

p(r) = V'(r)

(II.5)

We will (with an acceptable abuse of notation) denote

p(i) = V'(r)

if and only if i = V(r). In other words

p(i) = V'(v- 1 (i))

(II.6)

for all iel.

Intuition: For each reS, p (r) is the density function of the items in the source T - r and, for each iel, a (i) is the density function of the sources in the item i.

We have the following result:

Proposition H.l.1.1:

C(I)=

for every iel.

Proof: By (II.3)

K^)

(IL7)

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Power laws in the information production process: Lotkaian informetrics 1

"P(A-0 ' by (II.6).

D

Intuition: Because of (II.7), for every iel, p (i) is the density function of the items in the point A-i.

Our intuitive explanation that S = [0,T] is ordered in decreasing order of number of items is defined exactly by requiring p to be increasing (hence, by (II.7), also a increases). We furthermore suppose p (hence also a ) to be differentiate. Finally (although this is not necessary) we will always suppose p (0) = 1 (the lowest density of items in a source, i.e. the density in source T). Note that (by (II.5) and (II.6)) p (A) = p (T) is the highest density of items in a source (since V(T) = A). We denote this by pm .

II.1.2 General informetric functions in an IPP

We are now in a position to introduce, formally, the four informetric functions, introduced heuristically in sections I.I and 1.2. These basic functions are:

(i)

c itself

(ii)

g defined as g(r) = p ( T - r )

(II.8)

G(r)= f g ( r ' ) d r '

(II.9)

for all re[0,T] (iii)

G defined as

for all re[0,T] (iv)

f is defined as

m

~WUij

Basic theory of Lotkaian informetrics

105

forallje[l,p m ] (here gT denotes — , evidently). dr This is the fastest way to define the four functions. We now explain their meaning.

Intuition:

(i)

CT (i), being the density function of the sources in an item, is the continuous formulation of the Bradford formalism (not of the functional relation itself): there one looks, at packages of items with constant size, for the numbers of sources that produce the items in these packages, and one starts from the most productive sources (see Sections I.I and 1.2). Hence the function a is the continuous formulation of this formalism. Note already the power of this formalism: we attach a clear function a to it, something that is impossible in the discrete setting. In this connection, the function U(i), being I 0(i')di' (by (II.4) and since U(0) = 0 (by (II.3) and since V(T) = A)), is to be considered as the functional representation of the so-called (less popular) graphical form of Bradford's law (cf. Chapter I or see Wilkinson (1973), Narin and Moll (1977) or Vickery (1948)).

(ii)

g(r), being p(T —r) is the density function of the items in a source in re[0,T], hence we obtained the rank-frequency function, introduced heuristically in Section I.I. Note that g decreases since p increases,

(iii)

G is clear: it is the cumulative rank-frequency function, introduced heuristically in Section I.I.

(iv)

The function f is the size-frequency function introduced in Section I.I and is the basic function of this book. It requires explanation, however. From (11.10) it follows that

f(j) = -(g-')'(j)

(11.11)

(which is > 0 since g"1 decreases!). But, by (ii), g~' (j) = r = r(j) is the total number of sources with item density larger than or equal to j and, since g decreases, f(j) is the density of sources in j (by (11.11)). This corresponds to the size-frequency function f introduced in Section I.I.

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Power laws in the information production process: Lotkaian informetrics

We have, furthermore, that, by (II. 11),

g"'(j)= ff(j')dr J

(11.12)

J

for a certain c. But g(0) = p (T) = pm , hence g~' (pm) = 0. So c = p m . Hence r(j) = g-'(j)= P f ( j ' ) d j '

(H.13)

j

In the sequel, formula (11.11) can be used to find f, given g and formula (11.13) can be used to find g, given f.

The following proposition is a logical result from the definition of f but requires a non-trivial proof.

Proposition II.1.2.1: For every ie[0,A] we have

r < o jf(j)dj=i

(ii.u)

O1

Proof: Since p (0) = 1 we have that (11.14) is equivalent with

p(i)f(p(i))^(i) = l

but (g1 = M

by (11.10), forj = p(i):

(11.15)

Basic theory ofLotkaian informetrics

107

hence

-f(p(?))g'{&x[p{tf)) = \

(11.16)

Hence it suffices to prove that (by (11.15) and (11.16)):

g'igXOJH-pWftt

(ii.i7)

We have

g '« =

-f(T-r)

hence

8'(T-r) = -f(r) Also g(T — r) = p(r) (definilion of g) hence T - r = g~'(p(r)) = g"'(p(i)) (by (II.6), taking i = V(r)). So

g'lT-r^gVW)))

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Power laws in the information production process: Lotkaian informetrics

since i = V(r) and hence — = V'(r) = p(r) = p(i), proving (11.17) and hence (11.14). dr Intuition: Since f(j) is the density of sources in j , the number jf(j) expresses the item density in j . When integrating over the interval [l,p(i)] we have the cumulative number of items for item densities in [l,p (i)], hence i, by definition of p (i).

On the functions G and U we have the following result:

Proposition II.1.2.2:

G = IT'

(11.18)

Proof:

G(r) = / o r g(r')dr'

= / o 'p(T-r')dr'

= /oTp(r')dr'-/;-rp(r')dr'

G(r) = V ( T ) - V ( T - r )

But, by (II.3),

U(i) = T-V-'(A-i) . Hence

(11.19)

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109

T-U(i) = V"'(A-i)

V(T-U(i)) = A - i

i = A-V(T-U(i))

i = V(T)-V(T-U(i))

By (11.19):

i = G(u(i))

hence G = IT 1 .

Intuition: G(r) is the cumulative number of items in the sources belonging to [0,r] (by (II.9), (II.8), (II.5) and the definition of V). U(i) is the cumulative number of sources that produce the items belonging to the interval [0,i]. Hence U(G(r)) = r or G(U(i)) = i, hence G = U" 1 .

Back to the size-frequency function f. It follows from (11.13) and (11.14) that

J[ Pm f(j)dj = T

(11.20)

P'jf(j)dj = A .

(11.21)

and

*J 1

Indeed g(T) = p (0) = 1, hence

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Power laws in the information production process: Lotkaian informetrics

j;p"f(j)dj=g-'(i)=T

by (11.13) and pm = p (A), hence, by (11.14)

j| P "jf(j)dj = A .

Note that (11.20) and (11.21) also imply A > T, evidently.

Exercise H.l.2.3: Many other formulae can be proved, using the above formalism. Let us give one example: If (11.13) is valid (determining the relation between r and j), show that

G(r) = / o r g(r')dr' = ^ " j ' f ( j ' ) d j '

(hint: use formula (11.13)).

Chapter I explains the importance of the size-frequency function f in informetrics and especially in Lotkaian informetrics (which will be developed in Section II.2), where f is a power function. We close this general section with an existence theorem for f, given A (the total number of items) and T (the total number of sources).

II.1.3 General existence theory of the size-frequency function

Denote by

H= Y

the average number of items per source.

(H-22)

Basic theory ofLotkaian informetrics

111

Theorem II.1.3.1 (Existence Theorem for the size-frequency function f) (Egghe (2004b)): The following assertions are equivalent, given A > T > 0:

(i)

There exists a function f : [l,+oo[—> R + and a finite number pm > 1 such that

T=^fO)d J

A = J^'jf(j)dj (ii)

There exists a function f*: [l,+oo[—> R + such that

poo

^ =u < ^

T

'—

(11.23)

I f*0>J

We have, necessarily, that f* = Df with D > 0 a constant.

Proof: (i) ~ (ii)

By (11.22), (11.20) and (11.21), we must show that (taking f = f*)

rjf(j)dj _f f(j)dj

XW /"fojdj

which follows from the Lemma II. 1.3.2 following this theorem.

(ii) => (i)

Consider the function (for x > 1)

f jf*(i)dj cp : x - ^ -

(11.24)

112

Power laws in the information production process: Lotkaian informetrics Using several results of mathematical analysis (see Apostol (1957) or Protter and Morrey (1977)), we have that cp is a continuous function, hence attains all values between its minimum and maximum value, being (since f* > 0) respectively (using de l'Hospital'srule)

limm(x) = lim . . . V

*-i

'

*-i f

=1

x)

and

(since

1, we hence have the existence of a finite number, denoted pm > 1 such that

^ = ^ = dyP——

T

(H-25)

rnj^

Note that pm only depends on \i. From (11.25) it follows that, since f* > 0, there exists a constant D > 0 such that

f " j f ( j ) d j = DA

£f

(j)dj = DT .

Hence, the function f = — satisfies (11.20) and (11.21), hence (i).

Basic theory ofLotkaian informetrics

113

Lemma II. 1.3.2: For any integrable positive function f we have that 1 < x < y < oo implies that

fXuf(u)du rYvf(v)dv ' <J'

J

J ] f(u)du

J f(v)dv

(H.26)

Proof: It suffices to show that

(j|Xuf(u)du)(^yf(v)dv)<(j]yvf(v)dv)(j|Xf(U)du) But T fX(v-u)f(u)f(v)dudv

= J X J X (v-u)f(u)f(v)dudv + J y J X (v-u)f(u)f(v)dudv

= £ ^ vf(u)f(v)dudv-J^Xj|Xuf(u)f(v)dudv + JJJjX(v-u)f(u)f(v)dudv Since the first two terms are equal and since in the third term ue[l,x] and ve[x,y] the result follows from the fact that f is positive and that 1 < x < y.

Note that the Lemma above and (11.25) imply that p m (n) is an increasing function of \i, given a fixed size-frequency function.

The existence theorem will be used in the next section to actually determine which power functions (Lotkaian laws) are possible, given A > T > 0 as (given beforehand) the total number of items, respectively sources in an IPP. So, now on to the theory of Lotkaian informetrics.

114

Power laws in the information production process: Lotkaian informetrics

II.2 THEORY OF LOTKAIAN INFORMETRICS With the preparations of Section II. 1 we can now proceed with Lotkaian informetrics theory. This is, very simply, the theory that can be developed with a size-frequency function f (see Section II. 1) of the form

f:[l,P m ]->R +

j-f(j) = ^

,

(H.27)

where C and a are positive constants, hence a decreasing power function in the density j .

It is logical to start with the existence theory of such functions, given an IPP with A items and T sources. We will base ourselves on the general existence Theorem II. 1.3.1 for sizefrequency functions.

II.2.1 Lotkaian function existence theory

We will first give the easy theory when pm = oo (recall that pm = p (A) = p (T), the maximal item per source density), being a limiting case of the case pm < oo (the realistic case) which we will study in the next subsection. Most of the results go back to Egghe (2004b).

II.2.1.1 The case p m = oo

Let A > T > 0 be given and let f be of the form (11.27). We have the following result.

Proposition H.2.1.1.1 (Egghe (2004b)): The power function f as in (11.27) satisfies (11.20) and (11.21) for p m = oo and A < oo (hence T < oo) if the following relations hold:

9 A —T

cc = ^ — -

A-T

(11.28); V

Basic theory ofLotkaian informetrics AT

C = -^— A-T

115

(11.29)

which implies that a > 2, if A < oo.

Proof: (11.20) and (11.21) yield for pm = oo:

poo C

r-2_dj=A If a < 2 the last relation cannot be satisfied unless A = oo. If a > 2 we have the conditions

^ =T a-1

^ a-2

=A

from which (11.28) and (11.29) follow.

Note that (11.28) and (11.29) can also be read as

a =^

for ^ " Z i

H-l

(

C= -^- , H-l

cn.30)

a-2

(11.31)

where \x = — is the average production. Note that a only depends on \x while C depends on both A and fj. (hence on both T and A).

116

Power laws in the information production process: Lotkaian informetrics

Examples II.2.1.1.2:

1.

A=l,000 , T=500. Then n=2, a =3 and C=l,000 so that the function f(j) = - ^ r satisfies.

2.

A=10,000,T=l,000. Then n=10 and a = — sa2.11 , C = ^ ^ » 9 9 function f ( j ) = ' 2 U

3.

1,111 so that the

satisfies.

99 100 000 A=l 00,000, T=2,000. Then u =50 and oc = — PS2.02 , C = ' ^2,041 so that 49 49 the function f (j) =

] 202 (a "classical" Lotka law with a ~ 2) satisfies.

II. 2.1.2 The general case pm < oo

From Theorem II. 1.3.1 we obtain the following existence theorem for Lotkaian functions, given A > T > 0.

Theorem H.2.1.2.1 (Egghe (2004b)): Let A > T > 0 be given. Let a > 1.

(i)

If a < 2 then there always exists a number p m > 1 and C > 0 such that

satisfies (11.20) and (11.21).

(ii)

If a > 2 then the conclusion (i) is valid if and only if

Basic theory ofLotkaian informetrics 117

^ = M<^i T P a-2

(11.32)

Proof: The two parts will be proved together, using Theorem II. 1.3.1, if we show that

.DW

«-l

(n33)

P(j)dj"-2

(IL33)

if a > 2 and that the left-hand side of (11.33) is oo if a < 2 . This follows readily for all a>l.

D

Once the existence of f is proved, the next problem is, given A > T > 0 and a > 0, to construct the function f (and hence also to construct p m ). This will be executed now.

H.2.1.2.2 Construction off

Formulae (11.20) and (11.21) require

fp" c , r

T

yielding

7^-(p^-l) = T ' 1 —a v

(11.34)

-C_(p2-a-l) =A

(11.35)

for all a > 1 and

118

Power laws in the information production process: Lotkaian informetrics

if a * 2. For a = 2 we have

Clnp m = A

(11.36)

k\ Formulae (11.34) and (11.35) yield n = — :

1-a

pj-n-l

or

^2)

^ - ^

+1

- ^ ) = °

(IL37)

if a ^ 2 and (11.34) and (11.36) yield for a = 2:

lnpm+i-n = 0

(11.38)

Pm

Formulae (11.37) and (11.38) are the key equations that must be solved numerically. Because of Theorem II.2.1.2.1 we have that, under the conditions of this theorem, equations (11.37) and (11.38) always have a solution pm > 1. Once pm is determined, (11.34) or (11.35) (or (11.36) for a = 2) determine C, hence f. We will illustrate this with some examples. Note that, because of the extra parameter p m , the given A > T > 0 do not uniquely determine a but leave it as a parameter: for all a > 1 such that (11.32) is satisfied, we obtain a solution (contrary to the case p m = oo of the previous subsection).

H.2.1.2.3 Examples:

1.

a = 1.5 , A = 15,000 , T = 10,000 , hence u = 1.5. We have the equation (11.37) which takes the form (replacing pm by x)

Basic theory ofLotkaian informetrics

Y05

1.5

119

? 5

x-05+ — = 0 1.5

which can be solved for x > 1 (x = 1 is always an (imported) solution!) using e.g. the MATHCAD 4.0 software. We find x = 2.251 = p m . From (11.35) (but (11.34) also yields this) we find C = 14,990. In conclusion, the Lotka function

f(i) =

^

for je[l,2.251] yields the given A and T (for a = 1.5).

2.

a = 2 , A = 50,000 , T = 20,000 , hence \x = 2.5. We have the equation (11.38) which takes the form

25 lnx + — - 2 . 5 = 0 x which is solved in MATHCAD 4.0 as x = 9.322 = p m . From (11.36) or (11.34) we now have C = 22,398. Hence the Lotka function

f(j)=

22^398

for je[l,9.322] yields the given A and T (for a = 2).

3.

a = 2.5 , A = 20,000 , T = 10,000 , hence n = 2. Note that the condition ££

J

\i = 2 <

= 3 is satisfied so that there is a solution. We have now (11.37): a —2

-x-as-x-'-5-! =o 2 2

120

Power laws in the information production process: Lotkaian informetrics yielding x = 7.464 = p m . From (11.35) or (11.34) we find C = 15,773. In conclusion, the Lotka function

for je[l,7.464] yields the given A and T (for a = 2.5).

4.

a = 3.5 , A = 20,000 , T = 10,000 , hence \x = 2 (as above). Now

H = 2>1.6667 = — - . a-2

P

C So, according to the Theorem II.2.1.2.1, now there is no x = pm and function f = —^j that yields A and T. Equation (11.37) now has no solution in x = pm > 1.

The existence theory developed above can also be studied in Egghe (2004b). Note II.2.1.2.4: It follows from (11.23) (or, more simply from 11.14) that it is necessary for the existence of a size-frequency function f, given A > T > 0 and a > 1 that

A<J[°°jf(j)dj .

The right hand side of the above inequality is oo if a < 2 and hence is always satisfied. If a > 2 then the above inequality is

A < ^ a-2 or

Basic theory ofLotkaian informetrics

a < —+ 2 . A

121

(11.39)

This inequality was already noted in Egghe (1989, 1990a), see also Egghe and Rousseau (1990a).

From the results in this subsection it is also clear that the value of the exponent a = 2 is not only a "classical" value but also a "turning" point in the existence of the size-frequency function f. Throughout this book we will find more properties, implied by the Lotka function f, that have a = 2 as a turning point.

Note II.2.1.2.5: The results in Subsection II.2.1 could be considered as a mathematical "fitting" method for power functions, given A > T > 0, as opposed to statistical fitting (cf. the Appendix III on statistical fitting of a power law where formula (11.28) is re-used to quickly determine a , given a set of practical data).

We now turn our attention to the relations between the functions f, g, G and a as defined in Subsection II. 1.2.

II.2.2 The informetric functions that are equivalent with a Lotkaian size-frequency function f

We will consider the power functions

{

(i) = f

(H.40)

je[l,p m ], C > 0 and a > 0 and study the types of functions for g, G and a that are equivalent with f. We start with the "classical" case a = 2. We have the following theorem: Theorem H.2.2.1 (Egghe (1985,1989,1990a)) n : The following assertions are equivalent: (*) Preliminary, approximative results already appeared in Yablonsky (1980) and Bookstein (1984).

122

Power laws in the information production process: Lotkaian informetrics

(0 f(j) = §

(IL41>

, je[l, p m ], C > 0 a constant: Lotka's function with exponent a = 2.

(ii)

8

«=T4

(IL42)

, re[0,T], E, F > 0 constants: Mandelbrot's function with exponent P = 1.

(iii) G(r) = aln(l + br)

(11.43)

, re[0,T], a, b > 0 constants : Leimkuhler's function.

(iv) a(i) = M.Kj

(11.44)

, ie[0,A], M > 0, K > 1, M, K : constants: Bradford's function.

We, furthermore, have the following relations between the diverse parameters:

E = Pm

(11.45)

F = £a

(11.46)

a= - = C

(11.47)

Basic theory ofLotkaian informetrics

b= F= -^ C

(11.48)

M=— ab

(11.49)

K = e*

(11.50)

Proof: (iWii)

123

=» We use formula (11.13), yielding

r(j) = g-'(j) = j ; P " ^ d j '

j Pm

Putting j = g(r) we find

implying (11.42) with

E = Pm and

F=P- . C <= Conversely, (11.42) implies,

124

Power laws in the information production process: Lotkaian informetrics

which can be used in (11.10) to result in

E

f(j) = f which is (11.41).

(ii)^(iii)

=» using formula (II.9) gives

G(r)= W

f'—^dr'

J o l + Fr'

G(r) = | l n ( l + Fr)

implying (11.43) with

a= I = C F

Basic theory ofLotkaian informetrics and

C <= Reversely,

G(r) = aln(l + Br)

yields

£ which is (11.42) again.

G(r) = aln(l + br)

hence, calculating the inverse G"1 = U (by (11.18)), gives

U(i, = ^ l

So

a(i) = U'(i) = - U a(i) = M.K!

with

125

126

Power laws in the information production process: Lotkaian informetrics

M = -L ab

1

K = e»

(hence M > 0 and K > 1 since a,b > 0).

<= Since a (i) = M.K' we have (since U(0) = 0) that

U(i) = / o i a(i")di'

U(i) ^L(K'-l) W = lnK 1 I So, calculating inversely,

lnK^

Ink 1

/

I

by (11.18). Hence

G(r) = - L f l + ^ r l . V;

lnKl,

D

MJ

Note H.2.2.2: The parameter K is the universal Bradford multiplicator and is a real constant. Together with M, the parameter K builds the exponential function a (i) = MK1 (11.44) which is the simple functional relation that determines Bradford's law. There is no need to reproduce here the historic formulation of Bradford in terms of groups of equal item-size since it is a discrete approach and since it cannot be satisfied in practise (see the discussion in Chapter I). Furthermore the historic formulation (unlike the other ones: Lotka, Mandelbrot, Leimkuhler)

Basic theory ofLotkaian informetrics

127

of Bradford does not provide a function with fixed parameters. For those interested in the group-dependent formulation of Bradfords law, we refer the reader to Egghe (1989, 1990a) or to Egghe and Rousseau (1990a), where e.g. it is shown that the relation between the fixed multiplicator K above and the group dependent multiplicator k(p) (for p groups) is

K = k(p)A

(11.51)

(hence k(p)p is a constant).

Note also that, in case we have (11.44) for Bradford's function, we have that

U(i) = /o"a(i')di'

U(i) = J'M.K'di'

^~taK

"lnT

hence the "graphical formulation" of Bradford's law, represented by the function U can never be of the form M,KJ as suggested in Wilkinson (1973), cf. also the study of both laws (and also group-dependent and group-free) in Egghe (1991).

Very important is the characterization of the functions g, G and a in case we have (11.40) for general a . Above we treated already a = 2. In the next theorem we can use any a > 0 , a ^ l , a * 2 . The cases a = 0, a = 1 will be proved after this general case.

Theorem II.2.2.3 (Egghe (1985,1989,1990a), Rousseau (1988))°:

The following assertions are equivalent for a > 0, a * 1, a ^ 2 :

(*) Preliminary, approximative results already appeared in Yablonsky (1980) and Bookstein (1984).

128

Power laws in the information production process: Lotkaian informetrics

(i) f(j) = |

(H.52)

je[l,p m ], C > 0 and a constants: Lotka's general power function with exponent a.

(ii)

g(r)

=(Trkf

(IL53)

re[0,T], where

E = pm

F=

S^

C

(11.54)

(IL55)

Pm

P= ^ T a —1

(H.56)

are constants : Mandelbrot's function with general exponent p , cf. Egghe (1985).

(iii)

re[0,T]: Rousseau's extension of the Leimkuhler function (11.43) to general a s , cf. Rousseau (1988).

Basic theory ofLotkaian informetrics

129

(iv)

ie[O,A]: Egghe's extension of the Bradford function (11.44) (which itself- as a function - is also due to Egghe, cf. Egghe (1989, 1990a)).

Proof: (i)~(ii)

=> Formula (11.13) yields

r(j) = g-'(j) = j ; P " ^ d J '

C

/

i-a

hence, since j = g(r):

g(r) =

^-^r

which proves (11.53) - (11.56).

<= Formula (11.53) gives

g(r)=

(TT^r

and also

E J

" ( l + Fg-(j)f

-l-a\

130

Power laws in the information production process: Lotkaian informetrics

hence

1

iff"

These results can be put in (11.10) yielding

f(j)

^7(i%) =

E'+I? J _

"PEF

.Hi

hence (11.52), with the same parameter relations. (ii)~(iii)

=* using formula (II.9) gives

G(Ur)= J f ^ E °(l + Frf Note that, by (11.56), (3*1 since a * 2. Hence

P G(r)= , E ,((l + Fr)'~ -l). V ; ' F(l-p)\ V '

Using (11.54), (11.55) and (11.56) gives (11.57) after a small calculation.

g(r) = G'(r)

Basic theory ofLotkaian informetrics 131 from which

g(0 =

a-^r

easily follows.

(iii)^(iv)

- From (11.57) and the fact that U = G"1 (Proposition II. 1.2.2) we find

»0>=^p(r-(.4--^f. Hence

a(i) = U'(i) = ( p - - 2 z E L i ) ^ .

«= Since U' = a and since U(0) = 0 we have

U(i) = / o 'a(i')di'

u (i )^|(p---i^) 3 -Pr

Going to the inverse U~' we have

132

Power laws in the information production process: Lotkaian informetrics

whence

G(r) = U~>(r)

2-«

Pm

r

C )

'

showing all equivalencies and all relations between the parameters.

In

Theorems

II.2.2.1

and

II.2.2.3

we

could

have

sufficed

by

only

proving

(i) => (ii) => (iii) => (iv) =^ (i) . However the proof of (iv) => (i) requires implicitely all calculations

involved

in

proving

(iv) => (iii) => (ii) => (i).

Therefore

we

proved

(i) <S> (ii) <^> (iii) <s> (iv), hereby giving also direct proofs of equivalencies.

In Rousseau (1990a) some extensions of the above theorem are proved where other intervals than [l,p m ], [0,T],... are considered, trying to better conform with the discrete case but it is not clear that the results are useful in this case. A similar attempt is undertaken in Bookstein (1990a). Also the discrete index approaches of Chen and Leimkuhler (1986) and Hubert (1978) are not capable of establishing a far reaching informetric theory based on relations between the laws of Lotka, Bradford and Zipf. The early paper of Naranan (1971) has the merit to study rather rigorously the relation between the classical formulation of Bradford's law (i.e. in terms of groups) and functions of the type G.

Comments on the results of Theorems II.2.2.1 or II.2.2.3 can be found in Griffith (1988) and Wilson (1999).

Note H.2.2.4: From Theorem II.2.2.3 we see that, if a > 1 that

Basic theory ofLotkaian informetrics

133

with p > 0 and F > 0 (also E > 0 ). If a < 1 we have that

g ( r ) = E ( l + Fr)"p

(11.60)

with E > 0, F < 0 and -(3 > 0 (i.e. P < 0).

Although both cases could be proved in one time, regularities (11.59) and (11.60) are different: (11.59) represents the known Mandelbrot function; (11.60) was not included in the classical calculations and are due to Egghe (1985, 1989, 1990a).

Note H.2.2.5: From (11.58) it follows that (hence in the case a > 0 , a * l ,

P(i) = (l + i

^

(11.61)

Proof:

by Proposition II.2.1. Hence

i

i-\

( i a

J. 2 - a

.2-a"!^

P(.) = [p--A—+ i — j 1

M

f

.2-a^

p(i) = [l + . - ^ - j since (11.35) implies that

o*2)

134

Power laws in the information production process: Lotkaian informetrics

- ( 2 - a ) = p2-"-l.

D

Note also that, for all ie[0,A]

1+ i— - > 0

(11.62)

(as it should). This is proved as follows: If a < 2 then (11.62) is trivial. If a > 2 then

r , .2-a , . 2-a . mf l + i =1 + A >0 ie[0,A]

C

C

since (11.39) is valid for a > 2 . So

for all ie[0,A].

D

Exercise H.2.2.6: Show that, in case we have (11.52) in Theorem II.2.2.3 with p m = oo , that

i

pa-l

g(r)=

j_ (r(a-l))-

i f a > 0 , a ^ l , a ^ 2 . From Theorem II.2.2.1 it follows that this result is also valid if a —2 (i.e. g(r) = —) (hint: calculate lim g(r), with g(r) as in (11.53) or reproduce the proof of (i) <s> (ii) in Theorem II.2.2.3). Similar derivations can be made for G and a . As in Theorems II.2.2.1 and II.2.2.3, all functions obtained here are equivalent.

Basic theory ofLotkaian informetrics

135

We have now characterized the power size-frequency functions f(j) = — for all oc>0, a ^ 1. We will now complete the characterization with a surprising result for a = 1 and we will also present the case a = 0 for the sake of completeness. As is natural (see also Axiom 1.3.1.2 (ii)) for size-frequency functions fin informetrics, we will not consider increasing functions, hence we will not treat the cases a < 0 (although there might be a reason to do so which is not clear at the moment - the above developed techniques can - of course - then be applied).

For the case a = 1, we have the following (remarkable - for reasons to be explained further on) result.

Theorem H.2.2.7 (Egghe and Rousseau (2003b)): The following assertions are equivalent:

(i) f(j) = -

(11.63)

j € [l,p m ], C > 0 a constant: Lotka's power law with exponent a = 1.

(ii) g(r) = Hkr

(11.64)

i

re[0,T],

H = pm>l,

0
C

< 1 , H, k: constants: an exponentially

decreasing rank-frequency function

(iii) G(r) = N ( l - k r )

TT

refO.Tl, N = 1

J

> 0 , 0 < k < 1, N, k: constants. -Ink

(11.65)

136

Power laws in the information production process: Lotkaian informetrics

(iv) cj(i) = — - — W P + Qi

i e [0, A], P = -Nlnk > 0, Q = Ink < 0 : constants.

Proof: (i)<=>(ii)

=» From (11.13) we find

r(j) = g-1(j) = j : P ™ ^ J '

= Cln%] So

p(T-r) = g(r) = j = p m e " c = H k .

1

z

with H = p m > 1, k = e~ e ]0,l[. Hence (11.64) is proved. «- Conversely (11.64) implies g'(r) = H(lnk)k r

r'(i) - h ( l ) Hence, by (11.10) we have

s (g

(j)j

(11.66)

Basic theory ofLotkaian informetrics 137

\_\ Ink j

hence (11.63) with C =

(as found above). Ink

G(r) = / o r g(r')dr'

hence (11.65) with N = — — > 0. — Ink

g(r) = G ' ( r ) = - N ( l n k ) k r

hence we refind (11.64).

(iii)<=-(iv)

=> Since

G(r) = U - ( r ) = N ( l - k ' )

((11.18) and (11.65)) we have

U(i) U = — ln(l-i| Ink { Nj

whence

138

Power laws in the information production process: Lotkaian informetrics 1 CT(i)

= U'(i) = - S i p . ~N

, hence (11.66) with P = — Nlnk > 0 and Q = Ink < 0 (note that, because of this, a increases, as it should). <= Since we have (11.66) we have (since U(0) = 0):

U(i) = / o 1 a(i')di I

- Jof Pdi+'Qi' Q 1

PJ

So, inversely,

G(r) = U-(r) = ^ ( l - ( e « ) r )

p Q

which is (11.65) (e = k < l since Q<0 and since P > 0, N = — >0).

D

Finally the case a = 0.

Proposition 11.2.2.8 (L. Egghe, unpublished): The following assertions are equivalent

(i) f(j) = C

(11.67)

Basic theory ofLotkaian informetrics

139

j G [l,p m ], C > 0 a constant. So f is a constant size-frequency function

(ii) g(r) = R + Sr

re[0,T], R = p m > 0 , S =

(11.68)

< 0 : constants

(iii)

G(r) = f P ^ - ( P m - ^ ) 2

(11.69)

re[0,T]. (iv) c(i) =

r

(11.70)

ie[0,A]. Proof: The proof is contained in the one of Theorem II.2.2.3 which is also valid for a = 0. Of course direct, easy proofs can be given along the same lines.

It is "classical" to consider the function G(r) in function of log r (log = log10 or log = In) instead of r. As predicted in Egghe (1985) but proved exactly in Rousseau (1988) we have the following theorem, explaining the so-called Groos droop (Groos (1967)) and its opposite effect.

Corollary H.2.2.9 (Rousseau (1988)): The function

logr^G(r)

140

Power laws in the information production process: Lotkaian informetrics

has an inflection point if and only if a < 2 . In this case the inflection point is in the value

l +^ r =^ 2 —a C

"

(H.71)

r = ^ — P!T 2 —a

(H-72)

Proof: The function logr —> G(r) (where we use log = In, but, of course, any other log can be used) has the form (use (11.57)): for t = lnr

G(t) = ^

p--(p--i^e')^

From this it follows that

G"(t) = e ' ( p - - ^ A -

G"(t) = 0 if and only if

(i)

for a < l and (r = e')

,_a_l-a m

or

C

\

(11.73)

Basic theory ofLotkaian informetrics 141 (ii)

for all a r = e'=-^—p 1 "" 2-a (i.e. (11.72)).

Case (i) is excluded since (11.34) yields

e'=r
since a < 1.

Formula (11.71) follows from (11.72) using (11.61).

The above corollary makes it clear that for a > 2 we have no inflection point (since r > 0) and that G"(t) > 0 (hence G is convex). Only if a < 2 the value of r in (11.72) is finite and positive. In this point, the function G switches from convex to concave and this explains the Groos droop - see also Fig. II. 1. This shows, again, the special role of the exponent a = 2.

In Rousseau (1994b), some fittings of the curve of (11.57), including the ones with a Groos droop, are given.

We leave it to the reader to prove the following easy result, showing again the special position of the exponent a = 2.

142

Power laws in the information production process: Lotkaian informetrics

Fig. II. 1 Dependence of G on a. Existence of a Groos droop if a < 2 . Proposition II.2.2.10 (Egghe (1990b)): In case the size-frequency function f satisfies

f(j)=

7

je[l,p m ], oc>0, C>0, then the function a(i) satisfies:

(i) (ii)

(iii)

o'(i) —j-rf- strictly increases if a < 2 a(i) a'(i) —j^- strictly decreases if a > 2 a(i) —}^- = In K is constant if a = 2. a(i)

Basic theory ofLotkaian informetrics

143

Important remark II.2.2.11: If we compare the cases a = l and a = 2 and denoting temporarily pl,a,,p1,a2

for the

functions p and a in the respective cases a = 1 and a = 2 we see that, as functions

p,=02

(11.74)

a,=p2

(11.75)

This follows from (11.42), (11.44), (11.64) and (11.66), together with the relation (II.8): g(r) = p ( T - r ) or, what is the same: p(r) = g ( T - r ) , for all r e [0,T]. Indeed (11.42) has the form

p 2 (r) = g 2 ( T - r ) =

T T 1

|r-^

1 ~ 1 + FT-Fr

which is the same function as (11.66) since P = 1 + FT > 0 and Q = —F < 0 (replacing r by i, i.e. replacing sources by items). Also from (11.64) we have

p,(r) = g l ( T - r ) = HkT-'

kj which is the same function as (11.44) since M = HkT > 0 and K = — > 1 (again replacing r k byi).

In words: upon adding a translation r —> T — r in the Mandelbrot function, this function and the Bradford function are interchanged when going from a = 1 to a = 2 or vice-versa, a

144

Power laws in the information production process: Lotkaian informetrics

remarkable constatation ! Note that this result could only be revealed by defining the Bradford formalism in a continuous setting, hereby defining a Bradford function which is not defined in the historical formulation (Bradford (1934)).

It is furthermore remarkable that Lotka's power law (for a = 1) for f includes an exponential law for g, see formula (11.64) and not only Mandelbrot-type power laws (of 1 + Fr), see formula (11.53). In this sense, size-frequency functions (of power law type) comprise a wide, heterogeneous, class of rank-frequency functions.

This finishes the "classical" informetrics theory around the well-known functionalities of the type of Lotka, Mandelbrot, Leimkuhler and Bradford. In the next section we will investigate duality properties of these functions and we will make the constatation that there is an informetric function missing: the dual size-frequency function. We will introduce it here for the first time and derive its properties. We will also indicate possible applications.

II.3 EXTENSION

OF

THE

GENERAL

INFORMETRICS

THEORY: THE DUAL SIZE-FREQUENCY FUNCTION h From the previous sections (mainly Section II. 1) we can conclude that duality can be described as: interchanging the terms "item" and "source" and reversing the order.

We repeat the definition of U and V (Section II. 1):

U(i)

=

cumulative number of sources that produce the items belonging to the interval [0,i], for each ie[0,A],

V(r)

=

cumulative number of items in the sources belonging to the interval [T - r,T], for each r e [ 0 , T ] .

This makes a(i) = U'(i) and p(r) = V'(r) dual functions: both increasing and describing the same functionalities: density of sources in an item (for a) resp. density of items in a source (for p), which is also expressed by their relation (II.7). The functions g(r) = p(T —r) and

Basic theory ofLotkaian informetrics

145

G(r) — I g(r')dr' are simple transformations of p . This reveals the duality of three of the O 0

four functions, defined in Subsection II.1.2: a , g, G, leaving out our basic size-frequency function f. Since f is defined as (11.10) g' = — { drj

f(S

=-Jt?o))=- (8 " 1)1())

j € [l,Pm] and by the above remark, it is clear that another size-frequency function, now based on a , can be defined. Because p(r) = g (T — r), r £ [0, T] and invoking the duality between a and p , we hence formally define the dual size-frequency function h as a' = —

I h(s)=

,,

1

dij

,=(a-')'(s)

(11.76)

a'{a (s))

for s = a(i) e [a(0),l], 1 = G ( A ) (since we took p(0) = 1 and because of (II.7)).

Intuition:

f(j)

=

density of sources in the item density (in a source) j 6 [l,pm]

h(s)

=

density of items in the source density (in an item) s 6 [a (0), l]

and hence are dual functions.

Indeed, for f this has been explained in Subsection II. 1.2, using (11.11). A similar argument yields the interpretation for h: since a(i) is the source density in i € [0, A] (by (II.4)) we have that a"' (s) is the cumulative number of items with source density smaller than or equal to s (since CT increases). Hence h(s) = fa~' )'(s) (by (11.76)) is the density of items in the source

146

Power laws in the information production process: Lotkaian informetrics

density s.

In the discrete setting (which is modelled by the continuous functions above) f(j) is the number of sources with j items and h(s) is the number of items with s (< l) sources, i.e. the number of items produced by sources which have - > 1 items, using the fractional counting s system (see also Chapter VI for a thourough description of this system). Hence it is clear that the fractional counting system is the natural one if one studies duality in IPPs. The usefulness of the function h is clearly seen in the following IPP case: let us consider an IPP consisting of articles as sources and of authors of these articles (possibly multi-authored articles) as items: i.e. considering articles with 1,2,3,... authors. f(j) is then the number of articles with j=l,2,3,... authors and h(s) is then the number of authors with a fractional score s = - in one paper. The function h has already been used in this context to model fractional j frequency scores for authors - see the papers Egghe (1993b) and Egghe and Rao (2002a) but in these papers the function h was determined approximately (in the continuous setting in Egghe (1993b)) or exactly but in the discrete setting in Egghe and Rao (2002a), hence a clear relation with the dual informetric theory (Section II. 1) did not exist. We will present it here for the first time. We have the following theorem, generally valid in informetrics (even nonLotkaian).

Theorem II.3.1 (Egghe, unpublished): For every j e [l, pm ]:

'('I f(j) = - ^

or, what is the same: for every s £ fo(0),l|:

(11.77)

Basic theory ofLotkaian informetrics

147

f (I| h(s) = ~ ^

(11.78)

Proof: We start by the formula (II.7):

°(i) = ^

)

ie[0,A]. Hence CT' = —

,/.\

N dr

dp,™

1

CT(l)=

T(T^)d7(T-r)d[

since we defined p ( A - i ) = p ( T - r ) iff A - i = V ( T - r ) , see (II.6). Since g(r) = p ( T - r ) (by (II.8)) we have

_,m U

=

1

dg(r)dU(i)

g2(r) dr

di

since r = U(i) because this is equivalent with A — i = V (T — r) by (II.3). But

dU(i)

,,

1

1

1

^ = 0 i )=—-i—-=— r = -rr U p(A-i) p(T-r) g(r) di hence

(11.79)

148

Power laws in the information production process: Lotkaian informetrics

tf'(i) = - l ^

(H.80)

\G' = -£-, g' = - ^ .Hence, for every j e [ l , p m ] : 1, di dr)

i_

~~W) since j = g(r) (definition of g). Hence, by (11.80):

S{!)

=WWV)

since r = U(i) and s = o(i) (definition (11.76)). But, by (11.79)

'=°®=W)=T (11.82) and (11.76) in (11.81) yields

hence (11.77). From this (11.78) readily follows from (11.82).

(IL82)

Basic theory ofLotkaian informetrics

149

Note that, up to a constant, formula (11.78) already appears in Egghe (1993b) but where the

f

A)

formula is obtained via approximations. The constant m, = — appears in Egghe (1993b) since there one works with fractions (hence normalized in [0,1]) while here we have functions on r £ [0,T] and on i £ [0, A] which incorporate this constant. In any case it is remarkable that the approximate argument in Egghe (1993b) produces an exact result.

In Lotkaian informetrics we have the following corollary to Theorem II.3.1, hereby completing the equivalencies proved in Theorems II.2.2.1, II.2.2.3, II.2.2.6 and II.2.2.7.

Corollary II.3.2 (Egghe, unpublished): The following assertions are equivalent:

(i)

je[l,p m ], C > 0 , a > 0 constants

(ii) h(s) = Cs"-3

(11.83)

s £ [CT (0), l], C > 0, a > 0 constants.

Proof: This follows readily from (11.77) and (11.78) but can also be proved directly from definition (11.76), using the function a and Theorems II.2.2.1, II.2.2.3, II.2.2.6 and II.2.2.7.

Note that the power-type law of f (Lotka) implies a power-type law of h, and vice-versa.

150

Power laws in the information production process: Lotkaian informetrics

II.4 THE PLACE OF THE LAW OF ZIPF IN LOTKAIAN INFORMETRICS II.4.1 Definition and existence

In Section II.2 it is shown that, if a > 0 and o ^ l w e have equivalency of the law of Lotka

f(j) = ^

(H.84)

j € [l,p m ], and the law of Mandelbrot

8W

=07F7

(II 85)

-

r e [ 0 , T ] , with E = p m ,

F= ^ Cpm

(H.86)

P= ^ T oc-1

(H.87)

Definition II.4.1.1: The law of Mandelbrot for F = 1 is called the law of Zipf.

This conforms with the classical law of Zipf since, if F = 1 we have

.(0=^ r € [0,T], hence the classical function

aim

Basic theory ofLotkaian informetrics

g(r') = ^

151

(H.89)

r1 > 1 (note that r1 € [l,T +1]).

Can F = 1 be realized? It is clear that this requires, by (11.86) that

a = l + Cp^a>l

(11.90)

for all a . This implies, by (11.87) that Zipf s law is limited to the case (5 > 0 (contrary to the (3 in Mandelbrot's law), which is also the case in the classical definition. Equation (11.90) is the necessary and sufficient condition to have Zipf s law and determines implicitely the place of Zipf s law in Lotkaian informetrics. Indeed, the above shows that a part of Lotkaian informetrics is Zipfian. Reversely, if we have Zipf s law, we have Mandelbrot's law with F = 1 and hence Lotka's law (11.84).

If we have Zipfs law, Lotkaian informetrics is automatically limited (by (11.90)) to c o l . But for every a > 1 we have a case where Zipfs law applies, namely if (11.90) is satisfied, which can always be achieved: take

i

Pm = - ^ r ct-1.

(H.91)

Note that (11.91) produces a pm > 1 (as required) as the following argument shows: By (11.55) we have

cpir But (11.34) implies

152

Power laws in the information production process: Lotkaian informetrics

Hence (11.55) now becomes

F= — — a-1

(11.92) T

The condition F = 1 (for Zipf) implies

T=— a-1

1

(11.93)

Q

hence, since T > 0 we have that

> 1 and hence, by (11.91), pm > 1, since a > 1.

II.4.2 Functions that are equivalent with Zipf s law

From the general Theorems II.2.2.1 and II.2.2.3 we can deduce the functions that are equivalent with Zipf s law: as explained above, only the values a > 1 are possible.

Theorem II.4.2.1 (Egghe (1985,1989,1990a), Rousseau (1988,1990a)):

(i) f(j) = ^- (Lotka'slaw)

je[l,p m ],

P

=

f C]^

U-i.

Basic theory ofLotkaian informetrics 153 (ii) ( Z i p f S laW itSelf)

8 ( r ) = T^J

r€[O,T],E = p m , p = - i

(iii)

T

.

If a * 2:

"("-^-(f-^'f' i

r e [0,T], again with pm = - ^ j

.

If a = 2 :

G(r) = Cln(l + r)

(11.94)

re[O,T], the special form of Leimkuhler's law (b = 1) which can also be considered

as a

special

form

of Brookes'

G(r) = cln[d(l + r)] withd = l.

(iv)

If a * 2

i

i-\

{ 2-0. . 2 - a V - 2

1

f C I"1? i € 0, A , again with p = lot —lj

law (Brookes

(1973):

154

Power laws in the information production process: Lotkaian informetrics If a = 2: Bradford's law:

CT(i) = (lnK)K'

(11.95)

i € [0, A], K = e c > 1, being equivalent with the formula

U(i) = K ! - l

(11.96)

(graphical form of Bradford's law).

Proof: This follows readily from the results in Theorems II.2.2.1 and II.2.2.3 and the fact that the original (also called verbal) form of Bradford's law (the function a ) and the graphical form of Bradford's law (the function U) relate as

U(i) = / o i
forie[0,A]. For more equivalencies with Zipf s law, e.g. with the group-dependent forms of Bradford's law (which we consider as less useful since they do not define a clear informetric function), we refer the reader to Egghe (1991)).

Notes II.4.2.2:

1.

From the above it follows that, if Zipf s law applies then we have:

T —> oo if and only if pm —> oo.

Basic theory of Lotkaian informetrics

155

Proof: Formula (11.34) says

a —1v

'

Hence, by (11.91) we have

T = P r i — i r =Pm"'-i

(n-97)

Pm

and, as always when Zipf applies: a > 1 (see (11.90)). Hence T —> oo iff p m —> oo.

This result is not true for general Lotkaian systems. Indeed, as expressed by equation (11.37), p m only depends on a and \x = — .So if we let T and A go to oo such that \i is constant, and if we keep a fixed, we have that pm is fixed. A concrete example is given in the first example of II.2.1.2.3: for n = 1.5 and a = 1.5 we calculated that pm =2.251. There T = 10,000 and A = 15,000, but any common multiple of these values (hence for A,T —> oo) yields the same pm < oo .

From the above (if Zipfs law applies) and Proposition II.2.1.1.1 it follows that T = oo implies pm = oo, hence a > 2 , if A < oo. Hence, by (11.87), (3 < 1. So, Proposition II.2.1.1.1 is only valid for Lotka's law and not for Zipfs law (if it were valid then T = oo implies P > 2, which is false, as expressed above).

The above result (T —> oo iff pm —> oo in case Zipfs law applies) will become crucial in Chapter IV where we can, because of this result, develop concentration theory for Lotka's law and Zipfs law at the same time.

2.

Note that it follows from (11.97), (11.88) and the fact that E = p m and (3 = a —1 (Theorem II.4.2.1) that

156

Power laws in the information production process: Lotkaian informetrics

g(r) =

[

(11.98)

Vre[O,T].

3.

Because Zipf s law is a special case of Mandelbrot's law and because the latter one is completely equivalent with Lotka's law one is inclined to think that Zipf s law is not equivalent with Lotka's law. It is true that not all C and pm -values for Lotka's law imply Zipf s law (as is clear from the above) but, for the mentioned parameter values (i.e. (11.91)), we have equivalence of any Lotka power law with exponent a > 1, with any Zipf law with exponent P =

. a —1

4.

Note that, by (11.96), U(i) = K1 - 1 and not K ! , the latter one being the function in the "classical" graphical formulation of Bradford's law (in the group-dependent version see Wilkinson (1973), Narin and Moll (1977) or Vickery (1948)). It turns out that the pure exponential form of U can never be equivalent with the pure exponential form of G , contrary to the historical formulations - see also Egghe (1991).

Ill THREE-DIMENSIONAL LOTKAIAN INFORMETRICS

III.l THREE-DIMENSIONAL INFORMETRICS In Chapter I we introduced the informetrics theory by describing sources (the "producing" objects), items (the "produced" objects) and a matching function describing which source(s) produce(s) which items. In this connection we could talk of two-dimensional informetrics (sources/items or type/token) which is more than twice one-dimensional informetrics in which only data with respect to the sources, respectively items are studied (e.g. number of sources, number of items): in two-dimensional informetrics one also studies and quantifies which sources produce which items (the matching function) and yields, as a consequence, the informetric laws (functions).

Given an item set in two-dimensional informetrics we can determine different sets of sources that produce these items. Indeed e.g. for articles as items one can consider journals (in which these articles appeared) as sources but one could consider the authors of these articles as sources as well. Note that, in this chapter, we do not address the problem of multiple authorship: we will deal with this (intricate) problem in Chapter VI; in the present chapter we deal with different source sets such as journals and authors producing articles. Reversely (or in a dual way) we can consider a source set and determine two item sets of produced objects by these sources. An example is given by articles giving references but also receiving citations. Note that in the first example an article was an item while in the second example an article is a source. This leads us automatically to a third case: the case of two source sets and two item sets but where one source set is equal to one item set. Example: the source set of journals produces the item set of articles in these journals. This item set is then considered as our second source set producing a second item set, e.g. of references in these articles. Such

158

Power laws in the information production process: Lotkaian informetrics

systems are the object of study in three-dimensional informetrics. Again also here, threedimensional informetrics is more (is a "higher" discipline) than three times a one-dimensional informetrics theory or than two times a two-dimensional informetrics: the different matching functions are not independent from each other and, hence, their relations must also be studied. From the above introduction it is clear that we can determine three types of three-dimensional informetrics. Let us now describe them in more detail (cf. also Egghe (1989, 1990a) and Egghe and Rousseau (1990a)).

III. 1.1 The case of two source sets and one item set

This case can be symbolized by Si, S2 the two source sets, I = the item set and depicted as in Fig.III.l

Fig. III. 1

Three-dimensional informetrics: two source sets and one item set (—> means: produces).

The most classical example is the one given above: Si = {authors}, S2 = {journals}, I = {articles}, where we consider the authors and journals as producers of articles. Other example: I = {articles} produced by Si = {authors} but also by S2 = {research institutes}, i.e. the research institutes where these authors are employed. Of course, this example could also be considered in the "linear" way (see Subsection III. 1.3): institutes have authors and authors have articles. Of the same (double) nature are the examples of Lafouge (1995, 1998): volumes (of journals) contain articles and these articles are used (e.g. number of (inter)library requests). Although this example could also be classified in the coming Subsection III. 1.3, Lafouge also considers the "triangle" form (as in Fig.III.l) with two source-sets and one itemset. In our notation, his triangle looks like in Fig.III.2 (g,, g2 are the two rank-frequency

Three-dimensional Lotkaian informetrics

159

functions). Here h is the mapping of which article (in Si) belongs to which source (in S2) and hence the link between the two IPPs is made via the symbolic formula

g2(r2) = X-, (r2) g.( r .) dr ,

Fig. III.2

On-i)

Linking of two IPPs in the case of Fig.III.l (p r a l and p m 2 denote the maximal item densities p m in a source in the two IPPs - see Subsection II. 1.1).

(Lafouge uses a discrete setting and hence a 52 instead of an / ) . In this sense, use of the volumes is determined by a cumulation of the use of the articles that belong to these volumes. Lafouge then employs this model to determine size-frequency distributions for volume-use based on (non-Lotkaian - e.g. geometric) size-frequency distributions for article-use.

By duality, i.e. replacing items by sources and sources by items (i.e. replacing "produces" by "is produced by") we have that the above case is essentially the same as the next one on which more examples are known.

III.1.2 The case of one source set and two item sets

This case can be symbolized by S = the source set, Ii, h the two items sets and depicted as in Fig.III.3.

160

Power laws in the information production process: Lotkaian informetrics

Fig. III.3

Three-dimensional informetrics: one source set and two items sets.

Examples: Papers have references and receive citations, so papers produce references and produce citations; authors produce articles and produce article pages. Journals publish articles but also contain authors. So authors do not always have to be considered as sources (cf. the Subsection 1.4.4 devoted to authors as items of articles, i.e. where one considers the number of authors per paper). In this connection we also have the example: articles producing authors but also citations. From Coleman (1992) we obtain the example: labs are sources of scientists and of articles. In Yitzhaki (1995) one has the example of articles as sources for pages (or another measure of the length of an article such as words or even characters) and of references hereby also studying the relation between the length of an article and the length of its reference list. Kyvik (1990) studies authors as sources of articles and of ages. A bit strange is the following example that is implicite in Qin (1995) but stated explicitely in Egghe and Rousseau (1996c): authors as sources of publications and of co-authors (the latter variable for Lotka's function has also been studied in Gupta and Karisiddippa (1999)).

Although we have many examples, not many results are known in this case. We refer the reader to Egghe and Rousseau (1996c) for a first attempt to derive three-dimensional properties in this case. The results, however, are not far reaching, leaving open the general problem to describe the situation depicted in Fig.III.3.

Here we may give one indication on how one may proceed. Let g[ respectively g2 be the rank-frequency functions of the two IPPs (S,Ii) and (SJ2) respectively. As we indicated already, we must find the "added value" of three-dimensional informetrics above the description of these two-dimensional IPPs. It is clear from Fig.III.4 that the function

Three-dimensional Lotkaian informetrics

k= g2V

161

(ni.2)

shows the link between these two IPPs, hereby indicating the link between the items in Ii and the items in I2, being produced by the same source (that is why the rank-frequency functions g, and g2 are used and not the respective size-frequency functions, as was done in Egghe and Rousseau (1996c)).

Fig. III.4

Linking of two IPPs in the case of Fig.III.3 (p m l and p m 2 denote the maximal item densities pm in a source in the two IPPs).

Supposing power laws for g, and g2 (i.e. Zipf s law), it is obvious from (III.2) that also the function k is a power law and this might be regarded as a first (elementary) result on this type of three-dimensional informetrics.

III.1.3 The third case: linear three-dimensional informetrics

This case is, by far, the most important one and refers to the case of two source sets Si and S2 and two item sets Ii and I2 but where the second source set S2 equals the first item set l\. We can depict this situation as in Fig.III.5.

s,

Fig. III.5

>

i,-s2

>

i2

The case of linear three-dimensional informetrics: the item set of the first IPP is the source set of the second IPP.

162

Power laws in the information production process: Lotkaian informetrics

This means that the items in the first IPP become sources (producing items) in the second IPP. Fig.III.5 makes it clear why this theory is also called linear three-dimensional informetrics.

Examples: Researchers (sources in Si) produce articles (items in Ii) which become sources (in S2= Ii) of citations (or references) (items in I2). Of the same type is the "chain": journals —> articles —» citations. In Rousseau (1990b) one studies the linear three-dimensional problem: journals —> articles —> software programs (discussed in these articles). Linear three-dimensional informetrics problems were also (implicitely) mentioned in Rousseau (1992a) and Fox (1983). Rousseau (1992a) gives the example of a public library where one has categories of music, each of them having CDs available, being items of these categories but also sources of their borrowings. Of course many more examples can be given: the example in Subsection III. 1.2 of Coleman (1992) can be rephrased: labs (or countries) are sources of scientists (items) which become sources of articles. Also: authors produce papers and these papers have pages.

Unique for this case is (as is clear from Fig. III.5) that a "new" IPP is constructed, namely the IPP with source set Si and item set I2. A natural question arises: describe the rank- and sizefrequency functions of the "composed" IPP (S1J2) from the two IPPs (Si,Ii), (S2J2) (with I] = S2 of course). This will be done in the next Subsection III. 1.3.1 under the name "Positive Reinforcement" indicating the reinforcement of the production of the sources in Si via the composition with the IPP (Ii = 82,12). This problem will be completely solved for the (only important) case that the source-rankings r2 on S2 in the second IPP (S2J2) are the same as the rankings in S2 = Ii, induced by the source-rankings n in the first IPP (more explanation in Subsection III.1.3.1).

A bit more veiled but at least as important is another study that is possible in the case of Fig. III.5: Type/Token-Taken informetrics. In this theory (to be developed in Subsection III. 1.3.2) the informetrics of the use of items is described (instead of their existence). More details will be given in the sequel with many practical examples. Type/Token-Taken informetrics boils down to the study of the second IPP (S2J2) but where (as in Fig. III.5) S2 = Ii (and the influence of the IPP (S1J1) is incorporated) and where we keep the same item set: I2 = Ii (compare with positive reinforcement where the source set (Si) is kept fixed but where I2 is the item set). In this connection, we can consider Type/Token-Taken informetrics as the dual

Three-dimensional Lotkaian informetrics

163

of the positive reinforcement theory - see the next two subsections for more details).

III.1.3.1 Positive reinforcement

Positive reinforcement of an IPP is a feature that can be defined independently from threedimensional informetrics as the next definition shows.

Definition III.1.3.1.1: Let S = [0,T], I = [0, A] be an IPP with rank-frequency function g. Let I* = [0, A*]. We say that the IPP (S,I*) is a positive reinforcement of the IPP (S,I) if its rank-frequency function g* is given by g'=9°g,

(HI.3)

where (p is a strictly increasing function such that cp(x)>x for all x. By definition of g (II.8), the same relation is true if g is replaced by p (since T and r are the same in both IPPs).

The interpretation of the above definition is clear: the positively reinforced IPP (S,I ) has the same sources but with an increased productivity, given by (III.3).

We have the following basic result on linear three-dimensional informetrics, showing that the composed IPP (S1J2) (as described above) is a positive reinforcement of the IPP (Si,Ii), at least under the following (completely natural) restriction, which is the only important case of linear three-dimensional informetrics.

Restriction HI.1.3.1.2: We will restrict ourelves to the case that the source-rankings r2 in the second IPP are the same as the rankings in Ii, induced by the source-rankings r, in the first IPP. We will express this restriction in a mathematically exact way but first we give an intuitive interpretation: items in Ii are ranked according to the ranking we have in Si (as always, sources are ranked in decreasing order of production). Considering Ii = S2 as sources in the second IPP, we require that their productivity in the second IPP is such that the source-ranking r2 in this second IPP is the same as the one we have already in Ii. An exact formulation on this restriction is

164

Power laws in the information production process: Lotkaian informetrics

r 2 = / o r ' g , ( r ) dr

(III.4)

Note that, by (II.9) and (11.18), r2 is nothing else than G(r,) = IT 1 (r,) in the IPP (Si, Ii). Note also that (using (II.3)) r2 G It = S 2 , as it should.

Theorem IH.1.3.1.3 (Egghe (2004e)): Let the first IPP have source-set S, = [0,T,], item-set I, = [0,A,], size-frequency function fi and

rank-frequency

function

g,

and

let

the

second

IPP

have

S2 =[0,T 2 ] = I, =[0,A[], item-set I 2 =[0,A 2 ], size-frequency function

f2

source-set and rank-

frequency function g 2 . Then the composed IPP (under restriction III. 1.3.1.2) has source-set S = S,, item-set I = I2 and rank-frequency function

g =
where

^

9(j,) = g2(^ Pl Jfi(j)dj)

is a strictly increasing function. If j 2 = g 2

(in.5)

" g.(g. 0))

(HI.6)

I jf, (j)dj > j , then (p(j,)>j, for every j . .

Hence the composed IPP is a positive reinforcement of the first IPP. Its size-frequency function f has the form

Three-dimensional Lotkaian informetrics

165

(j £ [l,(p(p,)]j, where p, =: p m l = p(A,) = p(T,), the maximal item-density in a source in the first IPP.

Proof: Since g2 acts on the ranks r2 in the second IPP and since the restriction (III.4) defines the relation between r, and r2, we have that the composed IPP has the following rank-frequency function g:

g(rI) = g 2 (/ o "g 1 (r)dr)

(III.8)

Now we are looking for a function tp such that

g(r,) = (
(IH.9)

We will prove its existence via (III.8), show that cp strictly increases and hence that the composed IPP is a positive reinforcement of the first one.

If such a 9 exists then, (III. 8) and (III.9) combined yield

cp(g1(r1)) = g 2 (/ o "g,(r)dr).

Substituting r = g"1 (j) (see formula (11.13)) gives

since g, (0) = p m , =: p,, the maximal item density in a source in the first IPP and where r, =gj~ 1 (j 1 ), the existing relation between j \ and r, in the first IPP (see formula (11.13)). Hence

166

Power laws in the information production process: Lotkaian informetrics

-u-\\

q>(j 1 )=g 2 (j; i pi jf 1 (j)dj)

using (11.10), hence proving (III.5) and (III.6) . Since j 2 > j , we hence have cp(j,) > j , for all j , . That (p strictly increases follows easily from (III.5):

J

<" gl(g, (j))

g,(& 0,))

Since j , > 1 > 0 and since g2 and g, are negative (by (11.10)) we hence have that 0. Hence linear three-dimensional informetrics is an example of positive reinforcement.

Let us now determine the size-frequency function f of this positively reinforced IPP. Invoke (11.10):

r(i)

-i#M

We have, for the positively reinforced IPP: r, e [0,1;], j e [l,(p(p,)],g(r]) = j . So, by (III.9)

g'( r i) = (P'(g,(r,))gi(r1)

g'(g- i o))=
Butg = cp°g,, hence g~' = g,"10^^1 hence

gV'a)HVO))&(s>V(j)))

Three-dimensional Lotkaian informetrics

167

So, by (11.10)

f,.x

w

1

~

1

«P'(
W

"«P'(9-'(i))

for all j 6 [l,

Corollary III.1.3.1.4 (Egghe (2004e)): Let (x, be the average number of items per source (i.e. jx, = —L) in the first IPP and \i be the average number of items per source in the positively reinforced IPP. Then H>H,

(III. 10)

Proof: Since the number of sources in both the first and the reinforced IPP are the same (Ti) we have, by (II.9), (11.18), the definition of U and (III.9)

/ 0 T 'g( r >) dr . |i = ^

T,

_ J 0 T ' ( p(g l ( r i)) d r i T,

-

T,

168

Power laws in the information production process: Lotkaian informetrics

since j, for all j , .

The property j , is a natural requirement in positive reinforcement: it states that, in the composition, sources in S, produce more items than before the composition (in the first IPP).

For a discrete, stochastic attempt to model a 3-dimensional situation we refer to Burrell (1992c).

III. 1.3.2 Type/Token-Taken informetrics

The restriction of the source-item informetrics theory (Chapter II) (here also called Type/Token or TT informetrics) is that we only describe sources and the items in these sources; in other words : sources and their production. TT informetrics (important in itself of course) does not study the use of these items, hence these items as they are encountered by information professionals or other information users. Let us give some examples. Search keys (e.g. N-grams) are used by catalogers in a library network to quickly find out if a newly arrived book (hence that must be cataloged) is already in the database. For a description of search keys (or N-grams as a special case) see e.g. Egghe (1992, 2000a), Kilgour, Long and Leiderman (1970) - see also Section VII.6 for a thorough description of N-grams and their informetric properties. The distribution of the occurrences (tokens) of the N-grams (types) in the catalog will be different from the distribution of the occurrences of these N-grams as experienced by the cataloger. Indeed the more an N-gram occurs in the catalog, the more this N-gram will also be used by the cataloger. This will lead to larger (this will be shown in the sequel) average screen lengths than what would be expected from the occurrences of the Ngrams in the database itself. Catalogers are using ("taking") the occurrences (token) of the N-grams (type), hence this is an example of a Type/Token-Taken (TTT) informetrics.

Generalizing the above example : every user seeking information from a database, using a keyword, will use more a "popular" (i.e. heavily used) keyword than a less popular one and these popular keywords give rise to more hits in the database.

Three-dimensional Lotkaian informetrics

169

Averaged over the users, the number of hits per search will be longer (again this will be proved in the sequel) than the average number of documents containing this keyword (and hence, of course, the distributions from which these averages are derived are also different). This is, hence, another example of TTT informetrics. Some books in a library are borrowed heavily but many books are seldom borrowed. But a library user will seek these popular books more frequently than the other ones. So, on the average, the user will encounter more borrowed cases than the average number of borrowings per book. Also this is an example of TTT informetrics.

It is clear that the reader can add several other examples of the difference between TT informetrics and TTT informetrics. Surprisingly, before Egghe (1992, 2003), TTT informetrics has never been studied.

TTT informetrics studies the laws of informetrics as experienced by a user and in this sense could be considered to be more important than TT informetrics.

TTT informetrics can be considered as part of linear three-dimensional informetrics but where we only use one IPP in the composition in Fig. III.5 (therefore we will drop the indices): if (S,I) is the given TT IPP we will, for TTT theory, consider the IPP (S*,I*) where S* = I* = I: TT items are considered as TTT sources and the TTT items are just the TT items. While this might look complicated, it is easier to express TTT theory (versus TT theory) in terms of the size-frequency function.

Let f denote the TT size-frequency function of the given IPP (S,I). Then, denoting by f* the TTT size-frequency function, we have the following simple basic defining formula for the relation between TT theory and TTT theory:

f ( j ) = jf(j).

(III.ll)

j £ [l,pm] (P m is the maximal item density in a source in the given IPP (S,I) as well as in the IPP (S*,I*)). Formula (III.ll) is explained by the fact that the item set I becomes the TTT source set S*: f*(j) denotes the density of the tokens in the types with token-density j .

170

Power laws in the information production process: Lotkaian informetrics

Intuition: In the discrete setting this would mean (replacing j by n € N): the fraction of the items that belong to sources with n items.

If we denote S = [0,T], I = [0, A], it follows from formulae (11.20) and (11.21) that

f"

f(j)dj = T

(111.12)

f"jf(j)dj = A

(111.13)

O1

and from (11.22) that

A j>Q)dJ

\^ = ~ =hFi

T

Fw

.

(HI.14)

the average number of items per source in the given (TT) IPP (S,I). By the same arguments we have that the average number of items per source in the TTT IPP is given by u*, where

.

Pj2f(j)dj

Pjf(j)dj

n =4?——=4; j

( IIL15 )

j;-jf(j)dj

Let us denote by P and respectively P the size-frequency distributions derivable from the size-frequency functions f and f*, respectively: for j 6 [l,pm]

P(j)=

r

/

( j )

f f(k)dk *J 1

(HI.16)

Three-dimensional Lotkaian informetrics

r

^l^~- =7 ^ J ] f (k)dk J ] kf(k)dk

171

(OUT)

We have the following result, valid in general TTT theory.

Theorem IH.1.3.2.1 (Egghe (2003): n* > [i in all cases.

Proof: We express that the variance of the distribution P (equation (III.16)) is positive: from (III. 14) and (III. 16) we have:

o<J^"(j-n)2p(j)dj

= fmj2P(j)dj-2^r°jP(j)dj + n2 1

\J 1

= / P "j 2 P(j)dj-H 2

=rmJ)dJ-(j;>(j)df So

f-fpWdj^ff-jPO)^)2 hence

fmJ2P(j)dj r , /N ^h > I JP j dj

(111.18)

172

Power laws in the information production process: Lotkaian informetrics

Putting (III. 16) in (III. 18) yields:

JP"j2fQ)dJ

/P"jf(j)dj

^ifOjdj-^fOjdj hence yi' > [i by (III. 14) and (III. 15).

The result of Theorem III. 1.3.2.1 can be evaluated in informetrics terms as follows (cf. the examples given): the average number of items per source, as observed by the user, is larger than the real existing average number of items per source. Following the given examples, the cataloger will find larger screen lengths (on average) than the average number in the database (a negative constatation). The library user, seeking information from a database, using a keyword will find more hits than could be expected from the database itself (according to the IR evaluation criteria "recall" and "precision" - see e.g. Salton and Me Gill (1987) - this could be interpreted as a positive resp. negative constatation). The other example obviously gives a negative interpretation: the user will encounter more borrowed books (hence unavailable) than could be expected!

III.1.4 General notes

It is clear that we leave open the study of four-, five-, ... dimensional informetrics although the higher dimensional linear informetrics theories follow from making further compositions of the type that was studied in "positive reinforcement" (Subsection III. 1.3.1). An example is given by research institutes that have researchers that produce articles which produce citations (or references).

A different (but related) type of problem is the one of the determination of total influence between any two objects when one only knows the direct influence values between any two objects. An application in our field is given by the citing relation between two articles. Two articles (say A, and A 2 ) can be directly linked if A, cites A2 or vice-versa or can be indirectly linked if they do not cite each other but where there exists a "chain" of articles, starting

with

A,

and

ending

with

A2

(or

vice-versa)

of

the

form

Three-dimensional Lotkaian informetrics

173

A, —>Ak| —>Ak2 —>...—> AK —>A2 ( A k , i = l,...,n are articles from our article-set or bibliography under study and where —» means "influences" - in our example: "cites" (directly)). We also assume that the relation —> does not allow circuits, i.e. a chain of the form A —>... —> A (which is evident - in principle - in the case of "cites"). Let {A,,A 2 ,...,A N } denote our set of documents in which we want to determine the total influences based on the knowledge of the direct influences. For i, j 6 {1,...,N}, let a(j denote the (known) direct influence weight and c;j the (unknown) total influence weight. We define a(j = 0 if i has no direct influence on (does not cite) j and a^ = 1 if i has direct influence on (cites) j (but we can allow for general weights asj e[0,l], useable in other applications (or even to express the relative number of times article i is citing article j)).

Leti 5 j . Then as done in Rousseau (1987) we define c;j as the sum over all articles (all A k , k = 1,...,N) of the direct influence weight of Aj on Ak times the total influence of Ak on A. (see Fig. III.6).

Fig. III.6

Calculation of c(j.

Reprinted from Rousseau (1987), Fig.3, p. 220. Reproduced with kind permission of Kluwer Academic Publishers. In the form of a formula, this is (for i * j):

c^ZXs

(HI.19)

k=l

For mathematical purposes we define &~ = 0 and Cj. = 1 if i = j expressing a total influence of

174

Power laws in the information production process: Lotkaian informetrics

A, on itself of 1 but no direct influence of A; on itself (e.g. Ai does not cite itself!). This can be used in (III. 19) for aik if i = k and for ckj if k = j . Of course it also defines c^ if i = j . By (III. 19), we have now, for all i, j G {1,...,N}:

c .J =

E a * c «+ 5 , J

( IIL2 °)

k=l

where 5^ denotes the Rronecker delta (5^ = 1 if i = j and 5 ^ = 0 if i * j). Indeed, if i =*= j , formula (111.20) reduces to (III. 19). If i = j , then, if k ^ i , then aik = 0 if Aj does not influence Ak directly and if A; does influence A^ directly (so that aik ^ 0) we have that Ak does not influence Ai directly nor indirectly (otherwise we had a circuit, which we excluded), hence cki = 0. So

1X^ = 0 k=l k*i

but for k = i we assumed aki = 0 so that

1X^ = 0 k=l

and hence (III.20) reduces to c^ = 5^ = 1, which is in agreement with our assumption.

If we now denote C = (C;JJ, A = faijj, I = l8jjJ, the matrices of the c(j, afj and 8^-values respectively, (111.20) is nothing else than

C = A.C + I

or

(I-A).C = I

(111.21)

Three-dimensional Lotkaian informetrics

175

Hence (III.20) (or (111.21)) has a solution for C (i.e. for all total influence values c^) if the matrix I - A is invertible (this is called the Gozinto theorem - said to be developed by the Italian mathematician Zepartzat Gozinto) and we have C = (I-A)"'

(111.22)

This is one way of calculating total influences from direct influences. For more on this we refer the reader to Rousseau (1987).

We refer the reader to Egghe and Rouseau (2002, 2003a, 2004) and Egghe (2002b) for more graph-theoretic approaches of citation analysis and other hierarchical issues, which do not really belong to Lotkaian informetrics (at least as far as the author is aware).

This ends the report on the three-dimensional results that were obtained in general informetrics. We will now further develop these results in Lotkaian informetrics. As explained above we only have results on linear three-dimensional informetrics.

III.2

LINEAR THREE-DIMENSIONAL LOTKAIAN INFORMETRICS

III.2.1 Positive reinforcement in Lotkaian informetrics

The main result on positive reinforcement in general informetrics being already proved (Theorem II.1.3.1.3), we have the following easy consequence of this.

Corollary III.2.1.1 (Egghe (2004e)): If the first IPP is Lotkaian (i.e. if f, is a power law) and if (p is a power law then the positively reinforced IPP is also Lotkaian.

Proof: Let

cp(x) = Bx|i

(111.23)

176

Power laws in the information production process: Lotkaian informetrics i

1

P

(p>0) (hence (p- (j) = U] ) and

f

i0,) = -^

(m.24)

Ji

(a, > 0). Then, by (III.7)

I

\i) —

a

FT

B

[B.

f(J) = 7 B ^ ^ P

j

(111.25) 3

which is also a power function, hence a Lotka law with exponent

a =^P-

(IIL26)

Examples III.2.1.2: (i)

If P = 1, i.e. cp is linear, then a = a, as could be expected.

(ii)

For p = 2, we have that a = —

(iii)

If p > 1 then a < a, iff a, > 1 (which is so in most cases). This result will have

.

consequences for the concentration (inequality) theory of the reinforced IPP (versus the first IPP) (see Chapter IV). In the case of Lotkaian informetrics, we are able to present concrete relations between the average number |^ of items per source in TT theory versus the average number \x" of items per source in TTT theory. This will be done in the next subsection.

Three-dimensional Lotkaian informetrics 111 III.2.2 Lotkaian Type/Token-Taken informetrics

A direct, simple, relation between \i and \x* is obtained in the limiting case pm = +00. We have the following result.

Theorem HI.2.2.1 (Egghe (2004b)): Suppose that A < 00, where A denotes the total number of items in the TT IPP. Suppose that the TT size-frequency function f is Lotkaian:

f(j) = - |

(HI.27)

where j e [l,+oo[ (i.e. pm = 00). Then u.* = +00 iff a < 3 . If a > 3 we have

H P =

T 1 = 2T-A 2-n

(111.28) V '

where T(< A < 00) denotes the total number of sources in the TT IPP.

Proof:

It is easy to see that

rj2f(j)dj=~

if a < 3 and that rj 2 f(j)dj = - ^ a —3

(111.29)

Jl

if a > 3 .

Hence, if a < 3 and since A = /

jf (j)dj < 00 (this boils down to a > 2 - see Proposition

178

Power laws in the information production process: Lotkaian informetrics

11.2.1.1.1) we have, by (III. 15), that n* = oo. If a > 3 we have by (III. 15) and (111.29)

Using Proposition II.2.1.1.1 (existence theorem in case pm = oo) and formulae (11.28), (11.29) we can conclude:

T u. =

1 =

2T-A

2-n

and T and A are finite since a > 3 > 2.

Note IH.2.2.2: Note that (111.28) implies \i > \i, a fact that has been generally proved in Theorem III. 1.3.2.1. Note also that the relation (111.28) is limited to \i € ]l,2[, since A > T (hence n > 1) and since a > 3 implies, by (11.30), that \i < 2 . We hence have the functional relationship of p.* in function of \i as depicted in Fig. III.7. Note that the graph is tangent to the straight line y = x which visualizes the fact that \x* > \i. for \i > 1.

Fig. III.7

The function \i (JJ.) .

Three-dimensional Lotkaian informetrics 179

Now we will treat the general case. Here no direct formula between \x and ji* exists but we are able to prove a formula for n* in function of a parameter x which is the solution of an equation (in which (x appears) which is also given here and solved numerically. So the relation |x* (|i) can always be calculated.

We suppose (111.27) to be valid for j e [ l , p m ] ; see also Subsection II.2.1.2 for its existence. From this theory it followed that: from (11.20), (11.21), (11.34) and (11.35): if a > 1

T = J| P "'f( J )dj = T ^ - ( p ^ - l )

(111.30)

A=j;p"jf(j)dj=^-(Pr-i)

(HI.31)

and if a ^ 2

If a = 2 we have (see (11.36))

A = J ^ j f ( j ) d j = Clnp m

(111.32)

From this it followed that (cf. (11.37)) if a ^ 2

^ = A = iz£L i £ z l T

2 - a pL--l

and if a = 2 (cf. (11.38)):

Pm

(in.33)

180

Power laws in the information production process: Lotkaian informetrics

To calculate \x* we can argue: if a =*= 2, a * 3, then

J>U)dj =^(pL-a-l)

(IH.35)

jj P °j 2 f(j)dj = C ( P m - l )

(111.36)

j] P "j 2 f(j)dj = Clnp m

(111.37)

If a = 2 then

If a = 3 then

as is readily seen. Hence from (III. 15) it now follows that, if a =e 2, a ^ 3, using (111.31) and (111.35)

»

=

^

^

(in.38)

If a = 2 we use (111.32) and (111.36) to obtain

[i = ^ ^ -

(111.39)

If a = 3 we use (III.31) and (111.37) to obtain

u* = - ^ i j t_ Pm

(111.40)

Since neither (III.33) nor (111.34) can be solved exactly for pm and since p m appears in the formulae for ja* it is clear that a direct relation between n and \i is not derivable. But we

Three-dimensional Lotkaian informetrics

181

can solve (111.33) and (III.34) numerically which will yield indirect but concrete relations between \x and \x*. This will be done at the end of this subsection but first we prove an "exact" result.

Theorem III.2.2.3 (Egghe (2003)): For each a fixed we have that \x* is an increasing function of \i.

Proof: Denoting x = p m > 1 in (111.33) and (111.38) we have the following equations:

=

2^a 3-a

x^a-l x2-a-l

(a^2,a^3) V > ;

'

(111.42) v )

Let 1 < a < 2

Then u, increasing implies — increasing since < 0. Hence, by (111.41) (x a - 2 a —2 x2~a-l x'-a-l

decreases. By Lemma III.2.2.4 (below) (since 1 — a and 2 — a have opposite sign) we have that x increases. Again by the same Lemma we have that

x3-a-l x2-a-l

increases (since 3 — a and 2 — a have the same sign), hence by (III.42), \x" increases.

182

Power laws in the information production process: Lotkaian informetrics

2
On the same lines as the above proof, using the same Lemma below and (111.41) and (111.42), we now have

„

1 <x-l .

x2-a-l

fj,

x

V-/^-

~ V » ,-a

a —2

—1

„

,

/'=>x/'=>

x3-a-l

2_a \=>n x —1

. „

/.

a>3

Now

„

x2"a-l

1 a-1 .

A

x

x3"a-l

„

, ,

a=2

From (111.34) and (111.39) we have

V = ^T 1-x

(HI.43)

H*=^i lnx

(111.44)

- t - = J^—

= >0

since In x < x — 1 always (since ex > x + 1 , for all x > 1). Also

Three-dimensional Lotkaian informetrics

(lnx) 2

dx

since Inx > 1

183

. This follows from the fact that the function 9(x) = Inx -\ x

1 is minimal x

in x = 1 and 9 (l) = 0. Hence \i /"=> \x /*.

a =3

From (111.33) and (111.40) we have H= 2 ^ -

(111.45)

x+1

V=-^r1-x

(HI.46)

Obviously n strictly increases in x and by the above (j,* strictly increases in x. So

n/^nVLemma HI.2.2.4: The function

f(x) = ^ f ^

(111.47)

increases if a and a + 1 have the same sign and decreases if a and a + 1 have opposite sign.

Proof:

, , x a -'(x a + 1 -ax-x + a) f x = ^ z >[ }

(* a "l) 2

184

Power laws in the information production process: Lotkaian informetrics

So f'(x) has the same sign as

cp(x) = xa+1 - a x —x + a.

Now

cp'(x) = (a + l ) x ' - ( a + l)

which is zero in x = 1. Furthermore

cp"(x) = a(a + l ) x a - \

so

cp"(l) = a(a + l ) .

So if a (a +1) < 0 then cp has a maximum in 1 and cp (l) = 0. So

0 then ep has a minimum in 1. So cp(x)>0 for all x and hence f increases.

As will become clear from the sequel (Table III. 1) we conjecture the following, but we are not able to find a proof for it.

Conjecture IH.2.2.5: For each fixed |a we have that \i is an increasing function of a .

Finally, we will calculate the TTT average |j* in function of the TT average \x. The calculation of (i* in function of \i and a requires the solution (for x) of the equation (cf. (111.41)), if a ^ 2

Three-dimensional Lotkaian informetrics

(n a - 2

(a

185

a-2

andof(cf. (111.43)), if a = 2

lnx + ^ - p , = O. x

(111.49)

Once x is found we have then jx* using the formulae: if a ^ 2, a =* 3 (cf. (III.42))

'=i_a

3-a

x

J_

x 2 -"-l

if a = 2 (cf. (111.44))

H*=4-^

(HI.51)

lnx and if a = 3 (cf. (111.46))

V T ^ 1-x

(111.52)

In Egghe (2003a) we solved (111.48), (111.49) for x using the MATHCAD 4.0 software package. Note, however, that a and [i cannot be taken fully independent of each other: there is a lot of freedom but we have the restriction (11.32) of Theorem II.2.1.2.1, if a > 2 :

H < ^

(01.53)

a-2 In the next table (of \x' in function of ]x and a ) we hence have the following restrictions

186

Power laws in the information production process: Lotkaian informetrics

a = 2.5=>ji<3 a = 3=>n<2 a = 3.5 =>n< 1.6667

Table III. 1 Values of \x* in function of n and a . Reprinted from Egghe (2003), Table 1, p. 608. Copyright John Wiley & Sons Limited. Reproduced with permission.

JJ*

a = 1.5

a =2

a = 2.5

a =3

a = 3.5

H = 1.2

i- 2148

1.2151

1.2157

1.2164

1.2178

H = 1.5 ^=2

i- 5838 2.3317

1.5984

1.6177

1.6479

1.7010

2.4612

2.7320

-

-

3.2492

3.7279

5.8541

-

-

5.5996

-

-

8.3955

-

-

^ = 2.5 H= 3

^ = 3.5

4.3333 5.5894

Another way to produce (n,H*) -relations, that is simpler than the above method, is by inputting given values of x = pm > 1 and a > 1 into (111.48) (or rather (III.33)) or (111.49) (or (111.34)) for ]x and (111.50) or (III.51) or (III.52) for u*, so that no numerical solution as above is necessary. In this way we obtain Table III.2.

Table III.2 Values of n and |x* in function of x = pm and a .

M

a = 1.5

a =-2

a = 2.5

=3

a = 3.5

x=1.5

(1.225, 1.242) (1.216, 1.233) (1.208,1.225) (1.200, 1.216) (1.192, 1.208)

x=2

(1.414, 1.471) (1.386, 1.443) (1.359, 1.414) (1.333, 1.386) (1.309, 1.359)

x=3

(1.732, 1.911) (1.648, 1.820) (1.570,1.732) (1.500, 1.648) (1.438, 1.449)

x=4 x=5

(2,2.333)

(1.848, 2.164)

(1.714,2)

(1.600, 1.848) (1.505, 1.714)

(2.236, 2.745) (2.012,2.485) (1.821,2.236) (1.667, 2.012) (1.545, 1.821)

IV LOTKAIAN CONCENTRATION THEORY

IV. 1 INTRODUCTION Concentration theory studies the degree of inequality in a set of positive numbers. It is not surprising that the historic roots of concentration theory lie in econometrics where one (early in the twentieth century) felt the need to express degrees of income inequality in a social group, e.g. a country. Hereby one expresses the "gap" between richness and poverty. One of the first papers on this topic is Gini (1909) on who's measure we will report later.

The reader of this book will now easily understand that concentration theory takes an important role in informetrics as well. Indeed, as is clear from Chapter I, mformetrics deals with the inequality in IPPs, i.e. in production of the sources or, otherwise stated, the inequality between the number of items per source. As we have seen, Lotkaian informetrics expresses a large difference between these production numbers. Just to give the most obvious example: if we have Lotka's law (with exponent a = 2, just to fix the ideas): f(n) = — then n f (2) = ——, f (3) = ——, fw(4) = —— and so on, where fv(n) denotes the number of sources w V; 9 16 ' 4 with n items. It is clear that, expressed per production class n, there is a large difference between the number f (n) of sources in these classes. Zipf s law is also a power law, hence it also expresses a large difference but now between the numbers g(r), r = 1,2,3,..., where g(r) denotes the number of items in the source on rank r (where sources are ranked in decreasing order of their productivity). It is clear that all examples of sources and items, given in Chapter I, can be the subject of a concentration study. The skewness of these examples was apparent and hence one should be able to measure it.

188

Power laws in the information production process: Lotkaian informetrics

Generalizing the above examples, we can say that we have a decreasing sequence of positive numbers x,,x 2 ,...,x N , N e N, and we want to describe the degree of inequality between these numbers, otherwise stated, the degree of concentration: a high concentration will be where one has a few very large numbers x,,x 2 ,... and many small numbers ...,x N _,, x N . It is clear that this must be formalized. We will use techniques developed in econometrics but we will also report on the "own" developments that have been executed in informetrics itself. Under the "own" developments we can count the so-called 80/20-rule and the law of Price. The main part of this chapter, however, will be the study of the Lorenz curve which was developed in econometrics around 1905 (cf. the historic reference Lorenz (1905)).

Let us briefly (and intuitively) describe these concepts here, before studying them more rigorously in the further sections. The simplest technique is the 80/20-rule which states that only 20% of the most productive sources produce 80% of all items. Of course, this is just a simplification of reality: it is the task for informetricians, in each case, to determine the real share of the items in the most productive sources: 20% of the most productive sources might produce 65% of all items but this could as well be 83.7%! Also, we do not have to consider 20% of the most productive sources: any percentage can be considered. So, generalizing, we can formulate the problem: for any x € ]0,l[ determine 0 e ]0,l[ such that 100x% of the most productive sources produce 1000% of all items. We can even ask to determine 6 as a function of x. This "generalized 80/20-rule" could be called the determination of "normalized" percentiles since both x and 0 belong to the interval [0,l] while in the calculation of percentiles, one of these numbers is replaced by actual quantities (of items or sources). Since both x and 0 denote fractions this technique is (sometimes) called an arithmetic way of calculating concentration (see Egghe and Rousseau (1990a)).

In this sense we can call the law of Price a geometric way of calculating concentration. The historic formulation (see De Solla Price (1971, 1976) and implicite in De Solla Price (1963)) i

states that, if there are T sources, the vT = T ^ most productive sources produce 50% (i.e. a fraction —) of all items. For evident reason, this principle is also called Price's square root law. It is clear how to extend this principle: let 9 6 ]0,l[, then the Te most productive sources produce a fraction 9 of all sources. This is called Price's law of concentration and we will

Lotkaian concentration theory

189

investigate in what cases in informetrics this is true. Also this principle could be generalized stating that for 8 E ]0,l[ the top TE sources produce a fraction 9 of all the items and we can ask for a relation between s and 0.

Both general formulations of the 80/20-rule (in terms of x and 0) and of the law of Price (in terms of e and 0) involve two numbers. We could wonder if we can construct a function F such that, for any decreasing vector X = (x,,x 2 ,...,x N ), with positive coordinates, the value F(x) = F(x,,...,x N ) is a good measure of the concentration in X. It is clear that an "absolute" good value for F(X) does not exist but we can determine requirements for the value of F(X) in comparison with values F(X') for other vextors X' as above, i.e. to give relative valuejudgements. Let us formulate some "natural" requirements.

(i)

F(X) should be maximal for the most concentrated situation, namely for a vector X of the type X = (x,0,.. .,0) where x > 0.

(ii)

F(X) should be minimal for the least concentrated situation, namely for a vector X of the type X = (x,x,.. .,x) where x > 0.

In terms of wealth or poverty, (i) states that X = (x,0,...,0) must have the highest concentration value (given F), since one source (e.g. person) has everything and the other sources have nothing. Condition (ii) states that if everybody has the same amount (e.g. money), the concentration value should be minimal (and preferably zero).

(iii)

F(X) should be equal to F(cX) where, for X = (x,,...,x N ), the vector cX is defined as (cx,,...,cx N ), forallc>0.

Condition (iii) is called the scale-invariant property and is requested since describing the concentration of income (i.e. describing wealth and poverty) should be independent on the used currency (€, $, Yen,...) which all are interrelated via a scale factor. The next property is also very important:

190

Power laws in the information production process: Lotkaian informetrics

(iv)

F(X') > F(X) if X' is constructed from X by "taking (in X) an amount away from a poor person and giving it to a rich person". In other words, we require F(X') > F(X)incase X = (x,,...,x N ) and X1 = (x,,...,x i +h,..., X j -h,...,x N )

(IV.l)

if 0 < h < x r Condition (iv) is called the transfer principle and was introduced in 1920 by Dalton - see Dalton (1920). It is clear that the transfer principle is a very natural requirement and in Egghe and Rousseau (1991) we showed that it comprises properties such as

(v)

"If the richest source gets richer, inequality must rise": for all X = (x,,...,x N ) (as always, ordered decreasingly) and h > 0 we have that, if X' = (xj +h,x 2 ,...,x N ), then f(X') > f(X). The same can be said if the "poorest source gets poorer",

(vi)

The principle of nominal increase: if, given a vector X = (x,,...,x N ), the production of each source is increased with the same amount h > 0, inequality is smaller: for X' = (x, +h,...,x N + h ) we have f(X!) < f(X).

We have, deliberately, used the econometric terminology, just to illustrate these principles but it is clear that these principles are universally required in any application of concentration theory.

A genius invention of Lorenz, however, requires the construction of a Lorenz curve (denoted L(X)) of a given decreasing vector X and these Lorenz curves have the property that any measure F satisfying L(X) < L(X')=>F(X) < F(X') automatically satisfies all the above principles. In words: any function F on vectors X = (x,,...,x N ), which agrees with the Lorenz order, is satisfying the above requirements for concentration measures and hence can be called a "good" concentration measure. The Lorenz curves have, in addition, the property that any generalized 80/20-rule can be seen on these curves, hence they comprise this aspect of concentration theory as well.

Lotkaian concentration theory

191

Therefore, in the next section we will develop Lorenz concentration theory for vectors X = (x,,...,x N ) as above. Then we will check for the existence of such good concentration measures and we will describe their properties. We are then at the challenge of developing Lotkaian concentration theory, i.e. to calculate Lorenz curves and good concentration measures if the vector X is of power type, i.e. X = (f(l),f(2),...,f(n mas ))

(IV.2)

where f is Lotka's size-frequency function (nmax is the maximal production of a source) or if

X = (g(l),g(2),...,g(T))

(IV.3)

where g is Zipf s rank-frequency function (T is the total number of sources). We will see that exact calculations of these discrete versions of the laws of Lotka and Zipf are, also in this application, not possible. As explained in Chapter I, also here we will encounter problems of evaluating discrete sums, which is not possible. The previous three chapters fully showed the power of the use of the continuous versions of the laws of Lotka and Zipf, i.e. the functions (11.27) and (11.89). But we face here the need of extending Lorenz concentration theory from discrete vectors X = (x,,...,x N ) to continuous functions h on an interval [l,x m ]: for h = f (Lotka's function) we have xm = p m and for h = g (Zipf s function) we have xm = T + 1 (see Chapter II). Even for a general (non-Zipfian) rank-frequency function g, defined on [0,T] (see II.8) we can apply such a theory on the interval [l,T + l] by replacing the re[0,T] by 1 + r e [l,T +1] as we did for Zipf s law. If we can extend the Lorenz concentration theory to functions h : [l,xm] —> M+ , we can apply it to the functions f of Lotka and g of Zipf in an identical way since both f and g are functions like h. This will be done from Section IV.3 on: based on our insights in the discrete case we will define the Lorenz curve L(h) of a general function h on an interval [l,xm] and determine its special form if h is a power law. Our approach will be simpler than (but equivalent with) earlier approaches e.g. of Atkinson (1970) and Gastwirth (1971, 1972). We will determine three important good concentration measures based on L(h), namely the Gini index, the coefficient of variation and Theil's measure and calculate their values for power laws h. These results can then be applied to the power laws of

192

Power laws in the information production process: Lotkaian informetrics

Lotka (f) and of Zipf (g) and we will determine the crucial role of the exponents a and P in this matter ( a in (11.27) and P in (11.89)). We will be able to present concrete formulae for the mentioned concentration measures in function of a and P, hereby proving concentration properties in function of a and p . This is the real heart of Lotkaian concentration theory. From this Lorenz concentration theory we will also determine the generalized 80/20-rule in Lotkaian informetrics and we will also show that Price's law of concentration is (exactly) valid if we have Zipf s law.

We close this introduction with an important remark: although, once Lorenz concentration theory is developed for general power functions h, we can apply it to f (Lotka) and g (Zipf) in a mathematically identic way, the interpretation of concentration theory on f is completely different from the one on g. The latter theory is the most important one since it describes the inequality among the sources (expressed by their ranks r) with respect to their item production. It is this application that is always studied, also in econometrics. But, as said, in a mathematical equivalent way, we can study concentration aspects of the function f, hereby calculating the inequality between the different source productivity levels j (as in (11.27)). This is a functionality that - as far as we are aware of- only occurs in informetrics: Zipf s law also occurs in linguistics, econometrics, physics,... but Lotka's law is a regularity that is only studied in our field. Because of the fact that it comprises Zipf s law and even Mandelbrot's law (Chapter II) and the fact that Lotka's law is a simple power function, we think it plays a central role in informetrics (hence the writing of this book!) and hence we think it is worthwhile to study also concentration properties of Lotka's function, in addition to the study of the concentration properties of Zipf s function.

IV.2 DISCRETE CONCENTRATION THEORY In order to describe the concentration (inequality) of a vector X = (x,,x 2 ,...,x N ) we have to introduce the Lorenz curve as a universal tool in the construction of a wide range of concentration measures. We will suppose that all xt > 0 (although extension to the case that some X; are negative is possible - we will not go into this since we do not need it in this book) and that X is decreasing (this can always be achieved: if X is not decreasing we can reorder it decreasingly).

Lotkaian concentration theory

193

The Lorenz curve of X denoted by L(X), is constructed by linearly connecting the points (0,0) and

^,£ a j

(IV.4)

i = 1,...,N, where

aj=-j£-

(IV.5)

Note that the last point (for i = N) is (1,1). Since X decreases we have that L(X) is a concave polygonal curve that increases from (0,0) to (1,1). Its form is depicted in Fig.IV.l.

Fig. IV. 1

General form of a discrete Lorenz curve.

The power of this genius construction lies in the fact it describes all aspects of concentration in one graph: the higher L(X), the more concentrated X is. Let us illustrate this. The vector of "no" concentration is X = (x,x,...,x) with x > 0 and L(X) is the straight line connecting (0,0) with (1,1), the lowest possible curve (since a Lorenz curve is concave). The vector of

194

Power laws in the information production process: Lotkaian informetrics

"highest" concentration is x = (x,0,...,0) with x > 0 and L(X) is the curve connecting (0,0) with —,1 and —,1 with (1,1), the highest possible curve. It is easy to see that if X' is

IN

J

INJ

constructed from X via an elementary transfer as in (IV. 1), we have that L(X') > L(X) (here > means > and not equal as curves). A theorem of Muirhead (see also Egghe and Rousseau (1991)) says that, conversely, L(X') > L(X) implies that X' can be derived from X via a finite number of elementary transfers (as in (IV.l)). Further, we trivially see that cX = (cx,,...,cxN) (c > 0) has the same Lorenz curve as X. So, the Lorenz curve is the right tool to describe concentration. It is now clear that any function C that respects the Lorenz order, i.e. which satisfies, for all vectors X, X'

L(X)
(IV.6)

can be called a good measure of concentration. We are now in a position to construct good concentration measures: we only have to satisfy (IV.6) and then we are guaranteed of all the properties described in (i)-(vi) in Section IV. 1. Do such good concentration measures exist? It turns out that there are many (see Egghe and Rousseau (1990c, 1991) and references therein). One evident good measure is the area under the Lorenz curve or, in its normalized form: the Gini index G (Gini (1909))

G(X) = 2{area under L ( x ) } - 1

G(X) = 2 f ' L ( X ) ( y ) d y - l fj

(IV.7)

0

The next measures are based on a result of Hardy, Littlewood and Polya (1928, 1952) that states (in our teminology) that L(X) < L(X') implies that

f>( a i )(a;) i=1

i=l

(IV.8)

Lotkaian concentration theory

195

for all convex continuous functions (p. Here X = (x,,...,x N ), X ' = (xj,...,x N ) and (a,,...,a N ) and (aj,...,a N ) are defined as in (IV.3). From this result, applied to cp(x) = x 2 we have that

V 2 (X) = N ^ a f - l

(IV.9)

is a good concentration measure. Note that (IV.9) is equivalent with (by (IV.5)):

V2(X) = N £

-^-T-1 j=l

V

' u2 N t f '

K

V 2 (X) = - ^

(IV. 10)

being the quotient of the variance and the square of the average of X. For this reason one calls V = V(X) the variation coefficient and, because of the above, it is a good concentration measure. If we take cp(x) = xlnx we find Theil's measure (Theil (1967))

N

Th(X) = lnN + ^ a i l n a i

(IV. 11)

i-i

For other good concentration measures we refer the reader to Egghe and Rousseau (1990a, 1990c, 1991).

So far for general discrete Lorenz concentration theory. Since we want to have a Lotkaian concentration theory, we are at a point to interpret L(X), G(X), V2(X) and Th(X) for X as in (IV.2) or (IV.3). This is not easy for two reasons. First of all the linear parts in L(X) destroy the power law and make calculations on L(X) (e.g. for G(X)) difficult. Secondly, as already remarked in Chapter I, evaluating discrete sums as in (IV.9) and (IV. 11), ending up with

196

Power laws in the information production process: Lotkaian informetrics

analytic formulae, is not possible. Using continuous functions is a solution to both problems: if we can define a Lorenz curve for a continuous function, we may fully work with this function and we do not have to introduce linear parts and also, finite sums are replaced by integrals which can be evaluated. This will be worked out from the next section on.

In Egghe (1987b) we made an attempt to calculate G (or rather Pratt's measure

N

G

N —1 which is close to G for large N and which will not have a purpose in our continuous theory to follow) for some informetric functions including the ones of Zipf and Lotka. The results are, necessarily, approximative. Since we will obtain exact results in the continuous case, we will not go into these results here.

IV.3 CONTINUOUS CONCENTRATION THEORY IV.3.1 General theory

Let h:[l,xm]—->R+ denote any positive decreasing continuous function on the interval [l,x m ], where xm > 1. Based on the discrete case we define the Lorenz curve of h, denoted L(h) as given by the set of points, for x e [l,xm]

x _!

JLj

X

In other words, putting y = x

fh(x')dx'

r,J'

(IV.12)

n,-! J ^ h ^ d X 1

e [0,1] (hence x = y(xm - 1) + 1), the Lorenz curve L(h) of m ~1

h is the function

r(x-1)+1h(X')dx' L(h)(y)=JY. J h(x')dx'

(IV.13)

Lotkaian concentration theory

197

This approach on defining the Lorenz curve is direct (on h) and similar to the definition of L(X) for discrete vectors X. It is different (in its formulation) than the definitions in Atkinson (1970) and Gastwirth (1971, 1972) in econometrics but all definitions are mathematically the same. In addition, our approach allows for finite xm , which is not the case in the other approaches.

It follows that

L h

( )'(y)=

r

*mr!

I

h(y(x m -i)+i)

(iv.i4)

hfx'jdx'

and that L(h)"(y)= j X m " l ) 2 h'(y(x m -l) + l) f h(x')dx'

(IV.15)

J1

hence L(h) is a concavely increasing function from (0,0) to (1,1) (since h > 0, h' < 0). Its general form is depicted in Fig.IV.2. Since L(h)' is continuous we have that L(h) is a C1 (i.e. a smooth) function (see e.g. Apostol (1957) or Protter and Money (1977)).

Fig. IV.2

General form of a continuous Lorenz curve.

198

Power laws in the information production process: Lotkaian informetrics

As in the discrete case such a Lorenz curve can be used to measure the concentration in the (continuous) set of values h(x) for x e [ l , x m ] . Based on the discrete results, we can also define C to be a good measure of (continuous) concentration if we have the following implication for all functions h and h' as above:

L(h)C(h)
(IV.16)

(L(h) < L(h') means: L(h) < L(h') and L(h) ^ L(h') as functions).

It is now (as in the previous section) the task to find such good measures. In Egghe (2002a), based on an inequality in Hardy, Littlewood and Polya (1928) we proved that any measure of the type

C(h) = J o 1 (p(L(h)'(y))dy

(IV.17)

, where cp is a continuous convex function, satisfies (IV.16) (since L(h) is a C1 function). This result is the continuous extension of (IV.8) and yields the continuous extension of (IV.9) and (IV.11): we define, for h as above:

V2(h) = J o 1 [L(h)f(y)dy-l

(IV.18)

Th(h) = / o 'L(h)'(y)ln(L(h)'( y ))dy

(IV. 19)

Of course (IV.16) is also satisfied for the Gini index

G(h) = 2("L(h)(y)dy-l 0

being twice the area under the Lorenz curve of h, minus 1.

(IV.20)

Lotkaian concentration theory

199

Having (IV. 13) at our disposition, we can also give answers to the general 80/20-rule: determine, given xe]0,l[, the number 0e]O,l[ such that the 100x% of most productive sources produce 1009% of the items. Indeed, it suffices to take x = y and 0= L(h)(y) as in (IV. 13). Hence for all values x = y e ]0,l[, the generalized 80/20-rule is given by the value L(h)(y), read on the graph of the Lorenz curve of h. This will be applied on power law types so that the 80/20-rule for power laws follows from its Lorenz concentration theory, to be developed in the next subsection.

IV.3.2 Lotkaian continuous concentration theory

IV.3.2.1 Lorenz curves for power laws

It is the purpose to apply continuous concentration theory to the functions of Lotka (11.84):

f(j) = £

(IV-21)

g(r) = |

(IV.22)

a > 0 , je[l,p r a ] and of Zipf (11.89)

r £ [l,T +1]. Both are power laws and since T —> oo if and only if pm —> oo in case (IV.22) applies (see Note II.4.2.2 (1)), we have that both functions f and g are defined on a bounded interval (Tp m
h(x) = 4 X

xe[l,x m ], x m >l, K>0, y>0. By (IV. 13) we have that

(IV.23)

200

Power laws in the information production process: Lotkaian informetrics

L(h)(y)=(y(x-"!!+!r"1

av-24)

if y * 1 and

L(h)(y)=h'y(;°-1) mxm

+ 1

'

(IV.25)

if y = 1. In the previous subsection we have not dealt with the case xra = oo. Since we have the application on the functions of Lotka and Zipf in mind, where pm = oo and T = oo are also studied (cf. the existence theory for the function of Lotka - Subsection II.2.1) we have to include the case xm = oo. We define the Lorenz curve L(h) for h as in (IV.23) with xm = oo as the limit of expression (IV.24) for xm —»oo

lim(y(xm,-L>+i)--i

C-i It turns out that this limit is always equal to 1, except if y < 1. In this case we have

lim L(h)(y) = y'- r

(IV.26)

In the case of Lotka's law we hence can use (IV.26) only in the case a < 1 (implying T = A = oo, by (11.20) and (11.21) and the fact that here p m = xm —> oo). In the case of the law of Zipf we can use (IV.26) only in the case |3 < 1, hence for a > 2 since

P= ^ 7 a-1

(IV.27)

as was proved in Subsection II.4.2. We can immediately apply (IV.26) to the Zipf function to prove the generalized 80/20-rule in Lotkaian informetrics.

Lotkaian concentration theory

201

Theorem IV.3.2.1.1: Let p < 1, hence a > 2 . Let \s. denote the average number of items per source. Then

\_ limL(g)(y) = y''

(IV.28)

where g is the Zipf function (IV.22). In words: a fraction y e [0,l] of the top sources produces a fraction y11 of the items, i.e. the generalized 80/20-rule for power laws.

Proof: In formula (IV.26), for each finite T, y equals

X

l

=—-

(by (IV.22)), the

fraction (in [l,T + l]) of the top sources (since g decreases). By definition, L(g)(y) is the fraction of items produced by these sources. So L(g)(y) represents the generalized 80/20-rule. But

1-Y = 1-P

(for g)

_a-2 ~ a-1 by (IV.27). Invoke (11.30):

a-1

in the limiting case p m —> oo. But by Note II.4.2.2 (1): pm —> oo iff T —> oo which is the case here since we take lim . This proves (IV.28).

This result is not new: it appears implicitely in Gastwirth (1972) (p. 307, Table 1), we think for the first time. It can also be read (implicitely) in Burrell (1992b), p. 22. See also Egghe (1986) for a discrete version of this result in case a = 2 and Gupta (1989) for an application

202

Power laws in the information production process: Lotkaian informetrics

of it. Another reference on the use of Lorenz curves in the study of the generalized 80/20-rule (but not in Lotkaian informetrics) is Burrell (1985).

The following corollary to Theorem IV.3.2.1.1 can be read in Egghe (1993a).

Corollary IV.3.2.1.2 (Egghe (1993a)): Under the assumptions of Theorem IV.3.2.1.1, if we have two such situations with p., <\i2, then L(g,)(y,) = L(g 2 )(y 2 ) implies yl > y 2 . In words ; if both IPPs satisfy Zipf s law as in the theorem and if the average production in the first IPP is smaller than in the second one then, in order to have the same fraction of items, we need a larger fraction of top sources (which produce these items) in the first IPP than in the second one.

Proof: n,<|a 2 implies — > — . Further, by (IV.28), L(g 1 )(y,) = L(g 2 )(y 2 ) implies

whence y, > y2 (since y,,y 2 <1).

Table IV.1 shows the values of y in function of \i for L(g)(y) = 0.8 (i.e. a 80/100y-rule), based on (IV.28).

This table also illustrates Corollary IV.3.2.1.2. This corollary is also true in some nonLotkaian informetric theories as shown in Egghe (1993a) but in this paper also examples of non-Lotkaian informetrics theories are given where Corollary IV.3.2.1.2 is false.

Corollary IV.3.2.1.2 implies for instance that, in a library, the higher the average number of borrowings per book the smaller fraction of the (most popular) books is needed in order to have a fixed (say 80%) fraction of all borrowings, hence the larger fraction of the least popular books that yield the other fraction (say 20%) of all borrowings implying that in libraries with a high average number of borrowings per book (e.g. public libraries in

Lotkaian concentration theory

203

comparison with scientific libraries) one can weed more of the low-popular books, hereby only loosing a (low) fixed percentage of all borrowings.

Table IV.l Values of y in function of (I for 80% of the items. Reprinted from Egghe (1993a), Table 3, p. 373. Copyright John Wiley & Sons Limited. Reproduced with permission.

V-

y

1 2 3 4 5 6 7 8 9 10

0.80 0.64 0.51 0.41 0.33 0.26 0.21 0.17 0.13 0.11

Let us now go back to the general case xra < oo or xm = oo and prove the following theorem on Lorenz curves for power functions.

Theorem IV.3.2.1.3 (Egghe (2004c)): The Lorenz curve L(h) for a general power law as in (IV.23) is strictly increasing in y.

Proof; Since we keep xm fixed here it suffices to show (by (IV. 12)) that y, < y2 implies that

r x d x ' rx» dx'

Jl

X'V|

Ji

X'T2

r x d x ' rx» dx'

J]

X'T2

J,

x 'Ti

for all x € [l,x m ]. This is equivalent with

r x d x ' rx» dx'

pdx'

r x »dx'

204

Power laws in the information production process: Lotkaian informetrics

For this it suffices to show that, for all x' e [l,x] and all y' e [x,xm]

J x 'Tl

1_ y 'T2

_1 XTT2

l__ y-T,

or

Since y2 > y, we can denote y2 = y, + s where 8 > 0. In this notation, inequality (34) reads

y)

which is trivial since x' < y' always.

Important remark IV.3.2.1.4: It is clear that the above theorem applies to Lotka's law (IV.21) and to Zipf s law (IV.22). So we have that L(f) is strictly increasing in a (i.e. the inequality increases with a ) : hereby one examines the inequality in the numbers f (j),je [l,p m ]. As mentioned before: much more important, from an mformetric point of view (but mathematically the same) is the study of the inequality in the numbers g(r),r€ [l,T + l] since these numbers express the sourceproductivity. So, Theorem IV.3.2.1.3 gives that L(g) is strictly increasing in p , hence the inequality increases with (3. The latter result can be interpreted in function of a by using (IV.27). So we have the result that L(g) decreases in a . Since this is a source of confusion (see Wagner-Dobler and Berg (1995), Rao (1988), Yoshikane and Kageura (2004) (see (*) p. 438)) we state these results (in a ) explicitely as a corollary.

Corollary IV.3.2.1.5: Let f and g be as above. Then

Lotkaian concentration theory (i)

L(f) strictly increases in a

(ii)

L(g) strictly increases in p

(iii)

L(g) strictly decreases in a

205

It is clear from Rao (1988) that Rao refers to expression (i), so the criticism in Wagner-Dobler and Berg (1995) is not in order, especially since they discuss the inequality in Lotka's law (hence (i)) and not the one in Zipf s law (being (ii)). In the same way, Yoshikane and Kageura (2004) confuse (i) and (iii).

This dual interpretation of inequality is typical in informetrics: one can consider the sizefrequency function f but also the rank-frequency function g, hereby interchanging the role of sources and items. In this sense Corollary IV.3.2.1.5 is not surprising. We will experience the same differences when we will express the measures G, V2 and Th for g in terms of (3 and a .

IV.3.2.2 Concentration measures for power laws In this subsection we will calculate the measures G, V2 and Th for general power laws h on [l,x m ], xm < oo and xm = oo and then we will interpret the results to the functions of Lotka and Zipf.

IV.3.2.2.1 Calculation of the Gini index The easiest measure to calculate is the Gini index G (formula (IV.20)) being twice the area under the Lorenz curve, minus 1. Let y * 1 and xm < oo. Then (IV.24) implies for the Gini index of h, denoted G(h),

G(h) = 2/ o 1 L(h)(y)dy-l

G h

( )=^b/o'((y( x m - 1 )+ i r- l )dy-i

G(h) = ^ ^ [ ^ ( x - - l ) - ( x m - l ) ] - l

(IV.29)

206

Power laws in the information production process: Lotkaian informetrics

The limiting value for xm —> oo is

lim G(h) = —

1 = —5L-

(IV.30)

but here 7 is restricted to 7 < 1. This is in accordance with the calculation of G(h) using (IV.26) which is also restricted to 7 < 1 and which also yields (IV.30), as can readily be checked.

So, for Lotka's function f we have, if pm < 00 :

G(f) =

' (p a -i)(pL--i)[^ p ^- 1 )- ( p -- 1 ) ]- 1

(IV31)

and for Zipf s function g we have (if T < 00 )

G(g) in function of a yields (using (IV.27))

G

(g) = -7

2

-l^—^ 7 ^ f ( T + 1 ) ^ - l "T "I

(IV.33)

T(T + l p - l H 2 a - 3 ^

For the limiting values we have for g, based on (IV.30):

limG(g) = —2— T^OC

w

(IV.34)

2-p

= —-— 2a-3

(IV.35) V

;

Lotkaian concentration theory

207

((3<1, equivalently, a > 2 ) . Result (IV.35) was already proved in Burrell (1992b), where only this limiting case (and a > 2) is considered. These results are in agreement with Corollary IV.3.2.1.5, as they should be. We leave it to the reader to calculate G(h) for y = 1, using formula (IV.25).

We refer to Egghe (1987b) for some discrete calculations of Pratt's measure (Pratt (1977)) which is essentially the same as the Gini index (see Carpenter (1979)).

IV.3.2.2.2 Calculation of the variation coefficient We present two methods: one based on (IV. 18) for general h and one based on the formula V = —, which only yields V for Zipf s function g.

1. First method By (IV. 18)

V2(h) = / o 1 [L(h)'f(y)dy-l,

where L(h) is given by (IV.24) (y * 1) and (IV.25) (y = 1). If y * 1, we have

L(h)'(y) = , I T ( 1 1 7 ) | X " " , \ ) ,V K Y -i)(y(x m -i)+i)

(IV36)

If y = 1, we have

L(h) (y)

' >o(;k'- 1 ) + 1 )

Let now y *= 1, y *= - . Then, by (IV.36)

-

208

Power laws in the information production process: Lotkaian informetrics

P-yft*--1)2

v> ( h )= f

2dy-i

^^(Hi'k-r-i).,

}

(x--l) 2 (l-2Y)

If y = —, then we have V2(h)=^m~^lnX--l

(IV.39)

4(V^-l) If y = 1, then (IV.37) implies

(x - i f 2

Vm

/ 1 (IV.40) (inx ) x So we have for Lotka's function f, if p m < oo (only the case a v= — ,a ^ 1) is given) V (h)=

(l-ay(p.y-l) (p^-l)(l-2a)

and for Zipf s function g, if T < oo and (3 ^ — ,p * 1:

(i-p)2T((T+iy-2f>-i) 2

V (g)= l j

;

lV

^ '—I ((T+ l ) - - l ) 2 ( l - 2 P )

which takes the following form in a > 1, a ^ 2, a ^ 3

(IV.42)

Lotkaian concentration theory

(a-2f

209

T T 1

[( + F - l ]

V 2 V( g ) = / ( \ , } ' V( a - 1 A ( a - 3; {,

J--1 ,«=l f

(IV.43);

(T + l ) a - i - l

using (IV.27). Based on (IV.39) and (IV.40) the reader can give the analogous formulae for P = - , (3 = 1 (hence a = 3, a = 2).

2. Second method This method only applies to the calculation of V2(g) since we use the formula V2 (g) = — , where a 2 and (j. are the variance and average of the Zipf function g, which can be calculated f using the Lotka function — as the weight function (cf. formula (11.20)):

» =l

C~ f(j)

J^dj

(IV.44)

So

V 2 (g) = ^ - r P m j 2 f(j)dj-l

(IV.45)

where n = —, the average number of items per source. We have by (11.34) and (11.35), if a > 1, a ^ 2 that

210

Power laws in the information production process: Lotkaian informetrics

T=

I^ a ( p "" 1 )

(IV 46)

r - l )

(IV-47)

'

and that

A= ^ (

P

Also, if a * 3,

j;Pmj2f(J)dJ = XP"']grdj = ^ - ( p - - l )

(IV.48)

(IV.46), (IV.47) and (IV.48) in (IV.45) now gives

i8J

"(«-l)(a-3)-

v-(g)= ^ ^

2

(«-l)(a-3)

(p^-1)2

(pr-iKpr-i)^ 2 (pr_!)

and we have the task of proving that (IV.49) and (IV.43) are the same. Since we have Zipf s law, (11.91) implies that

,_a oc-1 Pm = - £ -

This and (11.34) or (IV.46) give

T+l=- ^ - =pr' a-1

(IV.50)

Lotkaian concentration theory

211

Formula (IV.50) is the link between (IV.49) and (IV.43). We leave it to the reader to consider the cases a = 2 and a = 3 .

3. Limiting case Forxm —> oo, formula (IV.38) gives, if y < —

lim V 2 (h) = ^ — l L - \ = jf— l-2y l-2y *»— V '

(IV.51)

This gives for V2(g) if p = y < — (implying a > 3):

limV 2 (g) = - ^ — = 1 T^OO W l-2p (a-l)(a-3)

(IV.52)

using again (IV.27). Note again that these formulae are in agreement with Corollary IV.3.2.1.5, as they should be. Formula (IV.51) also follows by direct calculation based on (IV.26) (here we also find the restriction y < — ).

IV.3.2.2.3 Calculation of Theil's measure We have to evaluate (IV. 19) with L(h)' as in (IV.36) (y *= 1) or as in (IV.37) (y = 1). We will limit ourselves to the general case y ^ 1, leaving the other calculation to the reader. The calculation is straightforward but tedious. We have

Th(h) = / o 1 L(h)'(y)ln(L(h)'(y))dy

J

° ( x ^ - l ) ( y ( x m - l ) + l)y

( x ^ - l ) ( y ( X i n - l ) + l)T]

212

Power laws in the information production process: Lotkaian informetrics

_(1- Y )(x m -1)

(1- Y )(x m -1) r'

^-1 +

dy

Jo (y(Xm _ l) + 1y

C-l

(i-r)(x - i ) j . ( y ( X m _ 1 ) +

irin|(y(Xm_1)+ir|dy

m

, »?-' 1 p^_ifa) r Vnnzdz x^-1

Jl

z

x^ T -lJi

'

using the substitution z = y (xm — l) + 1 .

Now we use the following integral results which can readily be checked r dz _ z'~T

J^"TT^ rhiz , _ J

7"

Z

1

lnz

1

~-^ T 7^T (y-l)2

We then obtain

mM-h[fiz^i|+^-0^-0

(IV,3)

The measure Th(h) for xm = oo (restricted to y < 1, see (IV.26)) can be obtained in two ways: using (IV.26) and noting that L(h)'(y) = (l — y)y"y and reperforming the above calculation of Th(h), or, more simply, take lim Th(h) in (IV.53). Both ways give the X m —toe

formula (for y < 1)

Lotkaian concentration theory

Th(h) = ln(l-y) + - ^ -

213

(IV.54)

For Zipf s function (IV.22) (the most important case) this gives (for (3 < 1)

Th(g) = ln(l-(3) + J - .

In terms of Lotka's exponent a we have B = {

(IV.55)

(hence a > 2): a-lj

Th(g) = l n f ^ ) + - L - . l,ot —1J

(IV.56)

a-2

Note also that Th(g) increases in p and decreases in a as predicted by Corollary IV.3.2.1.5.

Note: Theil's measure is (as the other measures G and V2) calculated on the Lorenz curve, hence on normalized data in terms of abscissa and ordinate (see Figs. 1 and 2). It is different from the classical formula for the entropy of a distribution. To see this difference, let us first look at the discrete case. Theil's measure of the vector X = (x,,...,x N ) is given by (IV. 11), where the up are given by (IV.5). In the same notation, the entropy of the vector X is defined as (use log = In)

N

H = H(X) = - J ] a i l n a i

(IV.57)

i=i

Hence we have the relation

Th(X) + H(X) = lnN

or

(IV.58)

214

Power laws in the information production process: Lotkaian informetrics Th(X) = - H ( X ) + lnN

(IV.59)

We notice two differences between Th and H : although they are linearly related, the relation is decreasing. In other words, H increases if and only if Th decreases. Now Th is a good measure of concentration (inequality), hence H is a good measure of dispersion (i.e. of equality), used e.g. in biology to measure diversity (see Rousseau and Van Hecke (1999)). Further |Th| ^ H due to the fact that Th is calculated on the normalized Lorenz curve while H is not.

In the same way Th and H are different in the continuous case. We calculated already Th above. In the same way Lafouge and Michel (2001) calculate H (in fact, our proof of the Th formula was based on their proof) via the formula (f = Lotka function)

H(f) = -Jj°°f(j)ln(f(j))dj

(IV.60)

H(f) = - l n ( a - l ) + - i - + l

(IV.61)

For a > 1, they find

(note that H(f) decreases with a , in agreement with Corollary IV.3.2.1.5 since —H(f) is a good concentration measure). The result, apparently, was first stated by Yablonsky (1980). For H (g), the same formula (IV.61) applies but with a replaced by (5 > 1 (hence 1 < a < 2).

IV.3.3

A characterization of Price's law of concentration in terms of Lotka's law and of Zipf s law

In Theorem IV.3.2.1.1 we expressed the arithmetic fraction of the sources in [l,T + l] by numbers of the form

boiling down to an arithmetic fraction in [0,T]. For a geometric

Lotkaian concentration theory

215

fraction (needed in the formulation of Price's law) we cannot use the interval [0,T] since T e ,6 € [0,l] has the minimal value 1 (for 0 = 0). This, once more, shows the value of the use of the interval [l,T + l]: here a geometric fraction is perfectly expressed by the form (T + l) , Q £ [0,1]. This explains the used intervals in the following theory on Price's law.

The following proposition is a continuous extension of a result in Egghe and Rousseau (1986).

Proposition IV.3.3.1: Denote by G(r), for r e [l,T +1], the cumulative number of items produced by the sources in the interval [l,r]. Then the following assertions are equivalent:

(0 G(r) = Blnr

(IV.62)

r 6 [l,T +1], where B is a constant,

(ii)

The law of Price is valid: for every 9e]0,l[ (hence also for [0,l]), the top (T +1) sources produce a fraction 0 of the items.

Proof: (i)=>(ii)

Since we have the top (T + l) (0 G ]0,l[) sources, their cumulative production is given by (definition of G and since (T + i f € [l,T +1]):

G((T + l) 9 ) = Bln((T + l) 9 )

G((T + l) 8 ) = eG(T + l)

216

Power laws in the information production process: Lotkaian informetrics

But G(T + l) denotes the total number of items, hence the top (T + l) sources produce a fraction 8 of the items.

(iiWi)

Let r e [l,T +1] be arbitrary. Hence there exists Q € [0,1] such that r = (T +1) 6 . By (ii) and the definition of G we have that

G ((T

+ l) 9 ) = eG(T + l)

for 0 e ]0,l[. This is also true for 6 = 1 and for 9 = 0 (since we work with source densities). But r = (T +1) implies lnr = 61n(T +1). Hence

G (Vr;) =

tar ,G(T + 1) ln(T + l) l '

G(r) = Blnr

G(T + 1) where B = —-. '-, a constant. ln(T + l)

Corollary IV.3.3.2 (Egghe (2004c)): We have the equivalence of the following statements

(i)

Price's law is valid

(ii)

Zipf s law (IV.22) is valid for (3 = 1.

(iii)

Lotka's law (IV.21) is valid for a = 2 and

Pm

=C

Proof: The equivalence of (ii) and (iii) was proved in Subsection II.4.2.

(IV.63)

Lotkaian concentration theory

217

As to the equivalence of (i) and (ii) it suffices, by Proposition IV.3.3.1, to show that Zipfs law for (3 = 1 is equivalent with G(r) = Blnr, r € [l,T + 1 where B is a constant. In other words we have to show that

E

I \

r £ [0,T] is equivalent with

G(r) = Bln(l + r)

r £ [0,T]. This follows from formula (II.9) in Subsection II. 1.2 or from Theorem II.2.2.1 and formula (11.48).

We have the following (for historic reasons - see further) remarkable corollary:

Corollary IV.3.3.3 (Allison, Price, Griffith, Moravcsik and Stewart (1976) for Price's square root law and Egghe (2004c) for the general case): If a = 2 then Price's law is equivalent with Lotka's law for which the following relation is valid:

Pm

= C

(IV.64)

Proof: This follows readily from the previous corollary.

The result in Corollary IV.3.3.3 was found in Allison, Price, Griffith, Moravcsik and Stewart (1976) after a long approximative calculation (and limited to Price's law for 9 = — , i.e. Price's square root law; that the result is valid for the general law of Price was first proved, exactly, in Egghe (2004c)). This paper apparently (see editor's note) grew out of lengthy and frequently heated correspondence between these authors on the validity of Price's square root

218

Power laws in the information production process: Lotkaian informetrics

law in case of Lotka's law. We hereby show that a long debate on this issue is not necessary since even a more general result can be proved in an exact way.

Note also that the results in Proposition IV.3.3.1 and Corollary IV.3.3.2 are exact formulations of a "feeling" of De Solla Price (in De Solla Price (1963)) that Lotka's law, Price's law and Zipfs law (there called Pareto's law) describe "something like the approximate law of Fechner or Weber in experimental psychology, wherein the true measure of the response is taken not by the magnitude of the stimulus but by its logarithm; we must have equal intervals of effort corresponding to equal ratios of numbers of publications" (p.50).

We refer the reader to Glanzel and Schubert (1985) for a discrete characterization of Price's (square root, i.e. 0 = —) law. Further discrete calculations on Lotka's law in the connection of Price's law can be found in Egghe (1987a). Practical investigations on the validity of Price's law can be found in Berg and Wagner-Dobler (1996), Nicholls (1988) and Gupta, Sharma and Kumar (1998). We also give the reference Bensman (1982) for general examples of concentration in citation analysis and in library use.

IV.4

CONCENTRATION

THEORY

OF

LINEAR

THREE-

DIMENSIONAL INFORMETRICS Three-dimensional informetrics was described in Chapter III, which was almost entirely devoted to linear three-dimensional informetrics. The composition of the two IPPs was proved to be a situation of positive reinforcement i.e. where the rank-frequency function of the first IPP, denoted by g,, was "reinforced" as g = cp°g, with 9 a strictly increasing function such that cp(x) > x and where g is the rank-frequency function of the composed IPP. In positive reinforcement we hence keep the source set and we reinforce the production of these sources. Also in Chapter III, one studies Type/Token-Taken (TTT) informetrics where the use (in the second IPP) of the items (in the first IPP) was studied. Both cases yield a new IPP of which we have proved informetric properties in Chapter III.

Lotkaian concentration theory

219

It is, hence, natural to investigate the concentration properties of the positively reinforced IPP as well as the concentration properties of TTT informetrics. This will be the topic of this section.

IV.4.1 The concentration of positively reinforced IPPs

The reader is requested to re-read Subsection III. 1.3.1. To put it simply: the rank-frequency function g[ of the first IPP is composed with a strictly increasing function cp given by (III.5) or (III.6). Let us denote by L(g,) the Lorenz curve of gj and by L(g) = L ^ g ^ the Lorenz curve of g =
Theorem IV.4.1.1 (Fellman (1976)): (i)

L (cp°g,) > L (g,) if — ^ is increasing

(ii)

L ((p°g,) = L (g,) if ^-L

is constant

X

(iii)

L (cpogl) < L (g,) if W

is decreasing

Strict inequalities apply in (i) and (iii) when (ii) is not the case.

Proof: Note that Ti equals the total number of sources in the first as well as in the composed IPP (only the productivity is changed, by cp). By (IV. 13) we have:

L(cp°gl)(y) = V ,

and

^ '

dV.65)

220

Power laws in the information production process: Lotkaian informetrics F T l + 1 I <\A ' I g,(x)dx L(g,)(y ) = J ' T l + '

J,

(IV

-66)

g,(^')dx'

for y e [0,l]. Note that here, as explained in section IV.1, we use g, as a rank-frequency function on [l,T, + l ] (replace r by r + 1) in order to be able to apply the general continuous Lorenz theory, the Lorenz curve being defined as in (IV. 13) for a general positive decreasing function h on an interval of the form [l,x m ]. Note that

f T ' + ' g l (x')dx' = A,

(IV.67)

<J 1

the total number of sources in the first IPP and that

(cpog,)(x')aV

I

= Jj T ' +1 g(x')dx' = A

(IV.68)

the total number of sources in the composed IPP. Also

JT +1 &( x ') dx ' = fg^y'frdy'

(IV.69)

and

/>yT,+l

J,

,

.

= / o y
(iv.7O)

Lotkaian concentration theory

221

(use the substitution x' = y'T, +1). Denote by \i, — —-, the average number of items per

source in the first IPP and by p. = —, the average number of items per source in the composed IPP. Then (IV.67)-(IV.7O) in (IV.65) and (IV.66) yield

L(cp°g1)(y) = i/ o y (p(g,(y')) d y'

(iv.7i)

L(g 1 )(y) = ^-/ o y g,(y')dy'

(IV.72)

Now

A(y) = L(
=

=

n

(
gi(y')dy,

r&M(
Jo

n

g,(y')

dv.73)

hj

Note that A(0) = A(l) = 0 and further

po&)(y)_jL| n gi(y) hj

A.(y)=ftM(

Suppose (i). Then, since that

^ ' increases, x

<

( IV .74)

, ' decreases since g. decreases. If we had g,(y)

222

Power laws in the information production process: Lotkaian informetrics

^

g

'/Y' < -t

81 (y)

(IV.75)

h

for all y e [0,l], then by (IV.73) and using continuity and positivity of the functions we would have that A ( l ) < 0 . If we had (IV.75) but with < replaced by > for all ye[O,l] then we would have A ( l ) > 0 . Since A(l) = O we hence have, by continuity of the function

', g,

the existence of a point y0 6 ]0,l[ such that

(
Since ^

(IV.76)

h

g ' ' \ y ' decreases we hence have, by (IV.76) and (IV.74), that A'(y) > 0 on ]0,yJ 81 (y)

and A ' ( y ) < 0 on ]y o ,l[. Hence, since A(0) = A(l) = O we have that A(y)>0 on [0,l], hence

L(< P °g 1 )(y)>L(g 1 )(y)

for all y e [0,l], proving (i). (iii) is proved in the same way and, by (IV.74) and (IV.76) (ii) is trivial. By the above proof, it is clear that strict inequalities apply in cases (i) and (iii) when (ii) is not the case.

The above theorem (in case —^—^ increases), restricted to the discrete case, was proved in x Rousseau (1992a) but Burrell (1992a) pointed out the existence of the general result of Fellman(1976). This theorem gives the complete situation of positive reinforcement. Based on Theorem IV.4.1.1 we do not always have a more concentrated situation after positive reinforcement. Indeed, it is clear that, although cp implies an increase in the number of items per source of

Lotkaian concentration theory

223

the first IPP, this does not always imply an increase in concentration. Indeed, examples abound: one can add items to each source so that they all have equal production, in which case we even have equality, hence the lowest concentration that is possible (and L((p°g,)(y) = y on [0,l]). Another, more realistic case is given by adding to all productions, the same amount of items so that we are in situation (vi) in Section IV. 1 (the principle of nominal increase), hence a decrease of concentration, hence an L((p°g1) which is lower than L(g,)-

The following corollary trivially follows from Theorem IV.4.1.1.

Corollary IV.4.1.2: Let 9 be a power law of the type cp(x) = bxa with a, b positive parameters. Then L ( c p o g l ) > L ( g l ) i f f a > l , L(
Proof: Note that here —^—^ = bxa~' which increases iff a > 1, is constant iff a = 1 and decreases iff x 0 < a < 1. Since these are mutually exclusive cases and since we comprise all cases (since cp is a power law), we have that the conditions are necessary and sufficient in this case (hence the validity of the "iff statements).

Note that we proved in Chapter III that a power law of

This is the time to give an example of positive reinforcement. It was presented in Rousseau (1992a), being at the same time a real life example of a Lorenz curve construction.

Table IV.2 shows the available number of CDs in a local record library and the number of loans during the year 1990.

224

Power laws in the information production process: Lotkaian informetrics

Table IV.2 Availability and 1990 loans of CDs in the Public library of Puurs (Belgium). Reprinted from Rousseau (1992a), Table 2, p. 393. Copyright John Wiley & Sons Limited. Reproduced with permission.

Category

Number of CDs

Number of loans

595 340 313 151 639 330 107

1,582 649 627 185 1,120 563 395

148 1,274 427 491 2,549 235 244 475

764 5,615 2,290 817 17,510 339 672 1,387

CLASSICAL MUSIC Orchestral Concertos Soloists (instr.) Ensembles Vocal, secular Vocal, religious Various NON-CLASSICAL MUSIC Spoken recordings Amusement music Film music Jazz Pop Ethnic music Country and folk Various

We first consider the loans per category. Ranking loan data from highest to lowest (for the construction of the Lorenz curve) yields the following sequence:

17,510; 5,615; 2,290; 1,582; 1,387; 1,120; 817; 764; 672; 649; 627; 563; 395; 339; 185.

Now we calculate cumulative partial sums, that is the number of loans in the highest category, the sum of the first and the second category, the sum of the first three categories and so on. This gives the following sequence:

17,510; 23,125; 25,415; 26,997; 28,384; 29,504; 30,321; 31,085; 31,757; 32,406; 33,033; 33,596; 33,991; 34,330; 34,515.

We see that the total number of loans equals 34,515.

The Lorenz curve of these data is the curve which connects the points with coordinates

Lotkaian concentration theory

225

(0,0), (1/15, 17,510/34,515), (2/15, 23,125/34,515), (3/15, 25,415/34,515), (4/15, 26,997/34,515),..., (15/15, 34,515/34,515) = (1,1)

Fig. IV.3

Lorenz curve of loans per category.

Reprinted from Egghe and Rousseau (2001), Fig. 2.15, p. 53. Copyright Europa Publications. Reproduced with permission from Taylor & Francis.

We now do the same for the availability categories.

We obtain the following decreasing availability sequence:

226

Power laws in the information production process: Lotkaian informetrics 2,549; 1,274; 639; 595; 491; 475; 427; 340; 330; 313; 244; 235; 151; 148; 107.

The corresponding sequence of cumulative partial sums is:

2,549; 3,823; 4,462; 5,057; 5,548; 6,023; 6,450; 6,790; 7,120; 7,433; 7,677; 7,912; 8,063; 8,211; 8,318.

Consequently, this Lorenz curve connects the points with coordinates:

(0,0), (1/15, 2,549/8,318), (2/15, 3,823/8,318),..., (1,1).

Here the availability curve is situated completely under the loans curve (see Fig. IV.4). This means that overall availability is more balanced (less concentrated) than loans.

IV.4.2 Concentration properties of Type/Token-Taken informetrics

In the positive reinforcement model, the rank-frequency function g, was composed to g = cp°g,, the rank-frequency function of the positively reinforced IPP and in the previous subsection we proved properties of L(g) versus L(g,).

In Type/Token-Taken (TTT) informetrics it was the size-frequency funcion that was changed. Let f denote the size-frequency function of the "start" IPP and f* the one in the TTT version. By (III. 11):

f*(j) = jf(j)

(IV.77)

for all j e [ l , p m ] . Note that in TTT informetrics we change the sources not the production densities j G [l,pm] which is valid for the start IPP and for the one in TTT version.

Lotkaian concentration theory

Fig. IV.4

227

Lorenz curves of loans (a) and availability (b) per category.

Reprinted from Egghe and Rousseau (2001), Fig. 2.16, p. 55. Copyright Europa Publications. Reproduced with permission from Taylor & Francis.

In this connection it is natural (cf. the dual approach of positive reinforcement versus TTT) to compare L(f*) with L(f), the Lorenz curves of f* respectively f. Unlike Theorem IV.4.1.1 we have only one possible result here: the TTT Lorenz curve is always below the Lorenz curve of the original size-frequency function.

Theorem IV.4.2.1 (Egghe): Let f be the size-frequency function of any IPP and let f* be the size-frequency function of its TTT version. Then

228

Power laws in the information production process: Lotkaian informetrics L(f*)
(IV.78)

Proof: According to (IV. 13) and (IV.77) we have

r("")+1f(x-)dx' L(f)(y) = ^ — - ^ —

(iv.79)

J]"f(x')dx' and Lf

/ *(y)=A7ii f

x'f(x')dx' (IV-8°) x'f(x')dx'

(note, as explained above, that both formulae use the same pm). We will have proved (IV.78) if we can show that

rx'=>'(P»-1)+! rx"=Pii.

I

I

I

S

I

rx'=y(p m -l)+l rX"=Pm

S

x'f x ' f x " d x ' d x " < /

I

I

S

I

S

x"f x 1 f x" dx'dx"

for all y € [ 0 , l ] . Deleting the double integral over [l,y(p m —l) + l] in both sides, this boils down to showing that

P»'=»(P.- | )+ 1

rX"=Pm

I

I

S

I

S

x'f(x')f x" dx'dx"

p -1)+1 px"=p

/

/ x'=l

Jx"=y( PlI ,-l)+l

x " f x ' f ( x " dx'dx" V /

V

(IV.81)

/

But, for all ( x ' , x " ) e [ l , y ( p m - l ) + l ] x [ y ( p m - l ) + l,p m ] w e have that x ' < x " , hence (IV.81) is trivially true and so is (IV.78).

Lotkaian concentration theory

229

Note that, in case

f(j) = f j e [l,p m ], the Lotka function, we have that

f(j) = pr so that Theorem IV.4.2.1 is in agreement with Corollary IV.3.2.1.5 (i): also from this corollary it follows that L(f*) < L(f) since a — 1 < a . Note, however, that Theorem IV.4.2.1 is valid for general size-frequency functions f.

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V

LOTKAIAN FRACTAL COMPLEXITY THEORY

V.I INTRODUCTION The most natural and exact way in which complexity of a system of relations can be described is by indicating its dimension in space. As an example we can consider a citation analysis study of how n journals are citing another k journals (possibly, journals can appear in both sets; both sets can even be equal in which case n = k but this is not a requirement). For each journal in the first set we can look at all the references that appear in the articles in this journal and in this way we can count the number of times one finds a reference to an article that was published in a journal that belongs to the second set. We hence obtain n vectors with k coordinates, hence an n x k matrix in which entry ay ( i e {l,...,n}, j e {l,...,k}) denotes the number of references, appearing in articles of journal i, to articles of journal j . Such a situation can be considered as a cloud of n points that belong to k-dimensional space Rk (R denotes the real numbers). We here have the problem of visualizing (or at least describing) this k-dimensional cloud of points. We will not describe the techniques of multivariate statistics that can be used here in order to reach this goal: one uses dimension-reducing techniques such as principal components analysis, cluster analysis or multidimensional scaling in order to visualise the cloud of points in two dimensions, i.e. on a computer screen or a sheet of paper - see Egghe and Rousseau (1990a) for an extensive account on this, including several examples from informetrics.

The purpose of the present chapter is not to reduce the dimension of such a situation but rather to describe it mathematically. We will also limit our attention to the description of the dimension of IPPs, henceforth called the complexity of IPPs. Our main goal in this book, of course, will be the study of complexity of Lotkaian IPPs, i.e. where we have a size-frequency

232

Power laws in the information production process: Lotkaian informetrics

function f that is of the power law type (11.27). So, here, we do not study (as in the example above) which source (journal) cites which source (journal) but the situation "how many sources (e.g. journals) have how many items (e.g. citations)" and this from the viewpoint of the "dimensionality" of the situation. In other words we will describe (Lotkaian) IPPs as fractals. A fractal is a subset of k-dimensional space lRk (k = 1,2,3,4,...) but, dependent on its shape, it does not necessarily incorporate the full k-dimensionality of R k . A simple example is a straight line in Rk which is a one-dimensional subset of Rk or a plane in Rk which is a two-dimensional subset of R k .

In the next section we will study dimensionality of general subsets of R k : e.g. explain why a straight line has dimension 1 and a plane dimension 2 but we will also study "strange" subsets of Mk for which we have to conclude that their dimension is not an entire number but a general number in K+ (the positive real numbers): such sets are called proper fractals. The reason for this study is their interpretation in terms of IPPs: it will be seen (in Section V.3) that IPPs can be interpreted as proper fractals. Lotkaian IPPs will be characterized as special self-similar fractals, i.e. fractals (subsets of R k ) which are composed of a certain number of identical copies of themselves (up to a reduced scale). Therefore our study, in the next section, of fractals will focus mainly on such self-similar fractals and we will see that, for such sets, it is easy to calculate their fractal dimension (also introduced in the next section).

As said, in Section V.3, we will interpret IPPs as fractals and show the special self-similarity property of Lotkaian IPPs interpreted as fractals. It is now clear that the Lotka exponent a must play a central role in this: we will show that a - 1 is the fractal dimension of the selfsimilar fractal associated with this Lotkaian IPP. This, again, shows the importance of power laws.

V.2 ELEMENTS OF FRACTAL THEORY In this section we will, briefly, review the most important aspects of fractal theory that we will need further on. For a more detailed account on these matters we refer the reader to Feder (1988) or Falconer (1990) or to Mandelbrot (1977a), the founding father of fractal theory and also the one that formulated the law of Mandelbrot - see Chapters I and II; in fact his

Lotkaian fractal complexity theory

233

argument given in Subsection 1.3.5 will lead us directly to the fractal interpretation of random texts as fractals, which we will generalise to general Lotkaian IPPs - see further.

The concept of a fractal and of its fractal dimension will become clear once we have studied the "dimensionality" (or fractal aspects) of simple sets such as a line segment or a rectangle or a parallelepiped. This will be given in the next subsection.

V.2.1 Fractal aspects of a line segment, a rectangle and a parallelepiped

Let us take an arbitrary line segment. Suppose we cut this line segment into M equal pieces (i.e. into M e N equal length line pieces which are non-overlapping). Each line piece has a length which is a fraction r = — < 1 of the length of the original line segment and we need M M = — such line segments. The number r is called the scaling factor: the original line r segment is reduced by a scale r which forms the smaller line segments of which we need N = M = - to cover the original line segment, r Suppose now we have a rectangle. We now reduce this rectangle with a scaling factor r = — < 1 as above (i.e. obtaining a similar rectangle of which the sides are scaled with a M

, (if UJ

factor r, with respect to the original rectangle). It is now clear that we need N = M = -

of

these rectangles to cover the original rectangle in a non-overlapping way (except for the boundaries). In the same way, the reduction of a parallelepiped with a scale factor r = — < 1 M needs N = M3 = — of these scale-reduced parallelepipeds to cover the original one in a non-overlapping way (except for the boundaries).

It is hence clear that dimensionality is recovered by reducing the scale of the figure and then by examining how many of these reduced-scale figures can cover the original one. This, obviously, only works with so-called self-similar figures, where the definition of self-

234

Power laws in the information production process: Lotkaian informetrics

similarity is as indicated above: there exists a scaling factor r < 1 such that a certain number of these "reduced" figures (with scale r) recover the original figure in a non-overlapping way (except for some boundary points). We will now see how this works for a less-trivial (but still self-similar) figure: the triadic von Koch curve.

V.2.2 The triadic von Koch curve and its fractal properties. Extension to general selfsimilar fractals

We first explain the construction of the triadic von Koch curve which is a figure in the plane E 2 . Its construction is depicted in Fig.V.I.

We start with a line segment (phase n = 0). For phase n = 1 this line segment is reduced to a line segment of length — of the original one (hence scaled with a factor r = —) and then applied 4 times as indicated in Fig. V.I. For phase n = 2, we do the same on each of the 4 line segments as we did when going from phase 0 to phase 1. We continue in the same way in the other phases n = 3, 4,... : in the limit we obtain the so-called triadic von Koch curve.

The curve has the property that a scale factor r = — = — reduces the triadic von Koch curve M 3 to a curve which is congruent with the first fourth part of the curve itself and hence we need N = 4 of these curves to cover the original one. Putting

as we did in the case of a line segment (Ds = 1), a rectangle (Ds = 2) and a parallelepiped (D s = 3) we now have 4 = 3 D s , hence

D s = —?«1.26186 In 3

(V.I)

a non-entire number. The number Ds is called the similarity dimension of the triadic von Koch curve (being indeed a self-similar fractal).

Lotkaian fractal complexity theory

Fig. V.I

235

Example of a self-similar fractal: the triadic von Koch curve.

Reprinted from Feder (1988), Fig. 2.8, p. 16. Reproduced with kind permission of Kluwer Academic Publishers.

236

Power laws in the information production process: Lotkaian informetrics

The similarity dimension of any self-similar fractal in any k-dimensional space Rk can be calculated this way: if we reduce the fractal with a scale r = — < 1 and if we then need N M identical copies of this reduced figure to recover the original figure we define its similarity dimension to be

s

lnN=Jn^ lnM -lnr

Although we have seen above that D s for line segments, rectangles and parallelepipeds really is the dimension of these geometrical figures, we still have to indicate what can be the definition of a fractal dimension in general and how (V.2) can be used in this connection for self-similar fractals. This will be done in the next subsection.

V.2.3 Two general ways of expressing fractal dimensions Let F be any (non-empty) subset of Rk (k = 1,2,3,4,...). We want to give an intuitively appealing definition of its fractal dimension, of course in agreement with our ideas on dimension for familiar sets such as "smooth" lines, surfaces, volumes etc. We will limit ourselves to two (non-equivalent) possible definitions.

V. 2.3.1 The Hausdorff-Besicovitch dimension

Let U; C Kk, i e N be a countable (or finite) cover of F, i.e.

FcQu, The cover (Uj)

is called an s-cover of F if, for each i e N, 0 < | U ; | < e , where |Us

denotes the diameter of U; in R k , i.e.

|Uj| = sup{||x - y|| | x,y £ U J

Lotkaian fractal complexity theory

111

where ||.| denotes the norm in K k . Define, for every s > 0 and 8 > 0:

r oo

1

<Jf(F) = inf r C I U j f K U ^

is an e - cover of F

(V.3)

This means: we look at all 8-covers of F and minimise the sum of sth powers of the diameters. It is clear that the numbers a^

increase for s decreasing since the latter reduces

the number of possible covers. Hence the following limit

<*Fs(F) = l i m ^ s ( F )

always exists as a value > 0 or oo. ^

s

(V.4)

(F) is called the s-dimensional Hausdorff measure of

F.

Let (U,).^ be an e -cover of F and let t > s > 0. Then, obviously

Du.Nfiunu.r i=l

i-l

Hence (V.5) implies, by (V.3), that

^"(F)<s'-S^S(F)

The inequality has an important consequence: if ^

(F) < oo then c^" (F) = 0 for all t > s

(let s —> 0). By reversing the roles of t and s in (V.6) we also see that if ^ t < s , t h a t <*r'(F) = +oo.

(V.6)

s

(F) > 0 and

238

Power laws in the information production process: Lotkaian informetrics

Thus the graph of ^" (F) in function of s (see Fig.V.2) shows that there is exactly one critical value of s at which ^

(F) jumps from +00 to 0. This critical value of s is called the

Hausdorff-Besicovitch dimension and is denoted as s = dimH (F).

Fig. V.2

Graph of
Reprinted from Falconer (1990), Fig. 2.3, p. 28. Copyright John Wiley & Sons Limited. Reproduced with permission. In other words: dimH (F) = inf {s | «TS (F) = 0} = sup {s | JVS (F) = +00}

(V.7)

The intuitive interpretation of this is clear. Let e.g. F be a line segment. Then s = dimH (F) = 1, of1 (F) = 0 for t > 1 (e.g. for t = 2: the "area" of a line segment is 0), <%?1 (F) = +00 for t < 1 and <^" (F) happens to be the length of this line segment. Let F be a rectangle. Then s = dimH (F) = 2 , ^Cx (F) = 0 for t > 2 (e.g. for t = 3: the "volume" of a rectangle is 0), <%fl (F) = +00 for t < 2 (e.g. t = 1: the "length" of a rectangle is +00) and atf1 (F) is proportional (proportionality factor equals — see Falconer (1990)) to the area of 71

this rectangle. This proportionality factor arises because, in (V.3), we use (e.g. for disks of radius r) (2r)s which equals 4r2 for s = 2 while the area of a disk is rcr2. Let F be a parallelepiped. Then s = dimH (F) = 3, afx (F) = 0 for t > 3 ,
is

Lotkaian fractal complexity theory

proportional (proportionality factor equals

239

see Falconer (1990)) to the volume of this 71

parallelepiped. This proportionality factor arises because, in (V.3), we use (e.g. for balls of radius r) (2r) which equals 8r3 for s = 3 while the volume of a ball is —7tr3. In general, c^ s (F), for s = dimH (F), equals cnvoln (F) where voln (F) is the n-dimensional volume of

F (if it exists) and cn = —^

n

' (see Falconer (1990)) but we do not need this result in this

book.

Hence dimH (F) represents the idea of a fractal dimension for any subset F of R", for every n e N . It is recognized as the perfect description of dimensional complexity of a set F. However, its disadvantage is that it is (usually) very difficult to determine dimH (F). The next definition of dimension is in this sense more popular, both from the point of its relative ease of mathematical calculation and its empirical estimation.

V. 2.3.2 The box-counting dimension

For a general non-empty bounded subset F of Rk (k 6 N) one can define a new dimension of F, inspired both by the similarity dimension for self-similar fractals and the HausdorffBesicovitch dimension. Note that, as in V.2.3.1, we do not need F to be self-similar. The rough link between the Hausdorff-Besicovitch dimension dimH (F) and formula (V.2) is as follows (an exact definition of the new dimension will be given after this heuristic argument). ^ S ( F ) (formula (V.4)) is, for small s, by (V.3), close to the N E (F)s s , N E ( F ) being the smallest number of sets of diameter (in R k ) at most s which can cover F. <%"" (F) itself is a measure of the "s-dimensional length" of F (called length for s = 1, area for s = 2, volume for s = 3, ...) which is a constant denoted c. Hence NE(F)ss~c

N E (F)^ce" s

(V.8)

240

Power laws in the information production process: Lotkaian informetrics

a power law of e . Hence

lnNs(F)^lnc-slne

(V.9)

and so

_

lnN E (F) — Ins

|

lnc Ins

Letting s —> 0 the above reduces to

—Ins

showing the link with (V.2) (M = N8 (F), the number of sets, related to the "scaling" factor r = s ). Therefore, we formally define, for any non-empty bounded set F C Kn

dim B (F) = l i m l n N e ( F ) E -° - I n s

(V.ll)

if this limit exists and we call dimB(F) the box-counting dimension. Formula (V.ll) clearly expresses the meaning of a fractal dimension: the number of sets (one shows that one can limit oneself to balls, cubes,...) of diameter E that intersect a set F is an indication of how spread out or irregular the set is when measured with a scale s (= resolution level). The dimension reflects how rapidly the irregularities of F develop as s — 0.

We mentioned already the popularity of the box-counting dimension. It can e.g. be used in the calculation of the fractal dimension (using dimB) of coast lines or country borders. Indeed, for a selected set of small s-values we graph the points (s,N E (F)) in a log-log scale. Based on (V.9) we see that we will find that these points lie (more or less) on a straight line with slope -s, where s = dim B (F) (by (V.ll)). Linear regression on these points then gives an

Lotkaian fractal complexity theory

241

estimate for this slope, hence for dimB (F). An example is given in Feder (1988) (see Fig.V.3) where this method reveals that the fractal dimension of the coastline of Norway is close to 1.52, a proper fractal: although we speak about a (coast)line, its fractal dimension is much larger than 1 and even above 1.5.

Fig. V.3

The number of "boxes" of size s needed to cover the coastline of Norway, as a function of s. The straight line is the result of a regression analysis yielding a slope —s « — 1.52 .

Reprinted from Feder (1988), Fig. 2.7, p. 15. Reproduced with kind permission of Kluwer Academic Publishers. This technique could be compared with the experience of flying over the coastline of Norway. The above shows that, when we lower our height, say with 50% (expressed by halving our measuring diameter e) we need far more than double of the squares to cover the coastline, expressed by (V.8):

Nf(F)~c(£[S N e (F)^cs~ s .2 s 2

N e (F)«N E (F).2 s ^2.87N E (F) 2

for s = dimB (coastline) « 1.52.

242

Power laws in the information production process: Lotkaian informetrics

In a way it is a pity that the Hausdorff-Besicovitch dimension and the box-counting dimension, although related, are not always equal. However for self-similar fractals (and we will only need this in Lotkaian informetrics) we have the following important non-trivial result.

Theorem V.2.3.2.1 (see e.g. Falconer (1990)): Let F be a self-similar fractal. Then

dim H (F) = dimB(F) = D s

(V.I 2)

For the intricate proof we refer the reader to Falconer (1990), p.118-120. The result in Falconer (1990) holds for more general fractals, namely where more than one similarity is allowed: here we only use that one similarity applies for the whole curve, in which case we find that the Hausdorff-Besicovitch and box-counting dimensions equal the similarity dimension - formula (V.2).

The above theorem makes life much easier since we will have no problems in calculating fractal dimensions, once we have shown the self-similarity of a system. This we will show to be the case in Lotkaian informetrics in Section V.3.

V.3 INTERPRETATION OF LOTKAIAN IPPs AS SELF-SIMILAR FRACTALS Let us reformulate the construction of the triadic von Koch curve (Subsection V.2.2) as follows: let each phase (phases n = 0, 1, 2, 3, 4, 5 are depicted in Fig.V.l) be interpreted as discrete time t e N . The construction in Subsection V.2.2 is equivalent with the following statements:

(a)

The number of line segments grows exponentially in time t, proportional with 4* (in fact equal to 41 for the triadic von Koch curve).

Lotkaian fractal complexity theory

(b)

243

of each line segment is the same for every line segment (for fixed t) and length grows exponentially in time, proportional with 31.

If we generalise the numbers above by replacing 4 (in (a)) by a, > 1 and replacing 3 (in (b)) by a2 > 1 we see that, up to the names "line segments" (to be replaced by "sources") and "

" (to be replaced by "number of items") we have the reformulation of the Naranan length

assumptions, given in Subsection 1.3.4, hence Naranan's assumptions describe a self-similar fractal. In this Subsection 1.3.4 we explained the derivation of Lotka's law as size-frequency function in such systems for continuous time t and we added our own argument yielding the same law of Lotka, using discrete time t € N . In both cases we showed that the size frequency function f was of Lotka (i.e. power) type as in formula (1.54), the Lotka exponent equalling <x = l + ^ lna 2

( V .l3)

°"' =J Tn

(see formula (1.55)). Hence

|n

- (d

From the above we can conclude that Lotkaian IPPs can be interpreted as self-similar fractals with (by Theorem V.2.3.2.1 and formula (V.2)) fractal dimension.

D s = <x-1

(V.15)

as given by formula (V.I4). This result is important enough to be formulated as a theorem.

244

Power laws in the information production process: Lotkaian informetrics

Theorem V.3.1 (Egghe (2004d), based on Naranan (1970)): Suppose we have an informetric system (IPP) of Lotkaian type, i.e. where the size-frequency function f satisfies

f(j) = ^

(V.16)

where C, a > 0, j > 1 (here we take the maximal density pm = oo as in Subsection II.2.1.1). Such a system can be interpreted as a self-similar fractal with fractal dimension D s = a — 1.

Note that it follows from the above theorem and Corollary 1.3.4.1.2 that in case the growth rate of the sources equals the growth rate of the items in Naranan's model), hence in case a = 2, that we have a self-similar fractal of dimension 1. This, once more, shows the special role of the exponent a = 2 within informetrics.

Similar with (but unrelated to) the theory of concentration, we can draw the conclusion that the fractal dimension increases (linearly) with a - see (V.I5). In terms of the Zipf exponent P , using (11.56) and (V.I5) we see that

P = -J-

(V.17)

From the above we see that a power function gives rise to a self-similar fractal where a (formula V.16) equals Ds + 1 if we interpret the power function as a size-frequency function (Lotka). In terms of the rank-frequency function (Zipf) we have formula (V.17). Hence in no case the exponent itself (a,—a, p or — (3) can be interpreted as a fractal dimension, contrary to the assertions in Katz (1999) and Stewart (1989). In Katz (1999), the author uses both interpretations (Lotka and Zipf) and hence, since a s = p , we cannot have that both a and (3 represent the fractal dimension of the system; in fact none of these do so and we have to use formulae (V.15) and (V.17) for their interpretation in terms of fractal dimensions (see also Fairthorne (1969) and Mandelbrot (1967)).

Lotkaian fractal complexity theory

245

In Subsection 1.4.3 we gave examples (from the literature) of social networks of which their connectivity size-frequency function (i.e. the number of nodes with k edges) is Lotkaian with high exponents a between 2 and 4. Hence these systems, by (V.I5), have fractal dimension between 1 and 3, which is rather high.

The obtained results are valid in any Lotkaian informetrics (i.e. IPP) interpretation. They are confirmed in random texts, which are a special example of an IPP (cf. the linguistics example in Section I.I) by Mandelbrot (1977b) - see also Mandelbrot (1967). We recall the conclusions of Subsection 1.3.5. Let us have N letters in our alphabet and that a text is a string of letters and blanks. Each letter has an equal chance of occurrence (simplification of reality!) denoted by p . Since there are also blanks we have p = P(letter) < —. Such a text is N considered as a self-similar fractal of which the "phases" are (cf. Egghe and Rouseau (1990a)): n = 0 : empty text n = 1 : the N letters = the N words, say A, B, C,... n = 2 : the N2 words

AA, AB, ..., BA.BB, ...,

and so on. At each level n we have P (word) = p" and Nn words. So this system can be considered as a self-similar fractal with a scaling factor p and a multiplicator N to go from each level n to the next level n + 1. By (V.2) we have that its fractal dimension is given by

D

s

= ^ — lnp

(V.18)

Combining (V.18) with the result (1.58), obtained in Subsection 1.3.5, we hence have that P = — , hence the same result as in (V.I7). Because Subsection 1.3.5 showed that the law of Mandelbrot holds in this case, hence, by Theorem II.2.2.3 also Lotka's law and formula (11.56): (3 =

, we hence refind that a —1 = D S , the fractal dimension of this special a —1

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Power laws in the information production process: Lotkaian informetrics

system, hence we refind formula (V.I5), which was, of course, more generally valid (Theorem V.3.1).

Theorem V.3.1 is a central result in Lotkaian informetrics. It expresses, in a mathematically exact way, that the power laws, which are characterized by their scale-free (self-similarity) property (Theorem 1.3.2.2 and Corollary 1.3.2.3) give rise to the systems (IPPs) in which the same size-frequency distribution is found at low or high productivity (number of items per source) values. To recall Huberman (2001) and referring to the size-frequency distribution of website sizes we can conclude that the same size-frequency distribution is found when looking at sites that have between 1,000 and 2,000 pages as when looking at sites that have a different size range, say from 10 to 100 pages. In other words (in intuitive fractal terms), zooming in or out in the scale at which one studies the web (as in the case of examining coast lines with a plane at different heights), one keeps obtaining the same result, as is also the case in every self-similar fractal.

VI

LOTKAIAN INFORMETRICS OF SYSTEMS IN WHICH ITEMS CAN HAVE MULTIPLE SOURCES

VI. 1 INTRODUCTION This is a unique chapter in many senses. This chapter deals with informetric systems in which items can have multiple sources. The most obvious application is, of course, the case where items are articles which are written by (possibly) more than one author (i.e. multiple sources). We underline that the goal of this chapter is not to study the informetric functions describing the number of authors per paper (for this, see Subsection 1.4.4) but the number of papers per author, although the former will be used in the latter. Note that the terminology of this application coincides with the terminology used in the historical paper Lotka (1926) in which Lotka's size-frequency function was introduced for the first time. As remarked in Chapter I, however, Lotka circumvented the problem of dealing with multiple authorship by only giving a credit (of 1) to the senior author (the other authors were not given any credit), hence, in fact, Lotka treated the articles as single-author articles. In the next Section VI.2 we will study other, more realistic ways, of crediting sources in systems where items can have multiple sources. Let us here just mention two possibilities: in an item (e.g. article) with n e N sources (e.g. authors), each author could be given a credit of 1 (each author of the article is hence fully recognized but in this way an article gives, in total, n author credits, so a different weight compared to articles with a different number of authors), called the "total counting" system or, alternatively, each author is given a credit of — (keeping hereby the total author "weight" n of each paper equal to 1 but making an author credit dependent on the total number of authors of an article), called the "fractional counting" system. We will see that the total counting

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Power laws in the information production process: Lotkaian informetrics

system is very different from the fractional counting system, which causes a serious problem, e.g. for evaluation studies. This fact and solutions for this problem will be given in Section VI.2.

Although interesting and important in itself, the topic of Section VI.2 will not be the main issue of this chapter. The fact that we encounter multiple sources for an item in informetrics makes this field rather unique: apart from artificial examples that could be produced in any field, we note that this problem is not encountered in other -metrics sciences (or at least we have not seen a paper introducing this topic): classical econometrics deals with wealth and poverty expressed e.g. by the income of people and it is clear that salaries are single sourced; in biometrics each animal belongs to one species; in linguistics the "token" (= item) is uniquely linked with one "type" (= source) since here a source is a word (as such) and an item is the use of this word (e.g. in a text). Even in informetrics, the possibility that items have multiple sources, is a rare phenomenon. In Bradfordian terminology, a source is a journal and an item is an article in such a journal: it is clear that the article is published in only one journal. In citation analysis a reference is uniquely linked with the citing article and the same for citations. In library sciences the borrowing of a book (= item) is uniquely linked with this book (= source), obviously. The reader can check the many other examples given in Chapter I to find out that they are all examples of single source items (except, as said, the case of authors and their publications (and some derived examples, see Section VI.2) and some artificial examples, discussed in Subsection 1.4.4). This special place of the author/publication system makes us think of a possible special informetrics theory that is behind it. In Subsections II.2.2 and II.4.2 we showed that Lotka's law is equivalent with functions such as the ones of Mandelbrot, Zipf, Leimkuhler and Bradford, but how can this be when Lotka's function represents a multiple-source system for items while the other functions represent single-source systems for items? Of course we can answer this question: in the theories developed so far (hence also in Subsections II.2.2 and II.4.2) we have always used the same informetric system (i.e. IPP) for all these laws, in fact comparable with what Lotka did by treating the articles as single-authored (as mentioned above). Of course, this leaves us with the problem of deriving informetrics theories for the multiple-source framework for items. Here we immediately encounter the problem of choosing the scoring system for sources in such a multiple-source framework for items, e.g. (but these are not the only possibilities) the total scoring system or the fractional scoring

Lotkaian informetrics of systems in which items can have multiple sources

249

system (briefly discussed above and more exhaustively in Section VI.2 to come). Let us discuss the problem and possibilities of solution for both scoring systems separately since they are completely different.

Problem VI.1.1 (Total scoring system)

To fix the ideas, let us consider the discrete case. In this case the only values for "the number of items per source" are the natural numbers n = 1, 2, 3, . . . . So the size-frequency function f(n), describing the number of sources with n items, has the same arguments as the one we have discussed already throughout this book. The same elementary observations, developed in Subsection 1.3.1 can be made here: f > 0 and f decreasing in n, hence also Proposition 1.3.1.1 is valid: there exists a constant D > 0 such that limf (n) = D . We can refer to Kretschmer and Rousseau (2001) where f(n), the number of authors with n papers, in the total counting system, is not decreasing but only in those cases where there are more than 50 authors per paper: in all other cases, where the number of authors per paper is below 40 (still very high!), f(n) is empirically seen to be decreasing and even of Lotka type.

We can remark here that Egghe (1995) extends the success-breeds-success principle, introduced in Subsection 1.3.6, to the case that items can have multiple sources and where, possibly, non-decreasing size-frequency functions can occur. The model is, however, theoretic and does not yield analytic forms of these functions (in a way understandable because of the intricate nature of such, exceptional, size-frequency functions - see e.g. Fig. 1, p. 612 in Kretschmer and Rousseau (2001)).

Also the following argument based on the results of Subsection 1.3.2 can be convincing in accepting that the size-frequency function of the total counting system is Lotkaian.

Argument VI.1.1.1 (Egghe): Lotkaian informetrics where items can only have one source implies Lotkaian informetrics where items can have multiple sources and where the total scoring system for sources is applied.

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Power laws in the information production process: Lotkaian informetrics

Proof: Let f denote the size-frequency function in the single source case and f, denote the sizefrequency function in the multiple source case. We make the following plausible assumption: there exists a number p > 1 such that the number of sources with n e N items in the singlesource case equals the number of sources with pn items in the multiple source case (one could even say that p will be around the average number of sources per item (>1!) in the multiple source IPP, but this is not needed further in the argument). Hence

f t (pn) = f(n)

(VI.l)

for all n £ N . Hence (VI.l) implies, for any n,m 6 N :

ft(pn.pm) = f(pnm)

= Ef(pn)f(m)

= E 2 f(p)f(n)f(m)

(VI.2)

using formula (1.29), Corollary 1.3.2.3 and Proposition 1.3.2.5 on f (given that f is Lotkaian). Hence (VI.2) implies, using (VI.l) again, that

f1(pn.pm) = Ff t (pn)f t (pm)

(VI.3)

where F = E 2 f(p), a universal constant in this multiple source IPP. We did this heuristic argument in m,n e N for reasons of clarity but we could, in the continuous setting, assume

f,(px) = f(x)

for all x € R+ leading to

ft(px.py) = Ff t (px)f,(py)

(VI.4)

Lotkaian informetrics of systems in which items can have multiple sources

251

for all x, y e R + . Since p > 1 > 0 we also have that pK+ = R + (this improves the above discrete argument since pN is not equal to N) so that we have: for every x,y e K+ :

ft(xy) = Ff,(x)f l (y)

(VI.5)

showing that ft satisfies the product property (1.29) and hence, by Proposition 1.3.2.5 and Corollary 1.3.2.3, we have that ft is Lotkaian.

The reason why we called the above observation an "Argument" and not a "Proposition" or a "Theorem" is that, formally, it is not possible to compare an informetric system where items have only one source with an informetric system where items can have multiple sources, since they do not occur in one IPP. One could, of course, consider an IPP in which papers can have multiple authors as an IPP in which papers have only one author, by simply replacing a paper with k authors by k papers with 1 author (each time we take one of these k authors). A similar approach was followed in Egghe (1994b), where the size-frequency function of the multiple author case was calculated based on the knowledge of the size-frequency function of the single author case. The used mathematical tool was: convolutions, a technique that we will encounter again in this chapter. In Egghe (1994b) we showed that a Lotkaian single author IPP implies (approximately) a Lotkaian multiple author IPP. We, however, think that the above Argument VI. 1.1.1 is simpler and more convincing to accept that a size-frequency function for IPPs where items can have multiple sources, with a total counting system, is Lotkaian (i.e. is a decreasing power function).

The most important topic in this chapter (and also the most intricate one) is the determination of the size-frequency function for informetric systems where items can have multiple sources and where we use a fractional counting system. Also this problem will be introduced here (but explained rigorously in Section VI.3).

Problem VI. 1.2 (Fractional scoring system)

To mention right away the most important feature of this problem: the size-frequency function is never decreasing. To put things right: we have an IPP where items can have multiple sources (e.g. articles can have more than one author) and where the source scores are

252

Power laws in the information production process: Lotkaian informetrics

credited in a fractional way. Example: an author A has published a paper authored by 3 authors (author A inclusive), a paper authored by 2 authors (author A inclusive) and a singleauthored (of course by author A) paper. Then this author receives a credit of — H

1-1 = —.

As is clear now, in principle, any score which is a positive rational number is possible. Denoting by Q + the set of positive rational numbers (i.e. numbers of the form — where b a,b 6 N), we hence have that, in such systems, the size-frequency function f is a function on Q + , namely q—>f(q), where q £ Q + . Without any experimentation we can already conclude that values such as

f — , f — , f — ,...

16

Ui

will be smaller than e.g.

Il2j'

f — , f(l), f(2) since a score of e.g. q = — can only come from one paper with 16 \2) 16 authors or (even more unlikely) 2 papers with 32 authors and so on, while a score of q = 1 can come from one single-authored paper, two papers with 2 authors, three papers with 3 authors, four papers with 4 authors or 3 papers: one with 2 authors and two with 4 authors (and so on). Many other examples can be given and this makes it very clear that f(q) cannot be a decreasing function of q. An immediate consequence of this is that f(q) cannot be a power function (i.e. Lotka's law), as remarked by Rousseau (1992b). This also shows that the heuristic argument in Bookstein (1990b), stating that if Lotka's law in one crediting system (e.g. total scores) is valid then Lotka's law is valid in any other crediting system (e.g. fractional scores), is not correct. We showed above that the difference between both sizefrequency functions is fundamental.

It now seems that we are beyond Lotkaian informetrics!

The interesting point is that we are in Lotkaian informetrics more than ever! In fact, as is clear from the above, two aspects are playing a role here: first of all we still have that, overall, for very large q (hence high productions), f(q) must be low; next, we can remark that rational values q with high denominators are the result of being a co-author in one or more papers with many authors (hence also leading to small values of f(q)). In other words, not only the number of papers (counted in a total way) per author is important here but also the number of authors per paper, the dual function (see Subsection 1.4.4).

Lotkaian informetrics of systems in which items can have multiple sources

253

As to the number of papers (counted in the total way) per author (more generally: the number of items per source), the Argument VI. 1.1.1 given under Problem VI. 1.1 sufficiently shows that we can suppose Lotka's law as size-frequency function in this case. As to the number of authors per paper, Subsection 1.4.4 sufficiently explained why also here a Lotka function can be used. Of course these remarks still leave open how to apply these two (dual) Lotka functions in order to model the fractional size-frequency function f(q) for q e Q + . This will be done in Section VI.3 in a, we think, satisfactory way so that this even sheds more light on the power of Lotkaian informetrics rather than it would indicate a breakdown of it (Rousseau (1992b)).

VI.2

CREDITING SYSTEMS AND COUNTING PROCEDURES FOR SOURCES AND "SUPER SOURCES" IN IPPs WHERE ITEMS CAN HAVE MULTIPLE SOURCES

As explained in the previous section, the most important example of an IPP in which items can have multiple sources is the case of articles (items) that can be written by more than one author (source). This, in turn, produces a substantial generalization: one could consider the country affiliation of each author and in this way considering countries as sources for articles. A credit for a country in an article (independent of the crediting system for authors - to be discussed in Subsection VI.2.1) is then the sum of the credits of the authors of this article that are affiliated to this country. In this way, countries are treated with a different crediting system than authors and it could also be considered as a generalization of the author crediting system: indeed, only when the country affiliations of each author in an article are different, the country crediting system coincides with the author crediting system. The same could be said when considering research labs, languages, . . . . This extension is basic and since we do not want to loose this important generalization we will consider it separately from authors as sources and call countries (labs, ...) "super sources". We will then develop a theory around super source credits which includes source crediting systems because of the above mentioned generalization. The next subsection, however, is devoted to the definitions of the different source crediting systems from which the super source credits can be derived.

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Power laws in the information production process: Lotkaian informetrics

VI.2.1 Overview of crediting systems for sources

We will present a rather exhaustive list of source crediting systems, for the sake of completeness. Since some of these definitions use the rank of an author in a paper, we will, for clarity reasons, use this terminology. Each time we suppose that we have an article with A authors.

VI.2.1.1 First or senior author count

First author count gives a credit of 1 to the first author and no credit (i.e. a credit of 0) to the other A - 1 authors (Cole and Cole (1973)). This method is also known as straight counting. This is, clearly, an imperfect system since every author should at least receive a positive credit. However it can be seen as an easy-to-use sampling method in a system in which author ranks are given randomly. If author-ranks are given according to the alphabetic order of the names, this method clearly causes a bias in the sample; the same is true with any other lexicographic (or even any non-random) ordering system. The popularity of the system also comes from the fact that the Science Citation Index presents citation data (not publication data) by first author only (this has been improved in the Web of Science (Web of Knowledge) where one can search on every author).

Senior author count is the same as first author count but here only the senior author receives a credit of 1 and the other A - 1 authors receive no credit. This method was used by Lotka in his historic paper Lotka (1926). In some disciplines, the senior author is the last one in the list. For a variant of this method: see VI.2.1.6.

VI.2.1.2 Total author count

This is the method in which each of the A authors of an article receives a credit of 1. This method is also called normal or standard counting. It is a fair method in the sense that each author receives the same credit. This is also the case in the next method. The present method, however, increases the weight of articles with many authors since the total author count is A. This is remedied in the next method.

Lotkaian informetrics of systems in which items can have multiple sources

255

VI. 2.1.3 Fractional author count Here each of the A authors receives a credit of — (De Solla Price (1981)). This counting A method is sometimes called adjusted counting. Each author receives the same weight and, independent of the total number A of authors, the weight of each article is 1. VI.2.1.4 Proportional author count

Less used is the proportional author count (Van Hooydonk (1997)). If an author has rank r in the author list of an article (with A authors in total, hence r 6 {l,2,...,A}) then this author receives a credit of

H\—^-\ A{

(VI.6)

A + lj

It is easy to see that (VI.6) is the normalized form of a score A + 1 - r (so r = 1: score A,...,r = A : score 1) so that the sum of all numbers in (VI.6) for r = 1, 2,..., A, is 1. So, also this method gives an equal weight of 1 to each article, independent of the total number A of authors.

VI.2.1.5 Pure geometric author count

If an author has rank r in the author list of an article (with A authors in total), then this author receives a credit of

2 A-r

A

(VI.7)

2A-1

V

;

(Egghe, Rousseau and Van Hooydonk (2000)). It is easy to see that (VI.7) is the normalized form of a score 2A"r (so r = 1: score 2A~',..., r = A: score 1) so that the sum of the numbers in (VI.7) for r = 1, 2,..., A, is 1. So also this method gives an equal weight of 1 to each article, independent of the total number A of authors.

256

Power laws in the information production process: Lotkaian informetrics

VI. 2.1.6 Noblesse Oblige

As in senior author counting, this author receives the highest score but does not get everything: here the senior author receives a score of — while the other A - 1 authors receive an equal credit of —, r. Also in this method, the total weight of an article is 1, 2(A-1) independent of the total number A of authors.

VI.2.2 Crediting systems for super sources The above Subsection VI.2.1 also yields the different crediting systems for super sources by simply adding the credits of the sources that belong to this super source. Example: let an article have 5 authors having a credit c, (i = 1, 2, 3, 4, 5) (c ; is determined by one of the methods discussed in Subsection VI.2.1). Suppose that authors 1, 2 and 4 belong to the same super source (e.g. country, lab,...) and the authors 3 and 5 do not belong to this super source. Then the score (in this article) of this super source is c, + c2 + c 4 .

This is the only way we will consider scores for super sources, derived from scores for sources. There is a variant of this method where a super source receives a credit of 1 if it appears in one or more of the sources and of 0 otherwise. This can then be normalized by replacing 1 by - , where s is the total number of different super sources. This is used in s Nederhof and Moed (1993) if s * 1. If s = 1 then they set the score of this paper source on 0 and is called the fractionated scoring system (used to study multinationality of papers).

VI.2.3 Counting procedures for super sources in an IPP

This subsection is devoted to introducing counting procedures for super sources (e.g. countries, labs, universities, or even authors themselves) in IPPs, i.e. where several items (e.g. articles) are involved. Note that this study automatically contains the study of counting procedures for sources (e.g. authors) as remarked in the introduction of Section VI.2. We will limit ourselves to the study (and comparison) of the total counting system, the fractional counting system and the proportional counting system.

Lotkaian informetrics of systems in which items can have multiple sources

257

In each case we suppose we have N items and that item i (i = 1,..., N) is produced by a; sources and we fix one super source c of which we want to calculate the "score" in the system (the classical application being: N articles and article i is written by a, authors and we will check which of these authors has an affiliation in country c). We will always suppose af > 0 (e.g. excluding anonymous articles).

The key variable in the study of the various counting procedures for super sources is djrc (i = 1,...,N; r = l,...,aj), where djrc = 1 if c occurs on rank r in article i and djrc = 0 otherwise. We have that

Ed, re =a,(c)

(VI.8)

r=l

denotes the number of occurrences of super source c in item i £ {l,...,N}.

VI. 2.3.1 Total counting

The total score of super source c in this system, using total counting (note the different uses of the word "total" here: total score refers to the sum of all scores of super source c, independent of the counting method per item and total counting refers to the counting method (VI.2.1.2)), denoted WT (c), equals

W

x(C) = E E d , r c = I > , ( c ) i=l

r=l

(VI-9)

i-1

Noting that the total number of scores in this system (over all super sources), denoted WT, equals

wT=Ewx(c)=EEEd,c=Eai c

i=l r=l

c

i=l

we can defme the relative score (or contribution) of super source c in this system as

(VL1°)

258

Power laws in the information production process: Lotkaian informetrics

N

QT(c) = ™

=^

WT

(VI.11)

£*

VI 2.3.2 Fractional counting

The total score of super source c in this system, using fractional counting, denoted WF (c), equals

WF(c)=EE^ = E f 1 i=l r=l

a

i

i=I

d

CVL12)

i

Noting that the total number of scores in this sytem (over all super sources), denoted WF, equals

c

i=l r=l

c

a

i

i=l

a

i

we can define the relative score (or contribution) of super source c in this system as

VFl

'

WF

Ntf

a,

;

VI. 2.3.3 Proportional counting

The total score of super source c in this system, using proportional counting, denoted Wp (c), equals (use (VI.6), where A is replaced by at for each item i)

Lotkaian informetrics of systems in which items can have multiple sources 259 N

a; j

N

a;

j

wP(c)=2££^-2£E-^ i=i r=i a,

i=1 r= ,

at [ai + I)

WP(c) = £ - a i ( c ) - ^ M i=i "-j

(VI.15)

a.j -|- l

by (VI.8) and denoting

R(i,c) = £rd, r e

(VI.16)

r=l

the sum of all ranks occupied by super source c in article i. Note that the total number of scores in this system (over all super sources), denoted Wp, equals

WP=£wP(c) = E E £ d i r c ^ l - - J _ a,

a,

N £Z/ d ,rc 2V-^V—r t f a,(a1+ l)

NE Z X c Wp = 2Y"^-^ P t i ' a,

W P = 2 £ ^ - 2 £ ^ =N i=l

a

i

(VI.17)

i=l ^ a i

since

EE^, r=l

c

r e

=^

(vi.i8)

Z

being the sum of all ranks in article i, i.e. the sum of the numbers 1,2,..., at. Hence we can now define the relative score of super source c in this system as

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Power laws in the information production process: Lotkaian informetrics

w = Q P (c)

wP ^

«-W-^-(c)-M?)

*W = « . W - 5 § ^

by (VI. 14).

Extensive examples, illustrating the large difference between these counting methods will follow. The following proposition will be handy in constructing examples for the proportional counting method, based on existing examples for the fractional counting method.

Proposition VI.2.3.1 (Egghe, Rousseau and Van Hooydonk (2000)): For any system of super sources (i.e. representing a system of collaboration between authors, countries, labs,...), denoted as system I, we can construct another one with the same super sources, denoted as system II, such that the proportional counting result in system II equals the fractional counting result in system I as well as in system II. Furthermore, also the total counting results are the same in both systems. In symbols: for every super source c:

Q'( c ) = Q'-(c) = Q?(c)

(VI.21)

Q ' ( C ) = Q; I (C)

(VI.22)

Proof: Take any system of N items. Consider the ith item. In this item, super source c appears a ; (c) times, by notation (VI.8), out of the total of a{ sources. Now mirror this situation so that we obtain an item with 2a; sources, where, if super source c has a rank r in system I it also has a rank 2a: - r + 1 in system II (and the existing rank r). Now super source c appears 2af (c) times in item i in system II. Do this for every item i = 1,...,N. Let us denote by Qx( c )

an

d

Lotkaian informetrics of systems in which items can have multiple sources

261

Q" (c) the relative scores of super source c in system I and II, where X can be T, F or P. We have, for every super source c:

P W

Ntf2a,

l W

2a;+l

where Rn(i,c) denotes the sum of all ranks of super source c in item i in system II. By construction, this is always equal to (2a; + l)a; (c). Hence

1

'

Nt^a^

lV ;

2ai+l

M =Q.(c)

=

N i=f 2af

F W

We also have

E2a,(c) E2a, i=l

VI.2.4

Inequalities between QT (c) and QF (c) and consequences for the comparison ofQ T (c),Q F (c)andQ p (c)

We have the following theorem, describing completely the interrelation of QT (c) and QF (c).

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Power laws in the information production process: Lotkaian informetrics

Theorem VI.2.4.1 (Egghe and Rousseau (1996b,d)): QT (c) > QF (c) if and only if the slope of the regression line of

a (c)

over a, (i = 1,... ,N) is

strictly positive. If we replace > by < then this inequality is equivalent with a strictly negative slope.

Proof: The formula for the slope of the regression line of the cloud of points

a.(c) 'v ' a.1

equals i=l,...,N

(see e.g. Egghe and Rousseau (2001) p.56, a reference on statistics applied to the field of library and information science)

lfaa,(c) N j ^ ' a,

h^h N

a, N

s2 (s2 denoting the variance of (a,,...,a N ), hence positive). So this slope is positive if and only if

i=l

I i=l

I

i=l

a;

whence

N

'^-i(c).5' i W

hence

Q F (c)
The other assertion is proved in the same way.

Lotkaian informetrics of systems in which items can have multiple sources

ale) ' increases in

As an immediate consequence of Theorem IV.2.4.1 we have that if

a

function of a ; , then Q T (c)>Q F (c) since then the regression line of

263

i

a(c) " ' over a; increases

(see also Egghe and Rousseau (1996b) for a proof of this evident fact). In Egghe and Rousseau (1996b,d) the above result was applied to determine the relation between the global and average impact factor (which is beyond the topic of this book) and to many other fields, even outside informetrics (see Egghe and Rousseau (1996d)).

It is now clear that a universally valid inequality between QT and QF does not exist. The following example makes this explicite. Example VI.2.4.2 Let N = 3 and suppose we have the following countries appearing as author affiliation in these 3 articles

Article 1

Countries c,

C

2

C

c,

3

C

C

2 2

3

3

c3

c3 C

3

c3

Then it is easy to see that QT (c,) = — > QF (c,) = - and that QT (c,) = — < QF (c,) = — .

Hence, using Proposition VI.2.3.1, we also have an example where Q?( c i)> Q" ( c i)

an(

l

Q?(c2) 1 for all i = 1,...,N. Then, by (VI.20):

264

Power laws in the information production process: Lotkaian informetrics

Qp(c) = 2 Q F ( c ) - - V ^ - < 2 Q F ( c ) - ~ V !

Since a: (c) = 1 for all i = 1,.. .,N, this implies that Qp (c) < 2QF (c) - QF (c) = QF (c). Take now a country c' for which R(i,c') = 1 for all i = 1,.. .,N. Then, again using (VI.20)

QP(c') = 2 Q

F

(c

0

-lE^

> 2 Q p ( - ' ) - ^ E 1 = QF(c1) -N i=i a;

again since a( (c') = 1 for all i = 1,... ,N.

We can even make examples that yield the following paradox: Paradox VI.2.4.3: There exist examples of countries a and b such that

QT(b)
(VI.23)

and the same is true with F replaced by P. Proof: Once an example showing (VI.23) is given, we also have an example of (VI.23) with F replaced by P, using Proposition VI.2.3.1. So we only show (VI.23). We will also indicate the system behind the construction of such non-trivial (but very realistic!) examples. First of all we will limit ourselves to authors (sources) (otherwise said: super sources are sources or, equivalently, a{ (c) = 1 for each i = 1,...,N).

Let us consider the general situation of three authors a, b and c in a system with x + y + z single-author articles, namely x with a as single author, y with b as single author, and z with c

Lotkaian informetrics of systems in which items can have multiple sources

265

as single author; x' + y' + z' articles with two authors, namely x' with b and c as authors, y' with a and c as authors and z' with a and b as authors: and finally a articles with a, b and c as authors. So, for this system N = x + y + z + x' + y' + z ' + a . We will try to determine the seven unknowns x, y, z, x', y', z' and a such that (VI.23) becomes valid.

Table VI. 1

A collaboration

pattern of three authors a, b and c, leading to

QT(b)
Article number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

a

b

c

X X X X X X X X X X X X X X X

X

X X X

X

X

X

X

X

X

X

X

X X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

This can be done by expressing QT (a), QF (a), QT (b), QF (b) in terms of the above variables and then requiring (VI.23). One (but there are many!) solution is given as follows:

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Power laws in the information production process: Lotkaian informetrics

a = 0, x = x' = 5, y = 7, y' = 8, z = 1, z' = 2, i.e. we have a collaboration pattern as in TableVI.l.

We have the following scores, as is readily seen

QT QF

QT

(a) 0.3488 = 0.3256

00

QF

QT

(c) 0.3256

QF

QT

Q 00 = 0.3750 (a) = 0.3571 (<0 = 0.2679

Note that QT (b) < QT (a) < QF (a) < QF (b), hence (VI.23) is satisfied. Also, author a ranks first according to total counts, and only second according to fractional counts.

So we have a real paradox here that, although QT (a) < QF (a), author a scores better in the T system than in the F system.

These results show the large difference between the total, fractional and proportional scoring systems. This means that one must be very careful when drawing conclusions on source or super source ranking according to one of these scoring systems since these rankings can be (completely) different in another scoring system. Yet, there are some positive results that can be mentioned, which can be handy to overcome the above mentioned difficulties in practical field work.

VI.2.5 Solutions to the anomalies

It is conceivable that scientists doing evaluations do not want to get involved in the problems and anomalies sketched in the previous sections. So, how can one exclude or at least diminish their influence?

We will show that Q T , QF and QP are close to each other if the number of collaborators is high. This gives a partial solution to the problems encountered in the previous subsection,

Lotkaian informetrics of systems in which items can have multiple sources

267

although we are not convinced that one should rely on it for a complete solution. Next we will show, based on earlier findings (Egghe and Rousseau (1996d)), that a complete solution exists if one replaces the arithmetic average by the geometric mean.

VI. 2.5.1 Partial solutions

The fact that QT (c) and QF (c) can be close to each other is shown in the unpublished note of Kranakis and Kranakis (1988) (the result can also be found in Egghe and Rousseau (1990a)).

Theorem VI.2.5.1.1 (Kranakis and Kranakis (1988)): For any super source

where m = min{a,,...,a N },M = max{a,,...,a N } .

Proof:

By (VI. 11) and (VI. 14) we have

—ya(c)
hence (VI.24). Inequality (VI.24) shows that, if m = M, i.e. if the number of coauthors is the same in all articles, then QF (c) = QT (c). Further, if m and M are approximately equal or if m (and hence M) is very large, then QF (c) RS Q T (C) . As to QF (c) versus Q p (c) we have the following result

268

Power laws in the information production process: Lotkaian informetrics

Theorem VI.2.5.1.2 (Egghe, Rousseau and Van Hooydonk (2000)): For any c, such that ai (c) = 1, for every i = 1,... ,N, the following inequality holds:

M-ZM^t^

(VI-25)

Proof: If &t (c) = 1 , for every i = 1,.. .,N, we have, by (VI. 14) and (VI.19):

i FV

VPV ;

'

N

1

N t f a,

Ntfa,

a, + l

where R(i,c)e {l,...^} , since a;(c) = 1, for every i= 1,...,N. So,

VFW

VPI )

N

Zj^

tfai(ai+l)

can be bounded as follows:

Consequently, since a: > 1:

IQ.(=)-Q,(=)I4E^.

°

Corollary VI.2.5.1.3 (Egghe, Rousseau and Van Hooydonk (2000)): Under the assumptions of the above theorem we have

Lotkaian informetrics of systems in which items can have multiple sources

Q F (c)-Q P (c)|< average of I—*—,..., - 1 - 1 [a,+l a N +lJ

269

(VI.26)

Corollary VI.2.5.1.4 (Egghe, Rousseau and Van Hooydonk (2000)): Let m = min{a1,...,aN} as before. Then

QF(c)-Qp(c)|<-L-. m+ 1

(VI.27)

Hence, QF pa Qp if m is high. Together with the earlier found result, we have that, if m is high:

QT~QF~QP-

(VL28)

VI.2.5.2 Complete solution to the encountered anomalies

In this section we will show that the use of geometric instead of arithmetic averages eliminates the occurrence of these anomalies. Recall that the use of a geometric mean was already suggested in (Egghe and Rousseau (1996d)). Let us first define these geometric versions (denoted with a g as superscript).

By dividing nominator and denominator of (VI.11) by N we have

arithmetic average of{a,(c),...,a N (c)} arithmetic average of {a p ...,a N }

Hence, we define (replace "arithmetic" by "geometric" in the above formula):

_i_

. . (a.(c)a,(c)...aN(c))N Q\{c)=y l W 2 W — ^ - . (a,a 2 ...a N )N

(VI.29)

270

Power laws in the information production process: Lotkaian informetrics

By (VI. 14), QF is:

Q F (c) = arithmetic average of - L ^ , . , . , a i

NV

'\ N J

a

Consequently, we define:

Q .( c)= f^M>M*. a,

a2

(VL30)

aN

It is clear that Q?(c) = Q«(c)

(VI.31)

for any system and any c. Hence all rankings based on the total counting method are the same as those based on the fractional counting method. Consequently, all ambiguities are gone!

We strongly advise researchers to start working with these geometric variants of QT and Q F , not only in this application but in all applications as described in Egghe and Rousseau (1996d).

VI.2.6 Conditional expectation results on Q T (c),Q F (c) and Q P (c)

The total counting system (T) is the simplest one from a probabilistic point of view: given N

N,a,,...,a N , we only need to know ^ a j (c), while for the fractional counting system (F) we i=i

need to know as (c) (see (VI. 11) and (VI.14)). In other words, for (T) we only need to know the total appearance of super source c in the system while for (F) we need to know how many times c appears in each item i = 1,...,N. So (F) implies (T) uniquely but, per (T)-situation, there are many (F) situations compatible with the given (T)-situation and hence also different values of QF (c), per fixed QT (c). We can now ask: what is the average (i.e. expectation) of

Lotkaian informetrics of systems in which items can have multiple sources

271

all the possible Q F (c) values, given the fixed Q T (c) value, in other words what is the conditional expectation, denoted E T (Q F (c)), given Q T (c). This conditional expectation is the same notion as the one encountered in Subsection 1.3.6.4 to define the evolution in time of stochastic processes, but the application is, of course, completely different.

We have the following theorem.

Theorem VI.2.6.1 (Egghe (1999b)):

E T (Q F (c)) = QT(c)

(VI.32)

for every super source c.

Proof: Using linearity of conditional expectations (being averages) we have:

E

(Q (C)) = E i f ^ 1 = - Y X ^iM)

denotes the averages of all a

i

(VI.33)

values, given Q T (c). Now Q T (c), by a

i

N

(VI.ll) is the overall probability, for c, to occupy one of the ^ a ; positions available in the i=i

N items. Then the probability to have a; (c) occurrences of c in a; places in item i is (binomial distribution)

a-w)Ql(o)-(l-Q,(c))"">

Hence

272

Power laws in the information production process: Lotkaian informetrics

ET*-M = a

i

a

-M

a

a;

a^cj

Si(c)=o

; Q T ( C r
But , s W

a,

a,-l ' 'a,(c)-l

aj(c)

as is readily seen. So, by (VI.33)

MQFW)4&,(C)£ N i=l

a

;~\ QT(cr'(i-QT(c))
ai(c)=l a i ( C j — 1

, the last sum equalling 1, being the total chance of a binomial distribution. Hence, E T (Q F (c)) = Q T (c).

The proportional and fractional scoring system can be compared in a similar way. Indeed, the proportional scoring system (P) is finer than the fractional (F) one: given at (c) (in (F)), there are several possibilities for the ranks that c in item i occupies, all leading to different Qp (c) scores, agreeing with one set as (c), i = 1,.. .,N. But given the ranks of c in each item i (i.e. the (P) situation), we can easily determine a^ (c) and hence the (F)-situation. As in the previous case we can ask: given a fixed (F)-situation ai (c), i = 1,... ,N what is the average of all Qp (c) values that agree with this (F)-situation? Again this is a conditional expectation which we denote by EF (Qp (C)) . We have the following theorem.

Theorem VI.2.6.2 (Egghe (1999b)):

E F ( Q P ( C ) ) = QP(C)

for every super source c.

(VI.34)

Lotkaian informetrics of systems in which items can have multiple sources

273

Proof: Using linearity of the conditional expectation (being an average), formula (VI.20) yields

, since Q F (c) is fixed here. Here E F (R(i,c)) denotes the average of the R(i,c) values, given the fixed (F)-situation i.e. given the numbers at (c), i = 1,.. .,N. If we can show that

E F (R(i,c)) = a f

i

(VI.36)

then (VI.35) yields

E F (Q p (c)) = 2 Q

F

( c ) - l X : ^ = Q F (c)

using (VI. 14). Hence we only have to prove (VI.36). For any situation, compatible with the given (F)-situation a s (c), i = 1,...,N and leading to R(i,c) we construct another situation, also compatible with the given (F)-situation ai (c), i = 1,...,N but leading to R'(i,c) (the sum of the ranks of c in item i in this second situation) for which we have that

R(1,c) + R'( i ,c)

= a ( c )

a^l

(VL37)

In this way all possible (P)-situations are partitoned in groups of two satisfying (VI.37) which is a constant, given the (F)-situation. Hence this shows (VI.36) since EF (R(i,c)) averages all a +1 these constants a (c)— . This leaves us with the task to construct the R'-situation such ,i ; 2 that (VI.37) is valid. Given the R(i,c)-situation (i = 1,...,N), denote by A(i,c)c {l,...^} the

274

Power laws in the information production process: Lotkaian informetrics

set of ranks that c occupies in this situation. The second situation is constructed as follows: let now c occupy ranks a; - r + 1, for every r€ A(i,c). The sum of the occupied ranks is now

R'(i,c)= £ ( a i - r + l) reA(i.c)

R'(i,c) = ( a , + l ) £ l - £ r reA(i,c)

reA(i.c)

R'(i,c) = (ai+l)ai(c)-R(i,c)

proving (VI.37) and hence the theorem.

Corollary VI.2.6.3 (Egghe (1999b)):

E T (Q P (c)) = Q T (c)

(VI.38)

for every super source c.

Proof: By the above theorems we have

ET (Qp (c)) = ET (EF ( Q P (C))) = ET (QF (c)) = QT (c).

Here we use the fact that ET°EF = ET since T is rougher than F, see also Chow and Teicher (1978) for a more theoretic treatment of this relation and of conditional expectations in general.

The above results allow us to use the simpler QT (c) values if we are dealing with averages E T (Q F (c)) or E T (Q P (C)). This was utilized in Egghe (1999b) to prove a relationship between the fraction of multinational publications (i.e. multi-authored publications where there are at least two different country affiliations) of a country and the fractional score of this

Lotkaian informetrics of systems in which items can have multiple sources

275

country, confirming a related empirical result of Nederhof and Moed (1993) on the relation between the fraction of multinational publications of a country and its fractionated score (cf. Subsection VI.2.2 for its definition).

This closes Section VI.2 where the different crediting systems and counting procedures for sources and super sources have been discussed. Although each of these systems have their own merits and pitfalls, we can - with little or no discussion - consider the fractional crediting system as one of the most important systems to count a contribution of a source or super source in the production of an item. Indeed, each source is contributing an equal weight, namely l/(the number of sources that produce an item) and we think this equality among the sources reflects, in modern times, best the contribution of each source in the production of an item (and even when equality is not the best way, it is, in most cases, not possible to derive a more optimal way). In addition, the fractional counting system yields a total weight of 1 to each produced item and, hence, the weight of each item, unlike the case of the total counting system, is independent of the number of sources producing it.

Therefore the next section is important. It gives an answer on the modelling of the sizefrequency function f, when fractional counting is applied. As explained in Section VI. 1 the numbers f(q) with q £ Q + (the positive rationals, being the set of possible production scores of sources in this fractional scoring system) are not decreasing, hence a discrete or continuous Lotka type function of the form

f(x) = J r

(VI.39)

with x G N, Q + or R + , is not the correct form of such frequency functions (cf. also the arguments in Rousseau (1992b)). Yet, the next section will show that we are able to model f(q) for the most important values of q e Q + , using Lotkaian informetrics. In fact, we will show that empirical data on f(q), in the fractional counting procedure, can be approximated very accurately by assuming even two Lotka functions: one for the number of authors with m € N articles, counted totally (cf. Argument VI. 1.1.1) and one for the number of articles with n authors (cf. Subsection 1.4.4). No doubt that we are here in an important part of Lotkaian informetrics: using Lotka' law both in its original and its dual formalism!

276

VI.3

Power laws in the information production process: Lotkaian informetrics

CONSTRUCTION OF FRACTIONAL SIZE-FREQUENCY FUNCTIONS BASED ON TWO DUAL LOTKA LAWS

VI.3.1 Introduction

We start by presenting the important data set of Rao (1995), see also Egghe and Rao (2002a), partially reproduced in the Appendix II for the reader's convenience, on experimental fractional frequency scores of authors in mathematics. It concerns data from the Mathematical Reviews of the year 1990. The table shows the total fractional score of an author in this year. This means, e.g. that an author with 3 articles, one with 1 author (of course this author), one with 2 authors in total and one with 3 authors in total will be appearing in this table as an author with a total fractional score of H

h - ~ l - 8 3 . The irregular shape of the graph

(Fig.A.l) (also in the Appendix II) is clear (note that we showed the nth fraction (abscissa) and its corresponding fraction of authors (ordinate) rather than the actual value of the n"1 fraction: this enchances the distinguishing level of the graph, which will be even more important in the comparison of theoretical and experimental values further on). Some intuitive explanations of values can be given. The value of 1 is the highest and the graph decreases in function of the entire values 1, 2, 3, 4, 5, . . . . If we look at the reciprocal scores: - , —, - , —, 1, we see that the graph increases. Both regularities can be explained using the size-frequency functions "number of authors with n = 1, 2, 3, ... articles" which is decreasing (see again section VI. 1) and the dual size-frequency function "number of articles with n = 1, 2, 3, ... authors" which is (or is supposed to be) decreasing (cf. Subsection 1.4.4) and certainly in fields where the number of co-authors is not very high (as is the case in mathematics -

in this sense,

mathematics ressembles fields in the humanities and social sciences, a fact that is also found to be true in citation analysis and aging studies!). The fact that the dual size-frequency function decreases implies that the number of authors with fractional score ...—, —, 1 is 3 2 increasing, at least for those authors with only one article in the database. The problem arises with the fact that authors can have several publications and that, as explained above, their fractional scores are added. This disturbs the simple reasoning given above, using the two size-frequency functions. Indeed, as also indicated in Rousseau (1992b), a fractional score of,

Lotkaian informetrics of systems in which items can have multiple sources

277

say — can come from one paper with 2 authors but also from two papers with 4 authors or even 4 papers with 8 authors or even 3 papers, two with 8 authors and one with 4 authors,... . This "decomposability" of rational numbers is linked with the number of divisors of the denominator (i.e. a score — can only come from papers with at least 7 authors since 7 is a prime number) but this is not the whole story: a score of — has again several decompositions (— = 1 + — but also — H

h — and so on). The point is that we can use a typical probabilistic

technique to handle all possible decompositions, namely convolutions. Intuitively (an exact argument will be given in Subsection VI.3.1) we proceed as follows. Let (p(j) denote the fraction of authors with j articles (counted in the total way, cf. Section VI. 1). Let vj/(i) denote the fraction of articles with i authors. q> and i|/ are the dual size-frequency functions that will be supposed to be Lotkaian, but we do not need this at the moment. It is clear that vj/(i) determines f[ (z) being the fraction of authors that have a fractional score z = - , supposing these authors to have written just one article. In fact, such an author receives a score z = - if i this author published one paper in which there are i authors. So f, (z) = f,

T

is related to the

occurrence of articles with i authors, hence with \\i (i). The exact relation will be given in the sequel. We then go from 1 paper to j papers: the j-fold convolution of f,, denoted f, ®... ® f, j times

is then the distribution that describes the fraction of authors, given that they have j articles, with a given fractional score z: (f, ®...® f,)(z) is the fraction of authors with a fractional j times

score z, given that they have j papers. This condition on the number of published papers can then be lifted using the theorem of total probability:

CO

f(z) = ^f 1 ®...®f 1 (z)«p(j) J=1

'

j~tim^

(VI.40)

278

Power laws in the information production process: Lotkaian informetrics

and here f(z) is the overall fraction of authors with a fractional score z. This is then, theoretically, the function we are looking for. The problem is to evaluate f qualitatively and quantitatively and to see how it can match irregular fractional frequency data such as the ones of Rao (1995), presented in the Appendix II.

In the next subsection we will consider the case z 61R+, i.e. studying fractional scores in a continuous way. We will present the exact formula for il in this case and we will confirm (VI.40). However, in this case, we are not able to explain the irregular values of the experimental f(q) for q 6 Q + . We indicate that, under some extra conditions, f is increasing on [0,l] and decreasing on [l,oo[ showing at least the "overall" tendency that was also discussed above. We consider these results (for z e R + ) as not satisfactory (unlike the other results obtained so far in continuous Lotkaian informetrics!) since we want to give an explanation for the large differences between the experimental values f(q) for q 6 Q + .

Of course, using only rational values, and knowing the irregularity of the values in q, we know that an analytical expression for f(q) is not possible. It is, however, also impossible to give individual results on f(q) for all q e Q + (or even limited, say to Q + n [0,2]) since these sets are infinite. In Subsection VI.3.3 we apply a special technique to approximate (VI.40) for grouped data, hereby having a finite number of possible f(q), q e Q+, each q being the midpoint of such an interval, representing each grouping. The technique is possible if the groups (of scores q) are not too small and if the values q £ Q + considered are not too large. We will indicate in Subsection VI.3.3 how to group data and in any case we will limit ourselves to q < 2 (the most important case since it contains the crucial value 1). We will see that the found values of (VI.40) (for this finite set of arguments) are very close to the data of Rao (see the Appendix II) when grouped in a similar way.

VI.3.2 A continuous attempt: z £ R +

Since we will reserve the notation f for our fractional size-frequency function that we want to determine, we will use another notation for the dual size-frequency functions that we will use. Let v|/ —> v]y(y): [l,pm] —> K+ denote the continuous size-frequency distribution describing the

Lotkaian informetrics of systems in which items can have multiple sources

279

density of papers with y as author density. From i|/ we can determine the function f, (z) being the distribution of authors with a paper density z in one paper, in the fractional scoring system. We have the following proposition (notations from Chapter II).

Proposition VI.3.2.1 (Egghe (1993b)): For every z £ [a (0), l] we have that

1

v f,(z) = - ^ -

(VI.41)

where (x is the average number of authors per paper.

Proof: Since i|/ = ij/(y) is a distribution of the density of papers with y as author density, we have that V(/T is the corresponding size-frequency function (T is the total number of sources, here papers). This is the function fin (11.78) and we have, by Theorem II.3.1:

h(z) = — ^

(VI.42)

for all z e [o(0),l], where h is the density function of items per source (i.e. h(z) is the density of authors with paper density z in case an author has only 1 article). Hence, since f, is a distribution, h = Af, where A equals the total number of items (authors) and by definition of f,. Hence

M1-

Af,(z) = — ^

280

Power laws in the information production process: Lotkaian informetrics

with u = —. v- T This result, hence, is a direct application of the new function h introduced in Section II.3. The only difference between the functions in II.3 and the ones here is that, in II.3, we are dealing with functions while here we are dealing with distributions (needed because we will have to apply some results of probability theory in the sequel). That explains the \x in (VI.41) which is lacking in (11.78). Note that (VI.41) was proved directly in Egghe (1993b) but with the use of approximations. The proof above shows that (VI.41) is an exact result in continuous informetrics theory, based on the new function h introduced in Section II.3.

We now go from f, to f, the overall fractional size-frequency distribution that we are seeking.

Proposition VI.3.2.2 (Egghe (1993b)): Let cp

: N —> M+ be the fraction of authors with n papers. Then, for every z £ R + we

have that the fractional size-frequency distribution equals

f(z) = £fi®...®f, (z)cp(j) J=>

j times

(VI.43)

'

where f, is as in Proposition VI.3.2.1 and where §3 denotes convolution and where we assume that the fractional scores distributions in a paper are independent and identically distributed (i.i.d.), being the distribution f,.

Proof: Let N be the random variable of the number of papers and Y(j) the fractional score from the j t h paper. Then

f (z) = P (overall fraction = z)

Lotkaian informetrics of systems in which items can have multiple sources

281

= P(Y(l) + Y(2) + ... + Y(N) = z)

= £ ) P ( Y ( 1 ) + Y ( 2 ) + ... + Y ( N ) = Z|N = J).P(N = J)

by the theorem of total probability (see e.g. Chung (1974), Blom (1989)). Hence

f(z) = £p(Y(l)+Y(2) + ... + Y(j) = z)q>(j) j=i

using independence of N with respect to the Ys. The distribution of Y(l) + Y(2) + ... + Y(j) is given by the convolution of the individual distributions, given the independence of the Ys (see again Chung (1974) or Blom (1989)) and this becomes the i-fold convolution of f,, using the assumption of identical distributions for the Ys.

Convolutions appear frequently in informetrics as a calculation tool, see Rousseau (1998) for an overview - see also Lafouge (1995), Egghe (1994b).

NoteVI.3.2.3; Although it is relatively easy to accept that all Ys have the same distribution, the fact that they are independent is less sure. It is indeed not certain that a certain score in paper 2 (say) is independent of the score in paper 1, due to collaboration habits. However, we have to suppose independence for the intricate model to work. The assumption can be considered as a simplification which is acceptable in this first attempt to model fractional frequencies.

In Egghe (1993b) one then shows that, under some realistic conditions, formula (VI.43) yields an increasing function f on [0,l] which is decreasing on [l,oo[. Since the result is technical and only valid under extra assumptions and since it does not reveal any of the irregularities of the fractional scores for q e Q + (as is also the case with the lognormal fitting of Rousseau (1992b) of such irregular graphs), we do not give the proof here and we go directly to the

282

Power laws in the information production process: Lotkaian informetrics

discrete variant of (VI.43) and then use groupings around certain rational numbers q, which will yield a good ressemblance with the irregular experimental fractional size-frequency data.

VI.3.3 A rational attempt: q £ Q +

We will now use the following discrete functions:

y = v|/(n) being the fraction of papers with n authors, n = 1, 2, 3, ... f, = f, (q) being the fraction of authors with a fractional score of q in 1 paper

(,-, i. }.-) cp =
We have the following variant of Proposition VI.3.2.1 which we will prove in two ways: one (heuristic) proof uses Proposition VI.3.2.1 and one proof (the easiest one) is direct: we will prove the discrete variant of (VI.41).

Proposition VI.3.3.1 (Egghe and Rao (2002a)): With f, and \\i as above we have

'I 1 ] f,(q) = - ^ uq

(VI.44)

1 1 A for all q = 1, —, —,... and where (a. = — is the average number of authors per paper.

First (heuristic) proof: In (VI.41) the continuous \\i, now denoted 4 to avoid confusion, is the density derived from our present function \\i, i.e. \ = — \\i' > 0 . Hence (VI.41) reads

Lotkaian informetrics of systems in which items can have multiple sources

283

.zzti z2

z\i

d m 1 dz

l^zj z^i

In our discrete case we replace — vi/ — by w — , hence

dz UJ

UJ

hence (VI.44) if we replace the continuous z by the rational q. This proof is necessarily heuristic since one cannot replace discrete by continuous variable functions and vice-versa, but it shows that (VI.41) and (VI.44) are not in contradiction with each other (as one could expect at first glance).

Second (exact) proof: f

/ \ '^ '

total # authorships with fractional score q in 1 paper total # authorships

— total # papers with — authors 5

f,(q) = ^ u

>-

total # authorships

total # papers . / N_ 1 q total # authorships

total # papers with — authors q total # papers

284

Power laws in the information production process: Lotkaian informetrics

f. (q) = — — . fraction of papers with — authors

q n

q

< W ~H Note that q = —, n e N , necessarily. This shows that the fractional score of an author in 1 n paper is directly derivable from \\i. Together with its dual function cp we will be able to derive the theoretical, discrete fractional frequency distribution.

Proposition VI.3.3.2 (Egghe and Rao (2002a)): Let f(q), for q£+ denote the overall fractional frequency distribution, i.e. the fraction of authors with a total fractional score q. Then

f (q) = £ J

~'

f,®...®f| (q)
(VI.45)

j times

where f, is as in Proposition VI.3.3.1. and where ® denotes (discrete) convolution. Here we assume that the fractional scores distributions in a paper are independent and identically distributed, being the function f,,

Proof: The proof is exactly the same as the one given in Proposition VI.3.2.2.

The model (VI.45) with f, as in (VI.44), in which both the size-frequency distributions cp and Y are appearing should be capable to explain the various irregular experimental f (q) data. The problem still is: how to deal with f (q), where q 6 Q + , an infinite set. First of all we will limit ourselves to q < 2 , hence to Q + n[0,2] since this contains the most interesting (and irregular) values, including the increase in the values —, - , —, 1 and then decrease to 2.

Lotkaian informetrics of systems in which items can have multiple sources

285

But Q + n[0,2] still is an infinite set. The trick is to allow only a few values q e Q + n[0,2]. This can be done by slightly deviating from the fractional counting system, which boils down to a grouping of scores. We will, hence replace f, by a function g,, which is derived from f, (in an exact way) and which only allows for a finite number of rational values q. We will show three degrees of refinement (and the increasing complexity) in the sense that, in the finer cases, we allow for more (but still a finite number of) qs. These theoretical models are then compared with the experimental results in which analogous groupings will be executed. The comparisons will speak for themselves.

Of course we still have to determine the distributions

, the dual size-frequency functions. As discussed and explained in Section VI. 1 we will use power laws (Lotka laws) for both of these functions and, to make things as simple as possible, we will use the power a = 2, hence since

(VI.46)

7i n

for n = 1, 2,... . Note that (p and \|/ are equal as functions but not in their interpretation: (p(n) is the fraction of authors with n articles and v|/(n) is the fraction of articles with n authors. Note that (VI.46) in (VI.44) gives that we use the following function f,:

f,(q) = ^

(VI.47)

The case i = 2: allowing an author score of 1/2 or 1 in 1 paper Although too rough, this case will very simply illustrate the methodology that we will apply in this paper to yield discrete fractional frequency distributions. Note that we will limit ourselves to fractional scores q < 2, as explained above.

In this simple model, an author receives a score 1 if he/she is an author in a single-authored paper. If he/she is author in a multi-authored paper, this author receives a score —. Let us call g, the author distribution of fractional scores in 1 paper. By definition

286

Power laws in the information production process: Lotkaian informetrics

g,(l) = f,(l)

41=84) gl(i)

= l-f,(l),

(VL48)

(vL49) (VI.50)

since f, is a distribution. Using (VI.47) this yields

g,(l)

J-

8.^1 = 1-4-

(VL51)

(VI-52)

We apply now (VI.45) but with f, replaced by g, and for the values q = —, 1, —, 2, the only possible scores (inferior to 2). This gives

f =gi
(i) (i)

(VL53)

f(l) = gl(l)(p(l) + [ g l ( i | cp(2)

(VI.54)

f ( | ) = 2g1(l)g1(l)cp(2) + ( g l (l|cp(3)

(VI.55)

f(2) = (g1(l))2cp(2) + 3(g,(i))2g1(l)cp(3) + ( g , ( 0 ( p ( 4 ) )

(VI.56)

Lotkaian informetrics of systems in which items can have multiple sources

287

where cp is given by (VI.46).

These values are then compared by the corresponding grouped data from Rao's table in the Appendix II, grouped as follows:

score — corresponds to grouping the data in the interval ]0,0.75] score 1 corresponds to grouping in ]0.75,1.25] score — corresponds to ]l.25,1.75] score 2 corresponds to ll.75,2.25].

It is clear that we take the interval ]0,0.75] for — (and not ]0.25,0.75]) since in our model, all fractional scores (in one paper), smaller than —, are transformed into —.

Of course, these groupings are not a perfect analogue of our simplified model since an author with an overall score of —, being the result of participation in 2 8-authored papers is 4 classified into the score 1 in the model while it is classified in the interval ]0,0.75] in the grouping. This difference is there but will diminish in the next cases where we allow for smaller fractions. Also, if we find good results in this setting, this will indicate that the above difference is not destroying the similar nature of both simplications.

The (only) parameter \x is determined by requiring f — I to be exact: 2

f(i)_(1

(2){

6 ) 6 _ # in ]0,0.75]_ 18,892

~^VJ^?~

total #

~ 46,853

288

Power laws in the information production process: Lotkaian informetrics

(see the table in the Appendix II). This yields \i = 1.80537576. We have now the Table VI.2 and the graph in Fig.VI.l, comparing theoretical and experimental fractional frequency distributions.

From the above we see, although we only compare 4 fractions, both theoretical and experimental graphs are following the same pattern. This will become more clear in the next (more important and more interesting) cases.

Table VI.2

Distribution of overall fractional scores (case of i = 2).

Reprinted from Egghe and Rao (2002a), Table 1, p. 794. Copyright John Wiley & Sons Limited. Reproduced with permission.

Fig. VI. 1

q

Theoretical (f)

1/2

0.4033047

0.4033047

1

0.2715154

0.345911681

3/2

0.0875965

0.079546667

2

0.0545977

0.081360852

Experimental

Theoretical and experimental fractional frequency distributions (case of i = 2). The numbers in the abscissa refer to the number of fractional scores (their ranks), not to the fractional score itself.

Reprinted from Egghe and Rao (2002a), Fig. 2, p. 794. Copyright John Wiley & Sons Limited. Reproduced with permission.

Lotkaian informetrics of systems in which items can have multiple sources

289

The case i = 3: allowing an author score of 1/3,1/2 or 1 in 1 paper

This will be the first interesting case. Here an author receives a score 1 if he/she is an author in a single-authored paper, a score — if he/she is an author in a 2-authored paper and a score - if he/she is an author in a j-authored paper, for all j > 3 . Now we have

g,(l) = f,(l) = ^

'GH-h^li-'-^

(VI-57)

(VL59)

Possible overall fractional scores (in lO,2l) are - , —, —, —, 1, —, —, —, - , —, 2 V J V 3 2 3 6 6 3 2 3 6 (theoretical) to be compared with grouped data (from the Table in the Appendix II) in the , „ 5 15 7 7 9 19 11 111 13 13 15 15 17 17 19 intervals 0,— , — , — , —,— , — , — , —,— , — , — , — , — , —,— , 12 Jl2 12 12 12 Jl2 12 Jl2 12 12 12 12 12 12 12 19 21 12'12'

21 23 12 ' 1 2 '

23 25 12~'l2~'

We have the following formulas from which the theoretical fractional frequency distribution can be calculated.

f

(l)

= gl

(l) ( p ( l )

f =Si cp(1)

(i) (i)

(VL60)

(VL61)

290

Power laws in the information production process: Lotkaian informetrics

f(f) = (&(0
f

(f) =2 g # # ( 2 )

f(l) = gl(l)cp(l) + (g1(!))
f

8

0H ##

(3)

(VI.62)

(VL63) (VI.64)

(VL65)

f (f) = 2g, (l)g, (|)9(2) + BJg, [ 0 g, (i)
(VI.66)

f ( | ) = 2g1(i)g1(l)cp(2) + (g 1 (i|cp(3) + 4( g l (l|g 1 (l)cp(4)

(VI.67)

f =3 gl

(f) ( (^)) Sl(1)(p(3)+6 ( gl ^f( 8l (0 (p(4)+ ( g '(l)) 5
(VL68)

f(y) = 6gI(i)gI(i)g1(l)9(3) + 4g 1 (0g 1 (l|
f(2) = (g1(l))2cp(2) + 3(g1(0g1(l)(p(3) + 4(gl(l|g1(l)cp(4) + [g1(0cp(4)

+ io(g, (j)) (g, ( | | cp(5) + (g, (j)) 9(6).

Again, the parameter |i is determined by

(VI.70)

Lotkaian informetrics of systems in which items can have multiple sources

291

i [l).f,-LR-' 'H. f [3j

[

TtVjTt2

total #

_ 6,508 ~ 46,853

yielding \x = 1.1819488123 . We obtain the remarkable Table VI.3 and graph in Fig.VI.2.

The agreement between the theoretical and experimental results is remarkable. This proves that the two dual Lotka laws are capable of modelling fractional frequency distributions although we are also convinced that other decreasing size-frequency distributions, to be used for 9 and \\i, can serve the same purpose as well. The model is also capable of explaining "generally felt" inequalities, i.e. inequalities of the type f (l) > f — , f — < f — and so on.

\2)

[6)

2)

For a list, see Egghe and Rao (2002a) but we must remark that some results might depend on the method of grouping. This is certainly the case for the following result: for all n e N it follows from (VI.57), (VI.58), (VI.59) and (VI.45) (with f, replaced by g,) that

= 0,

q-f

limf(q)

(VI.71) _jl_ ~6n2'

_n q

~3

The explanation for this last regularity is as follows: for extremally high \x, the chance to have a paper with less than 3 authors is very small. In this case one can only receive a fractional score of — per paper. Hence the only overall scores that are possible are q = — if an author has n papers. The probability for this last event is

292

Power laws in the information production process: Lotkaian informetrics

Table VI. 3 Distribution of overall fractional scores (case of i = 3). Reprinted from Egghe and Rao (2002a), Table 2, p. 795. Copyright John Wiley & Sons Limited. Reproduced with permission.

Fig. VI.2

q

Theoretical (f)

Experimental

1/3

0.1389322

0.1389322

1/2

0.1562373

0.2530468

2/3

0.0079701

0.0112693

5/6

0.0294265

0.0236911

1

0.323324

0.3125520

7/6

0.0027311

0.0097112

4/3

0.0389478

0.0192304

3/2

0.0417687

0.0340426

5/3

0.0059575

0.0049303

11/6

0.0129368

0.0072781

2

0.0483318

0.070959

Theoretical and experimental fractional frequency distributions (case of i = 3). The numbers in the abscissa refer to the number of fractional scores (their ranks), not to the fractional score itself.

Reprinted from Egghe and Rao (2002a), Fig. 3, p. 795. Copyright John Wiley & Sons Limited. Reproduced with permission.

Lotkaian informetrics of systems in which items can have multiple sources

293

More universal inequalities between fractional score frequencies are obtained in Egghe (1996), where the SBS principle is extended to systems in which items can have multiple sources and where one uses fractional scores. The model leads to irregular fractional sizefrequency functions but a match with experimental data is not really possible.

Table VI.4

Distribution of overall fractional scores (case of i = 4).

Reprinted from Egghe and Rao (2002a), Table 3, p. 798. Copyright John Wiley & Sons Limited. Reproduced with permission.

q

Theoretical (f)

Experimental

1/4

0.036241009

0.036241009

1/3

0.1039555

0.102233051

1/2

0.1564714

0.250570935

7/12

0.0030927

0.003052099

2/3

0.0044441

0.01274198

3/4

0.0046533

0.00420464

5/6

0.013455

0.019827973

11/12

0.0003526

0.001323288

1

0.3220503

0.309905449

13/12

0.0010631

0.001686125

7/6

0.0015435

0.005698675

5/4

0.0100909

0.003905833

4/3

0.0290481

0.018654088

17/12

0.000205

0.001899558

3/2

0.0417575

0.053358376

19/12

0.0024309

0.000661644

5/3

0.0035013

0.003478966

7/4

0.0033829

0.001430004

11/6

0.0093664

0.007000619

23/12

0.0004929

0.000277464

2

0.0476329

0.0697428726

294

Power laws in the information production process: Lotkaian informetrics

Fig. VI.3

Theoretical and experimental fractional frequency distributions (case of i = 4). The numbers in the abscissa refer to the number of fractional scores (their ranks), not to the fractional score itself.

Reprinted from Egghe and Rao (2002a), Fig. 4, p. 798. Copyright John Wiley & Sons Limited. Reproduced with permission.

In Egghe and Rao (2002a) we further discuss the cases i = 4 and i = 5 but the complexity rapidly increases. The agreement with the experimental fractional frequency values remains remarkable. We only show the results in the case i = 4, i.e. where we allow an author score of - , -, - or 1 in 1 article (Table VI.4 and Fig.VI.3). 4 3 2 An attempt, comparable with the concrete calculation of discrete convolutions, has been undertaken in Burrell and Rousseau (1995) is some special cases (e.g. articles are written by 1 or 2 authors or a situation where articles can only have 1, 2, 4 or 8 authors). All these restrictions are needed to limit the number of rational numbers q to be considered, a necessity. The approach in Subsection VI.3.3 is more fruitful since it can handle any real situation.

The next Chapter VII (Section VII.4) gives another application of Lotka's law in the case that articles are written by multiple authors. There we calculate the author rank distribution e.g. in case that the number of authors per paper follows a Lotka function. Since this topic is totally different from the ones discussed in this chapter and since Chapter VII is devoted to Lotkaian applications we deferred this study to Chapter VII.

VII FURTHER APPLICATIONS IN LOTKAIAN INFORMETRICS

VII.l INTRODUCTION The purpose of this book was to present a unified theory of Lotkaian informetrics so that the informetrics community (and even beyond informetrics) has a tool that can be used in further applications. Indeed, although the law of Lotka and its equivalent expressions (Chapter II) are interesting in themselves, we should always try to apply them in other informetrics issues. If this can be done we are far ahead of research that always starts from zero and tries to explain certain regularities without referring to earlier results. In this sense the Chapters III, IV, V and VI could be considered as applications in Lotkaian informetrics but, because of their special importance and far-reaching consequences in informetrics, we devoted a special chapter to them.

This book could end here were it not that there exist further interesting applications of the Lotka function in diverse topics. They are gathered in this closing chapter. We start, in Section VII.2 with a warning that some regularities found in informetrics (exact or approximate) are not explained using informetric laws but simply by plain mathematics. We give two examples: one on the well-known arcs at the end of a Leimkuhler curve which are exact exponential functions on a log-scale (i.e. linear functions of the original variable) and one on a relation (called type-token identity) - see Chen and Leimkuhler (1989), which is not correct but which is - mathematically - explainable in an approximative way.

Section VII.3 deals with a regularity that is, in essence, also not informetric in nature but a probabilistic argument leads to the empirically verified power law. It concerns the graph of Wallace (1986) on the relation between the number of articles per journal and the journal

296

Power laws in the information production process: Lotkaian informetrics

median citation age. We will prove that the graph is below a decreasing power function with exponent 2, but this is not a consequence of Lotkaian theory but of the Central Limit Theorem in probability theory. Only the fact that the cloud of points is becoming thinner for high number of articles per journal is explained by Lotka's law.

Section VII.4 deals with another topic on multi-authored articles (cf. Chapter VI), namely on the distribution of the rank of an author in such an m-authored paper. We show that, if the number of authors per paper follows Lotka's law (cf. Subsection 1.4.4 and Chapter VI), the author rank distribution follows the same Lotka law. We further determine author ranks, using author seeds, i.e. a universal number indicating the general place of an author name as e.g. expressed by the alphabetic order.

A very important application of Lotkaian informetrics is given in Section VII.5. There we determine the so-called "first-citation distribution", i.e. (e.g. in a bibliography) the overall cumulative distribution of the time period between the publication of an article and the time it receives its first citation. We can explain it using an exponentially decreasing age function (i.e. time distribution of all citations) and a Lotka law for the number of citations for an article in this bibliography. Note that both functions are applicable in one model, due to the arguments given in Subsection 1.3.4.2. The application is remarkable since it is capable of explaining concave as well as S-shaped cumulative distribution functions and - what is even more interesting - the difference between the two types is characterized in terms of the Lotkaian exponent a . Hence Lotka's size-frequency function is decisive in this matter. In this section we also mention an application of aging functions (as the exponential one) to the explanation of the relation of the Price index in function of the mean and median reference age.

The closing Section VII.6 deals with the intricate problem of determining the rank-frequency and size-frequency functions for N-grams and N-word phrases. We derive these functions using Zipf s function as rank-frequency function for the constituting parts of the N-gram (letters) or of the N-word phrase (words) and using a technique of N-product space, being the Cartesian product of the IPPs of the constituting parts. The theory presented here is exact in the sense that no approximations are used and in this sense improves earlier results of the author. Using the size-frequency function we will also determine (as in Chapter III) formulae

Further applications in Lotkaian informetrics

297

for the (Type/Token) average nN and the Type/Token-Taken average \in of N-grams and Nword phrases and the values of uN and \x'N are compared. This final section also gives rise to (hard) problems concerning N-grams and N-word phrases.

VII.2 EXPLAINING "REGULARITIES" Real regularities need explanations. Sometimes these explanations are elementary and are not of an informetric nature (i.e. we do not need Lotka's law or subsequent results to explain them). The researcher should be able to make a distinction between the various types of explanations. In this section we will give two examples of regularities that can be explained via plain (simple) mathematics.

VII.2.1 The arcs at the end of a Leimkuhler curve

One of the simplest regularities ever found in informetrics, but which is not an informetric regularity at all, is the fact that, at the end of a Leimkuhler curve, one detects "arcs". One obtains a Leimkuhler curve when graphing the cumulative number G(r) of items in the first (largest) r sources, versus log r (any log can be used, e.g. In). The graph looks as in Fig.VII. 1 and can be found, for example, in Warren and Newill (1967), Brookes (1973), Praunlich and Kroll (1978), Wilkinson (1973) and Summers (1983).

Fig. VII. 1

A Leimkuhler curve, with arcs for large r.

Although the graph (without the arcs) has an equation of the form (see formula (11.43))

298

Power laws in the information production process: Lotkaian informetrics

G(r) = aln(l + br)

(VII. 1)

which is certainly an informetric regularity (see Chapter II or Egghe (1989, 1990a), Egghe and Rousseau (1990a)), the arcs, apparently deviating from (VII. 1), are not informetric in nature. Indeed, there are frequently several high-ranking sources that provide the same number of items: there might be a large number of sources with three items, a larger number with two items, and an even larger number of sources with only one item each. Because increases of G(r) at these ranks are linear in r (per group of equal productivity), the graph of G versus log r is exponential (per group of equal productivity). These exponential graphs become more visible as the groups of sources with equal productivity become longer. This explains these arcs near the end of a Leimkuhler curve.

Hence, this phenomenon has a purely mathematical consequence and has nothing to do with informetric aspects such as formula (VII. 1) or the so-called Groos droop (see Groos (1967)).

VII.2.2 A "type-token identity" of Chen and Leimkuhler

In our terminology a type-token identity could also be named a source-item identity. In our notation (A = total number of items, T = total number of sources) the Chen and Leimkuhler identity (see Chen and Leimkuhler (1989)) is the following (any log can be used):

l A

+

i2il = l

(VII2)

log A

Chen and Leimkuhler comment on this identity as "approximately true". But, both as a possible exact or approximate identity, we are interested in its explanation, certainly since such a possible identity completely belongs to the informetric theory developed in this book.

Let us suppose Lotkaian informetrics. Theorem II.2.1.2.1 says that for any A > T > 0 and any 1 < a < 2 , there exists a number pm > 1 such that the Lotka function

Further applications in Lotkaian informetrics

f(i) = £

299

(vn-3)

j e [ l , p m ] , satisfies (11.20) ad (11.21), hence that, even in the special case of Lotkaian informetrics (and even restricted to 1 < a < 2) any value A > T > 0 can occur. This shows that (VII.2) is not valid since we can take A as large as we wish, when keeping T fixed. So, informetrically we strongly have to reject (VII.2) since we have a proof of the contrary result even in a special part of informetrics (namely Lotkaian informetrics).

Yet Table 1, p.46 in Chen and Leimkuhler (1989) is remarkable since the 38 IPPs (texts in this case) considered there all have a right hand side for (VII.2) between 0.946 and 1.036! We checked the values for T and A of these 38 texts and we can conclude that, approximately, A pa 20,000 and T ~ 3,500 (magnitude estimation). So we could say that A = |aT with \x sa 6 (a relation that is more or less also true for their Table 2). But A = \xT implies for the left hand side of (VII.2)

1 , logT H logu + logT

= - + -;— V- l Q g^ | i logT

(VII.4)

which takes the value 0.987 for T = 3,500 and \i = 6. Furthermore, in all cases where (x is large and logT S> log(i we have that (VII.4) is close to 1 (= the limit of (VII.4) for [i —> oo and — logn

> oo). Hence the regularity (VII.2) is indeed approximately true due to

mathematical properties of the surface (in coordinates (x,y,z) £ R 3 )

z=*+^ y logy

(VII.5)

300

Power laws in the information production process: Lotkaian informetrics

It is our feeling that further informetric assumptions to prove (VII.2) are not necessary: in Chen and Leimkuhler (1989), SBS arguments (see Subsection 1.3.6), although yielding an approximation of (VII.2), only serve to yield A = yCX with \x large and with log^i < logT and again, the real explanation comes from the mathematical properties of surface (VII.5).

VII.3

PROBABILISTIC

EXPLANATION

OF

THE

RELATIONSHIP BETWEEN CITATION AGE AND JOURNAL PRODUCTIVITY In Wallace (1986) one can read the following hypothesis: "For a given subject literature, the median citation ages of the journals contributing to that literature will vary inversely with the productivity of those journals, where productivity is measured in terms of the number of articles contributed by each journal". The validity of this hypothesis needs verification, not in the least because the hypothesis describes an important property because it would mean that the size of the journal (number of articles) (which is visible at first glance) implies the scientific active life span (in terms of citations to a journal) of a journal, which is not so easy to see. Wallace produces the graph in Fig. VII.2 and comments "that the observed relationship is clearly a systematic one" although a functional relation does not seem to be the case here (an abscissa can have many ordinates).

This graph can be described as follows (Wallace (1986)). The highly productive journals all have relatively low median citation ages, while the journals with high median citation ages are all relatively unproductive. Unproductive journals do not all have high median citation ages (in fact they form a minority!) nor are journals with small median citation ages all highly productive (again: they form a minority in fact!). In fact, what Wallace describes is an area filled with points with increasing density for smaller coordinates and where there is a decreasing upper curve that looks hyperbolic, or, more general, a decreasing curve going from +oo to 0 when the abscissa goes from 0 to +oo. Denoting such a hypothetic curve by y = f(x) we hence are looking for a mathematical explanation of a relation of the form y < f (x) with f of the form as described above.

Further applications in Lotkaian informetrics

Fig. VII.2

301

Journal productivity plotted against journal median citation age.

Reprinted from Wallace (1986), Fig. 3, p. 143. Copyright John Wiley & Sons Limited. Reproduced with permission.

In our study we will replace the abscissa by the "journal mean citation age", leaving open the problem as such, but we are convinced that a solution to the problem where "median" is replaced by "mean" also explains the original problem to a great extent. We also fix the subject, represented as a set of journals, as did Wallace. Considering all mean citation ages of each article in each journal, we can calculate their average: the mean citation age of the field, denoted \x.

Let us now consider the journals in this set that consist of A articles (A e N , fixed). Let us call this the A-subfield of the entire field. This A-subfield again can be considered as

302

Power laws in the information production process: Lotkaian informetrics

consisting of articles (as we did above with the entire field). As above, this yields the mean citation \xA of the A-subfield. The hypothesis of Wallace, rephrased in this terminology, is that |i A is a decreasing function of A. To show that this hypothesis is not necessarily linked with Figure VII.2, and that the regularity in Figure VII.2 can be explained in a noninformetric way, we will assume that \iA is a constant of A, i.e., that p.A does not depend on A. In other words: nA = \i. We assume this for the sake of simplicity but also (and more importantly) because explaining Figure VII.2 with this assumption is the most "spectacular" explanation because, in our arguments, we deny Wallace's hypothesis, but we will nevertheless be able to explain the regularity of Figure VII.2. In the same way we assume that the variances a A are also A-independent: <JA = a 2 , for all A g N .

For each A G N fixed and for each journal with A articles (hence belonging to the Asubfield), we have that this journal's mean citation age is a number which is the mean of a sample of A articles. The Central Limit Theorem (CLT) (see e.g. Blom (1989), Chung (1974), Chow and Teicher (1978)) then yields that this mean belongs to a 100(1 — a)% confidence interval (around p.) of the form

) , CT , VA —1

(VII.6)

where Z(a) is the abscis such that, on the graph of the Gaussian distribution (i.e. the standard normal distribution), the tails, determined by Z(a) and -Z(a) have a total area of a. More concretely, for example, for a = 0.05, a 95% confidence interval is given by

6 ,

g

VA-I In general, the values of Z(a) can be read from the table of the standard normal distribution, which is available in any book on statistics or probability theory.

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303

Expression (VII.6) contains the key of the explanation of the graph in Figure VII.2. Indeed, for each fixed A £ N (i.e. ordinate in Fig.VII.2) we have that the "sample" journals with A articles have a mean citation age between the values given by (VII.6), for 100(1 —a)% sure. The lower A, the larger this confidence interval. From (VII.6), it follows that the deviation from \x to the right is equal to

m = Z(a) , " VA —1

(VII.7)

where m is the abscissa in Figure VII.2. Because A is the ordinate, we will invert (VII.7), yielding

m

or, more simply

A = -% + l, m

(VII.8)

where Ea is a constant, decreasing with a. Equation (VII.8) is the decreasing graph at the right side of Fig. VII.2 (described above as y = f(x)). It turns out to be a decreasing power function with exponent 2, but, as indicated, unrelated with Lotka's function.

The "fading away" effect in Fig. VII.2 has two causes. First of all, in the abscissa, when going from low values of m to high values of m, horizontally at each fixed A, the effect is explained by the different values of a and corresponding probabilities 1 - a. Of course, the left part of (VII.6) is cut off by the requirement that m > 0 (in (VII.7)). Secondly, in the ordinate, when going from low values of A to high values of A, the effect is explained by the law of Lotka on the number of journals with A articles (hence decreasing in A). This is the only informetric part in the explanation!

Note also that from (VII.8) we have that

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Power laws in the information production process: Lotkaian informetrics

lim A = 1

(VII.9)

limA = oo

(VII. 10)

and

all in agreement with the graph in Fig.VII.2. In short, Fig.VII.2, but with "median" replaced by "mean", is explained by high variances of small samples and, conversely, by small variances of large samples, as expressed by (VII.6).

Unrelated to this topic we note that in Egghe (1997) another probabilistic explanation is given for the relation between the Price index (the fraction of references that are not older than a certain fixed age) and the mean (or median) reference age. Lotka's function is not appearing in this explanation (therefore we do not give the argument here) but we can indicate that we are also dealing here with a thick cloud of points (as in the above problem), hence with a relation and not a function. The overall functional tendency is explained by using a decreasing exponential aging distribution (for the number of references to t years ago), having only one parameter but the full relation (the cloud) is explained by replacing the exponential aging distribution by a lognormal aging distribution which contains two parameters (this extra parameter allows for the explanation of the full relation and not just the functional tendency).

VII.4

GENERAL

AND

LOTKAIAN

THEORY

OF

THE

DISTRIBUTION OF AUTHOR RANKS IN MULTIAUTHORED PAPERS VII.4.1 General theory

It is clear that, if there are m authors in a paper, that each of the m author ranks is occupied, hence

that

the

distribution

of

author

ranks

is

a

uniform

distribution,

being

p(k|m) = —, k = l,...,m. If one is interested in the overall author rank distribution, m

Further applications in Lotkaian informetrics

305

incorporating a variable number of authors per paper, the distribution of author ranks is not uniform anymore since an author rank m cannot occur in k-authored papers with k < m.

We now wonder what the author ranking distribution is. It is clear that the distribution v|/ = v|/(m) of the papers with m authors ( y is the same as the function studied in Subsection VI.3.3) is involved here. Indeed, we have the following easy Proposition (all the results in this section are proved in Egghe, Liang and Rousseau (2003)).

Proposition VII.4.1.1: The author ranking distribution, p(k), k = 1,2,..., is given by

P

(k)=E^

(vim)

Proof: This result immediately follows from the theorem of total probability (see e.g. Grimmett and Stirzaker (1985) or Feldman and Fox (1991)). Indeed:

p(k) = £p(k|m)H/(m) = f ; ^ m=k

m=k

(VII.12) m

where p(k|m) denotes the conditional probability of rank k, given that there are m > k authors. As explained above p (k | m) = — for all k = 1,... ,m, the uniform distribution on the m set {l,...,m} . Note that ^ p ( k ) = 1, as is easily seen, using that y is a distribution. k=l

Of course, expression (VII. 11) is hard to evaluate since a discrete infinite sum is appearing. We can, however, proceed as follows.

Definition VII.4.1.2 (see e.g. Apostol (1957)): Consider two sequences (a(n))

and (b(n))

. One writes that

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Power laws in the information production process: Lotkaian informetrics

a(n)= O(b(n))

(VII. 13)

if there exist numbers n0 and C such that, for all n 3 n 0 :

a(n)|£ C|b(n)|.

Intuitively, this means that the sequence (a(n)). (b(n)).

(VII. 14)

does not grow faster than the sequence

and, in the case that limb(n)= 0 (which will be the case in our application), that

nl ¥

the sequence (a(n)).

n® ¥

ressembles the sequence (b(n)).

in the sense of variability in n

(and, of course, then also lim a (n)= 0). The O-notation is originally due to E. Landau. n®¥

We have the following result. Proposition VH.4.1.3: If the distribution of numbers of authors per article, y(m), is given by a Lotka distribution (cf. Subsection 1.4.4), then the author ranking distribution, p(m), is related to \|/(m) by

p(m)= 0 ( v ( m ) )

Proof: Let \\i be the Lotka distribution

V(m)= -^r

with a > 0 then, by the above proposition, ¥

p(m)= a -

(VII.15)

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307

Hence, by the integral test (cf. Apostol (1957) or Protter and Morrey (1977)) and the fact that * the series a

c

n=i n a +

r *s convergent (since a > 0), we have:

o n= m

c

n

v

CQX _

A

c

m

So,

or

p(m)£

JL|L + 1& ^SL + 1?=

v(rn|L+

This proves the theorem. If we adopt the continuous model of author densities, as in Subsection VI.3.2, we have the following result. Proposition VII.4.1.4: In the continuous setting, the author ranking density distribution p(k), k I | + , is given by

p(k)= 6 ^ ^ d x

where\|/ is the continuous size-frequency distribution of Subsection VI.3.2.

Proof:

(VII.16)

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Power laws in the information production process: Lotkaian informetrics

Proof: The theorem of total probability is also valid for density functions (see e.g. Grimmett and Stirzaker (1985)) so that

p(k) = J k °°p(k|x) V (x)dx

where p ( k | x ) is the conditional density function for continuous rank k, given we can pick ranks in the interval [l,x] (ranks are > 1 since ranks are item densities and all item densities are > 1, see e.g. formula (11.10)). Hence p(k | x) =

and (VI.16) follows. Note that X

1

riP(k)dk=i as it should and as follows easily from an application of Fubini's theorem (see Apostol (1957) or Protter and Morrey (1977)) and the fact that vj; is a distribution.

If \j/(x) = —— = — — , X

the continuous Lotka distribution (c = a —1

since i|/ is a

X

distribution), (VII. 16) has the form

p

«=/r(£r^ dx

(VIL17)

which we cannot evaluate, let alone relate it to \\i as was done in the discrete case (Proposition VII.4.1.3).

VII.4.2 Modelling the author rank distribution using seeds

Assume that each author, A, has a characteristic number sA £ [0,l], where sA is equal to the probability that an other author comes before A in the byline of an article. This characteristic number will be called a 'seed'.

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309

We will next solve the problem of determining p(k,s): the probability for an author with seed number s to be the kth author (in general); or more specific p(k,s | m ) : the probability of an author with seed s to be kth author in a publication with m authors. In this connection we have the following result.

Proposition VII.4.2.1:

p(kjS|m) = [ m _" i 1 js k - 1 (l-sr k

(VII.18)

p(k,s) = gfm"1 ^ ( l - f V H

(VII.19)

where v|/(m) is the probability that a paper has m authors.

Proof: Consider an author A, with seed s. Hence

P(an author is before A in an author list) = s P(an author is after A in an author list) = 1 - s

Author A has rank k in an article with m authors, m being at least equal to k, if and only if k — 1 authors precede A, and m - k follow A. We can describe this as follows. As the article has m authors, this means that m - 1 co-authors are choosen at random. They end up before A with probability s (we refer to this as 'success' in a Bernoulli trial). So author A ends up at rank k if there are k - 1 successes (and consequently m - k 'failures'). This shows that the situation can be described by a binomial distribution.

p(k,8|m) = [°_- 1 )."(l-.r

310

Power laws in the information production process: Lotkaian informetrics

The second formula, where the total number of authors of an article is not given, follows by the law of total probability:

p(k,s) = £ p ( k , s | m ) f ( m ) m=k

=£(" 1 " 1 1 ] sk " 1 ( l - s r^H

n

m=k( l K ~ 1 )

These formulae, based on seeds, are very important since it is relatively easy to determine seeds for authors, e.g. based on alphabetic ranking of authors' names. This is explained in the next subsection.

VII.4.3 Finding a seed based on alphabetical ranking of authors

In this section we introduce a method of finding a seed for an author. First we will introduce an injection between an author's name and a number in the set [0, l] D Q.

We will work with the standard western alphabet, consisting of 26 letters, but the method applies to any other alphabet or another lexicographical ordering consisting of symbols with a fixed rank. We add a 0 (zero) symbol to the alphabet, so that we have an alphabet of 27 symbols. Let S denote the set of all concatenations of a finite or infinite number of symbols. Then S r ..S n e S (a concatenation of a finite number of non-zero symbols) represents an arbitrary name. The injection

f : S -* [0,1]

is defined as:

f(S,S2...Sn) = ^ ^ 4

(VII.20)

where |Sf| denotes the rank of the symbol Sj. An equivalent way of defining the function f is:

Further applications in Lotkaian informetrics

311

i \tixz>1...s>n) — u. s,|...p n |

where O.|S,|...|Sn| denotes a number in the 27-ary number system. For clarity's sake each number SJ must be expressed by two digits, otherwise a 1 followed by a 2 could be confused with 12. Hence 1 must be written as 01, 2 as 02, and so on.

Examples: 1.

f(A) = 0.|A| = — = 27"' V ; I I 27

2.

f(AZZZZZZZZZZZ....) =

- + £ -

=-

L

+ - ^ ^ = - + - = - = a|B| = f(B) 27 j

This second example shows that, in the same way as we identify 0.19999... with 0 . 2 e R , AZZZZZZ... is identified with B.

3.

f(zzzz....) = V — = I-I

l l

,26 . = i . 27 1 - —

{

4.

Note that f (000...0 * 000....) =

21) rankf*) M-, where * denotes any symbol from the alphabet,

placed after (i — l) zeros. Of course, this symbol does not represent a real name. Clearly, the limit of this expression, for i —> oo, is zero.

5.

The image of any real name belongs to [0, l] n Q.

The function f, restricted to 'real names', (the subset of finite symbols of the form S,...SN) is an injection. Indeed, let

312

Power laws in the information production process: Lotkaian informetrics

f(S,S 2 ...S n ) = f(T 1 T 2 ...T m )

Hence,

i=l

Z /

j=lZ

/

Assume now that S,...Sn ^ T,...Tm (this is: assume that f is not an injection). Let k e |l,...,min(m,n)j be the first rank for which Sk ^ T k . There is no loss in generality in assuming that |Sk | > Tk + 1 . Hence

Is I

TI

1

^4-^4 >-*27k

(V11.21)

27" ~ 27k

But, we always have that n k 00 T£ T/; m IT I ynl_vn_
j=k+l

77 L

I

97

j=k+l ^ '

77

j=k+l ^ '

97

^'

+

|

1 1 L_ = _L 1 k

L

9l7

'

omm ^ '

27 which is in contradiction with (VII.21) and the fact that f (S,S2...Sn) = f (T,T2...Tm). Hence the function f is an injection on the subset of'real names'.

How can the actual occurrence of letters be taken into account? One suggestion is to use a telephone directory. This suggestion, however, only works for scientists 'speaking' the same language, because the use of letters differs in different languages. Otherwise one needs an 'international' directory (perhaps that of a city such as New York or Los Angeles). Then a name listed on page 345 of the 1,557 (this is just an example) would get a seed equal to 345/1,557 = 0.22158. Further refinements (using lines within a page) are possible. Averaging would be necessary for popular names.

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313

Having determined a seed for an author, Proposition VII.4.2.1 then gives the probabilities for such an author to be at rank k. So these distributions can be tested by using a group of scientists with the same surname (hence the same seed) in a field where alphabetic ranking of authors is customary. Such is generally the case for pure mathematicians, logicians, statisticians and theoretical physicists (see Harsanyi (1993)).

This model is valid in all cases where a seed can be given, not just in the case of alphabetical name ordering. Indeed, there exist many ways and conventions for ranking co-authors (see Harsanyi (1993), Hoen, Walvoort and Overbeke (1998) or Feger (2000)). A seed can, e.g., be derived from the importance of the author. Indeed, assume that author ranking always occurs according to 'importance'. 'Importance' could then be calculated from the number of publications, the number of citations, or even the age of scientists (Liang, Kretschmer, Guo and Beaver (2001)).

Another suggestion is to calculate a seed from 'older' publications (calculating an average rank) and to use this to 'predict' the author rank distribution in 'newer' ones.

Note that a seed is always a number in the interval [0,l], so that observations must always be transformed to the unit interval in order to obtain a seed.

In the next subsection we will discuss an application of Lotka's function to the cumulative distribution of first-citations in a set of articles. It is one of the most intriguing applications of Lotka's law and, what is more, this application of Lotka's law also shows its explanatory value for the different forms of such a first-citation distribution, again showing that Lotka's power law is at the basis of many informetric regularities.

VII.5

THE FIRST-CITATION DISTRIBUTION IN LOTKAIAN INFORMETRICS

VII.5.1 Introduction

The fact that an article receives citations is an indication of its visibility and its use by other researchers. The importance of citation analysis is apparent from the many articles on this

314

Power laws in the information production process: Lotkaian informetrics

subject, e.g. in the journals Scientometrics, Research Policy,... . It is indeed so that this visibility (in terms of the articles of a research group, in a certain time period) is measured using citations and that these quantitative data are used by policy makers in research for the determination of the degree of subsidizing this research. Important research tools in this matter are the citation indexes produced by the Institute for Scientific Information (ISI), founded by E. Garfield. They can be used, e.g., to determine the times at which an article receives its citations and hence one can also determine, for each article in a field, the time of the first citation to an article. Over the whole field on can then talk about the first-citation distribution which (in its cumulative form) is the topic of this section. The first-citation distribution is the cumulative distribution of the time period between publication of an article (in a field) and the time it receives its first citation.

Knowing first-citation times is very important since it measures "response times" and the speed of scientific communication in a field. The first-citation time t[ is also very important for an article since, at this time, the article shifts its status from "unused" to "used" and the smaller t, is, the more we can say - in general - that the article under study is important and early visible in the scientific world. One could also say (with Moed and Van Raan (1986) and Schubert and Glanzel (1986)) that t, is a measure of immediacy, however not directly related to the immediacy index, defined by ISI in their Journal Citation Reports (JCR®) - see also Egghe and Rousseau (1990a), where also general aspects of citation analysis are treated.

The reason why we will deal with cumulative first-citation distributions is that they are easier to deal with (yielding also easier formulae - see further), it is more common to do so (cf. Rousseau (1994a) and Gupta and Rousseau (1999)) and, finally, cumulative processes are more important here since their limiting supremum value is the fraction y of eventually cited articles, hence 1 — y is the fraction of the articles that are never cited, an important indicator.

Although the first-citation distribution is important, there are not many papers involved in the study of this distribution. In Glanzel (1992) and Glanzel and Schoepflin (1995) one studies ith Harmonic Mean Response Times, being the harmonic means of the time elapsed between the publication date and the date of the i' (i = 1,2,...) citation of the papers. Of course, i =1 represents first-citation. Basically there is only Rousseau (1994a) who develops a model for the first-citation distribution. His arguments are based on the definition of two differential

Further applications in Lotkaian informetrics 315 equations leading to two different models (the one not implied by the other). One model is based on the differential equation

©'(t^Be-^l-d^t,))

(VII.23)

where ^ ( t , ) is the cumulative first-citation distribution and B and p are constants, leading to

(VII.24)

where k and b are other constants.

This model fits well in situations where the first-citation data are concave in t, as is, e.g., the case for the data (derived by Rousseau) appearing in Motylev (1981) on references in the Russian scientific literature to Russian language library science periodicals. But, as is easy to see, model (VII.24) can only handle concave cases since R " ( t , ) < 0 always. There exist, however, cases where the first-citation process is S-shaped, i.e. starts in a convex way, then proceeds in a concave way, both parts being separated by an inflection point. In fact, Rousseau himself points this out by collecting data on first citations of JACS articles in JACS (JACS = Journal of the American Chemical Society). This resulted in an S-shaped cloud of points (see Figs. 2 and 3 in Rousseau (1994a) or see further Fig.VII.4 where the data have been re-used) for which (VII.24) is unsuited. This leads Rousseau to develop a second model based on the differential equation.

cD1(t1) = Be-pt'(l-
(VII.25)

leading to

O(t,) =

L 1+

, pt

MBe- '

(VII.26)

316

Power laws in the information production process: Lotkaian informetrics

where M, b and p are constants. This function is capable of fitting the S-shaped cases very well, as can be seen in Rousseau (1994a).

We want to make the following remarks on these models. The work of Rousseau has some explanatory value since it is derived (in a mathematical way) from differential equations (VII.23) and (VII.25). This technique is custom in sciences like physics and chemistry, giving indications about the dynamics of first-citation processes but often these equations are only expressions of widely accepted and experimentally verified properties. In equation (VII.23), the first-citation rate is proportional with the fraction of the (at that time) yet uncited articles, but with proportionality factor decreasing with t,. In (VII.25), this rate is multiplied by R (t,) itself, for which a rationale seems to be missing.

The most important drawback of Rousseau's results is that neither of the two models is capable of modelling all existing first-citation relations: the first one is needed to model the concave cases while the second one is needed to model the S-shaped cases. This leads to two different rationales (coming from the different differential equations (VII.23) and (VII.25)) for the two types of first-citation relations. Not that this is wrong in itself but in this way we lack the rationale for this different behavior.

Last but not least, one can wonder if the observed first-citation regularities cannot be explained by using elementary informetric tools. For citation times we have the well-known aging distributions, the simplest ones being the exponential aging distribution, see e.g. Egghe and Rao (1992a) or Subsection 1.3.4.2 and for number of citations we can use a sizefrequency function of the power type (i.e. Lotka). This will be done in the next subsection: in other words, we will put the theory of first-citation distributions within Lotkaian informetrics. There we will also see that Lotka's exponent a plays a crucial role in the explanation of the different shapes (which happen to be concave or S-shaped) of the first-citation distribution.

In Subsection VII.5.3 we will then make explicite calculations of the model for the existing data-sets, i.e. the ones that were already used in Rousseau (1994a) in the test of his models (VII.24) and (VII.26).

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317

In Subsection VII.5.4 we will extend the first-citation distribution to the n th (n = 2,3,- ) citation distribution and to general aging distributions.

VII.5.2 Derivation of the model

Let

c(t) = ba'

(VII.27)

(b > 0, 0 < a < 1, a,b : constants and t > 0) be the probability density function of citations to an article, t time after its publication. Since c is a probability density function we have

J

f°°c(t)dt = — = 1 ° lna

(VII.28)

(since 0 < a < 1), hence b = — lna > 0. If an individual article in the bibliography (or field) receives M citations in total we have that

n(t) = -(lna)Ma'

(VII.29)

describes the number of citations, t time after publication, to this article. The distribution of the M-values will be taken as Lotkaian

(VII.30)

for Me[l,+oo[, hence with maximal item density pm = oo as in Subsection II.2.1.1. We will take a > 1. This implies that, since cp is a probability density function,

r°(p(M)dM= f°°—dM = - 5 - = l. Ji v v ' Ji Ma a-1 (VII.31) in (VII.30) gives

(VII.31)

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Power laws in the information production process: Lotkaian informetrics

cp(M) = ^ i

(VII.32)

Note that we do not only cover the case of Proposition II.2.1.1.1 ( a > 2 , hence A < oo) but also the case 1 < a < 2 where we have A = oo but T < oo, hence |i = — = oo; this is no objection for this model: only a > 1 is needed in order to have the existence of (p as in (VII.32). We have the following remarkable result.

Theorem VII.5.2.1 (Egghe (2000c)): If we have an exponential aging function as in (VII.29) (supposed to be the same for each article) and a Lotka size-frequency distribution as in (VII.32), then we have that

(VII.33)

where Ct> is the cumulative first-citation distribution (over all articles), t, > 0 and y is the fraction of ever cited articles.

Proof: Let the article have M citations in total. Then the time t, at which this article receives its first citation is defined by

Pn(t)dt = l. t/ 0

This yields:

M = —*— 1 —a'

(VII.34)

Since (p(M) is the fraction (among the at least once cited articles) of those articles with M citations, y
Further applications in Lotkaian informetrics

319

articles with M (given by (VII.34)) or more citations will have their first citation at t < t, (by (VII.29)). Hence the cumulative first-citation distribution, ( t>(t l ), is given by (using (VII.32))

(VII.35)

with M replaced as in (VII.34). This yields, since a > 1:

O»(t1) = y(l-a t ') a " 1 . We have the following important property of <J>.

Theorem VII.5.2.2 (Egghe (2000c)): The cumulative first-citation distribution O is concave if and only if 1 < a < 2 and is Sshaped (i.e. first convex, then concave) if and only if a > 2 .

Proof: It is easy to see that 0 for all t, > 0 and that ®" (t,) = 0 if and only if

'4—) t =

l » l> lna

(VII.36)

and that 2 then the value in (VII.36) is positive. Any value of t, below this yields O" > 0 and any value of t, above this yields (0) = 0 and

lim©(t,) = y

as it should (since we consider © with respect to all (also the not cited) articles).

(VII.37)

320

Power laws in the information production process: Lotkaian informetrics

That an S-shaped curve occurs only for large values of a and that the S-shape becomes more and more apparent the larger a is (as is clear from (VII.36)) is intuitively clear. The higher a the more inequality we have (in this case between the different total number of citations per article) - see Corollary IV.3.2.1.5, so, relatively speaking, there are fewer articles with a large number M of citations. In other words, using (VII.34), there are not many cases of low values of t, and hence the first-citation process starts in a convex way, meaning that increases very slowly. This remark makes clear the involvement of Lotka's a in the study of the firstcitation process, besides the aging rate a, whose involvement in the first-citation process is much more evident.

The model (VII.33) is easy to use in practical examples. We will try to find the parameters in (VII.33) that are best in the sense of nonlinear regression, given a set of data. We can estimate y from the data, hence using (VII.33) as a 2-parameter model (in a and a ) . Of course, we can also consider (VII.33) as a 3-parameter model where also y is determined by the optimalization process.

VII.5.3 Testing of the model

We will present two examples: one where the empirical data imply that O is entirely concave and one where the empirical data imply that ct> is S-shaped. This way we can perfectly test all aspects of the model.

VII. 5.3.1 First Example: Motylev (1981) data

These citation data concern references in the Russian scientific literature to Russian language library science periodicals, published by Motylev (1981). The data were re-used in Rousseau (1994a). We will use them here for testing our model. The data are presented in Table VII.1.

Note that in this case only about 30% of all articles is cited. Here a substantial improvement occurred when using a 3-parameter fit above a 2-parameter fit. We obtained a = 0.956, a = 1.746, y = 0.486. The latter value is far above 0.303 which is clear by visual inspection of the graph: after 16 years the graph is far from being horizontal; hence adding further years

Further applications in Lotkaian informetrics

321

will result in more cited papers (increasing the low number of 30%!). The calculated model is shown in Fig. VII.3. The fit is very good, giving an R2 (in nonlinear regression) of over 0.97.

Table VII. 1

First-citation data of Motylev.

Reprinted from Egghe (2000c), Table 2, p. 353. Reproduced with kind permission of Springer Science and Business Media.

Year

Number of articles

Fraction of

Cumulative fraction

cited for the first time

column 2

of first-citation

10 33 18 17 14 4 0 10 7 7 8 9 4 13 6 6

0.018 0.060 0.033 0.031 0.026 0.007 0.000 0.018 0.013 0.013 0.015 0.016 0.007 0.024 0.011 0.001

0.018 0.078 0.111 0.142 0.168 0.175 0.175 0.193 0.206 0.219 0.234 0.250 0.257 0.281 0.292 0.303

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fig. VII.3

Three-parameter fitting of the Motylev data.

Reprinted from Egghe (2000c), Fig. 2, p. 354. Reproduced with kind permission of Springer Science and Business Media.

322

Power laws in the information production process: Lotkaian informetrics

VII.5.3.2 Second Example: JACS to JACS data of Rousseau

These data were collected by Rousseau in JACS (Journal of the American Chemical Society) of the year 1975. The first citations to these articles are followed in JACS during a 4-year period (in biweekly issues). Since the table is long (see Rousseau (1994a) or Egghe (2000c)) we do not present it here. It follows from this table that, after 4 years, 67.6% of the articles are cited. Hence we use y = 0.676 in a 2-parameter fitting (since the 3-parameter fitting was very similar to the 2-parameter one, due to the fact that the data are more complete than in the previous example: for large t,, the graph is nearly horizontal). We have a = 0.955, a = 3.641 with an R (in nonlinear regression) of 0.999. See Fig.VII.4 where the S-shape is clear and the model fits very well.

Note that the high value of a = 3.641 is typical in the sciences (genetics, chemistry,...) and is higher than in e.g. library sciences as considered in the previous example (there a = 1.746). Note (Chapter IV, V) that high values of a imply high levels of concentration (inequality) and of fractal complexity. This is another way of interpreting "hard" sciences!

Fig. VII.4

Two-parameter fitting of the Rousseau data.

Reprinted from Egghe (2000c), Fig. 3, p. 355. Reproduced with kind permission of Springer Science and Business Media.

Further applications in Lotkaian informetrics

323

VII.5.4 Extensions of the first-citation model

If we look at the exponential aging distribution (VII.27), hence with b = — In a (see (VII.28)) we see that its cumulative form equals:

C(t)= ( " ( - i n a V ' d t ^ l - a '

(VII.38)

0

and hence the cumulative first-citation distribution
«p(t1) = r(C(t1))°"1

(VII.39)

Denote = —, which is the cumulative first-citation distribution, where one only considers Y the population of ever cited articles. Note that now, because of (VII.37) we have lim
We now have

O-(t,) = ( C ( t 1 ) r

(VII.40)

This result will be extended to general aging distributions C and to *, the cumulative nth citation distribution, as follows.

Theorem VII.5.4.1 (Egghe and Rao (2001)): Let <£>*n be the cumulative nth citation distribution, i.e. the distribution of times when an article receives its nth citation and where we only consider the articles that, eventually, receive n citations. Let C be the cumulative aging distribution, i.e. the distribution of the times of the citations to an article, supposed to be the same for all articles. Let the distribution of the number of citations received be given by (VII.30) (Lotkaian). Then we have

o;(t) = (C(t))""1

(VII.41)

324

Power laws in the information production process: Lotkaian informetrics

Proof: The nth citation is received if (M = total number of citations to an article)

MC(t) = n

(VII.42)

and at that time all articles with M' > M citations evidently (since M'C(t) > n) have at least n citations, hence received their nth citation earlier. Its fraction (among the at least once cited articles) is

JJ°
1

= / — - d x = M'-a JM

(VII.43)

X°

Hence (VII.42) and (VII.43) imply that

Cft) " ' —^ n

(VII.44)

is the cumulative nth citation distribution, conditional with respect to the set of at least once cited articles (because we do not use y here as we did in Theorem VII.5.2.1). For obtaining the distribution
It suffices to calculate

lim -i^ t-= n J

=JL n

(VH.45)

Hence (VII.44), divided by (VII.45) is the cumulative nth citation distribution, conditional with respect to the set of (eventually) n times cited articles. So

Further applications in Lotkaian informetrics

a>;(t) = (c(t))-\

325

n

Note that O | = <£>*, by notation. We have the following remarkable corollary.

Corollary VII.5.4.2: 1.

<£'n = * ; for all n e N

2.

If a = 2 then O* = C, i.e. the cumulative nth citation distribution equals the cumulative citation distribution.

Proof: This follows readily from the previous theorem. The previous results allow researchers to check the nth citation models for other than exponential aging functions. This was done in Egghe and Rao (2001) using the lognormal distribution as aging function c. The results are good but not better than the ones of Egghe (2000c) and the calculations are, of course, more intricate (note also that the lognormal distribution has 2 parameters while the exponential one has only 1; it is always better to use as few parameters as possible, if good fits can be obtained).

In Egghe and Rao (2002b) the same study was done but where one does not consider firstcitations but most-recent-references, the dual aspect. The advantage of this is that it is more easy to collect data: one can collect reference data from each journal article while for the firstcitation data, the citation indexes of ISI are needed. In addition, following earlier work of Stinson (1981) and Stinson and Lancaster (1987), one can expect the same results when studying references (synchronous study) as when studying citations (diachronous study).

These assumptions were tried out in Egghe and Rao (2002b) and confirmed: most-recentreference distributions, using (VII.33) (but with y = 1) and using c as a synchronous aging distribution were tested on most-recent-reference data from JASIS (Journal of the American Society for Information Science) and from JACS (Journal of the American Chemical Society): the results are similar to the ones in Egghe (2000c), also with very good fits of the obtained models.

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Power laws in the information production process: Lotkaian informetrics

VII.6

ZIPFIAN THEORY OF N-GRAMS AND OF N-WORD PHRASES: THE CARTESIAN PRODUCT OF IPPs

VII.6.1 N-grams and N-word phrases

N-grams are strings of N letters; N-word phrases are strings of N words. In this sense both objects can be compared: they belong to an N-fold Cartesian product of the space of letters (for N-grams) and to the N-fold Cartesian product of the space of words (for N-word phrases). Symbolically, N-grams and N-word phrases can be symbolized as a vector (a,,a 2 ,...,a N ) where each a; is an arbitrary letter or an arbitrary word.

It is clear that N-word phrases are very important: they form the basis of linguistical expression and allow for more complex ideas than the single words on their own. This, in turn, implies that their use in information retrieval (IR) is basic in the refinement of searches: if the 2-word phrase "electric vehicle" is used in a query (supposing truncation at the end of each word) then this query is superior to a Boolean input "electric AND vehicle" in which one not only receives information on electric vehicles (as wanted) but also on electricity in vehicles which is a totally different subject and which, most probably, gives the majority of the retrieved articles. Hence, pre-coordinative indexing techniques for such N-word phrases (concepts) and post-coordinative retrieval techniques with proximity operators are basic in any literature database construction (see e.g. Salton and Me Gill (1987)).

N-grams are less known in concept and in their use. Let us indicate some elements here. Ngrams can be generated from words or, more generally, from texts (i.e. consisting of several words). They are generated (from a word) as follows. Let us consider the word SUBSTRING. From this word on can generate the following 2-grams:

*S, SU, UB, BS, ST, TR, RI, IN, NG, G*

or the following 3-grams:

**S, *SU, SUB, UBS, BST, STR, TRI, RIN, ING, NG*, G**

Further applications in Lotkaian informetrics

121

These are examples of "redundant" N-grams (Willett (1979)) in which one shifts from left to right with 1 symbol and where one completes the beginning and ending N-grams. These can be compared with non-redundant N-grams (e.g. for N = 3: SUB STR ING). The advantage of redundant N-grams over non-redundant N-grams is that all letters in a word (or text) appear in every place i = 1,...,N of an N-gram and hence that the overall occurrence of letters in redundant N-grams is independent of the given place ie{l,...,N} (this is also why, in the beginning and at the end, one makes use of the asterisk *). Since this will be a crucial property in our models (to be developed in this section) it is clear that we will, henceforth, only consider redundant N-grams. Also in applications (which we will describe now) redundant N-grams are the most important. We refer to Cohen (1995), Damashek (1995), Robertson and Willett (1998), Grossman and Frieder (1998) (Section 3.4), Yannakoudakis, Tsomokos and Hutton (1990) and Nelson and Downie (2001) for a more elaborate description of the applications on N-grams, which we will briefly indicate now.

The basic idea behind N-grams is that - for a word or a text - one creates a sequence of Ngrams which contains more information than each word in itself: e.g. similar words (e.g. woman, women) are different but their N-gram pattern is very similar. This similarity can be expressed in different ways e.g. by constructing, for each word or text, a vector in which each coordinate is linked with one N-gram (hence the length of this vector equals the total number of different N-grams) and the coordinate value equals the total number of times this N-gram occurs in the word or the text. In this way two words (or texts) can be compared using vector similarity measures such as (e.g.) the cosine measure of Salton (see e.g. Salton and Me Gill (1987), Blair (1990)). This gives the following opportunities of application.

1.

In indexing: documents can be clustered in the indexing process by using a similarity measure for their titles (clustering is done using multivariate analysis techniques - see e.g. Egghe and Rousseau (1990a)). Also, from the N-gram pattern of a document one can automatically generate a series of key words ("highlights", "stems") for indexing purposes. This technique generalizes stemming using truncation (e.g. "compression" and "textcompression" will be similar by using N-grams, not by using truncation).

2.

In IR: via given key words one generates similar key words using the N-gram technique, with which one can expand the query. Also this can be considered as a generalization of

328

Power laws in the information production process: Lotkaian informetrics truncation. N-grams (or more general search keys) can also be used in catalographic retrieval - see Subsection III. 1.3.2.

3.

Error detection and correction: if there is an error in a word, its N-gram pattern is very similar to the one of the correct word and hence the system can propose a correction. This is also applied for the correction of surnames.

4.

Compression of texts: unused byte-codes, reserved for 1-grams, can be linked with frequently occurring N-grams (N > l).

5.

Identification of languages: different languages have different N-gram patterns.

6.

Authorship determination: different authors have characteristic N-gram patterns in their texts.

7.

Classification of subjects: as for languages and authors, different subjects have characteristic N-gram patterns.

8.

Speech recognition: here one uses N-grams of words (i.e. N-word phrases) instead of Ngrams of letters.

9.

Indexation and retrieval of music: in monophone music one delimits intervals indicating the change (up or down) of one tone to another one. N-grams are formed by these intervals (instead of letters) as building blocks.

N-grams are extremely useful (but not exclusively!) in Asian languages because of their special structure (truncation is not so efficient there). Note also that all these techniques are language or subject independent (in contrast with many indexing and IR-techniques).

This should convince the reader that N-grams and N-word phrases are important tools in information science. Hence their informetric properties should be revealed. It is, however, clear that N-grams and N-word phrases consist of N objects each of which also have informetric properties. Indeed N-word phrases consist of N words and the informetric (linguistic) properties are covered by Zipf s law, hence, by Section II.4, are of Lotkaian nature. We will show in Subsection VII.6.3 that letters, building N-grams, can also be supposed to satisfy Zipf s law certainly in cases where there are many letters such as is the case in Asian languages. We can now wonder if, from Zipf s law for words or letters, we can deduce a rank-frequency function for N-word phrases and N-grams, by interpreting them as "Cartesian products" of the single objects. This will also be done in Subsection VII.6.3, while in Subsection VII.6.4 we will deduce the size-frequency function derived from this rank-

Further applications in Lotkaian informetrics

329

frequency function (using the general relations in Chapter II). This size-frequency function will then, in Subsection VII.6.5, be used to calculate the average number jiN of N-gram or Nword phrase occurrence per N-gram (N-word phrase) type as well as its Type/Token-Taken variant (x^ (cf. Chapter III) and a comparison between JI N and \iN for various values of N and Lotka's exponent a (being the size-frequency function for single letters or words) is made. In the next Subsection VII.6.2 we will present an argument for 2-word phrases, based on Mandelbrot's argument for words in random texts: i.e. we will extend the argument in Subsection 1.3.5 to 2-word phrases. This has interest in itself but will also yield material to compare with the results of Subsection VII.6.3 which will be far more general. Since the arguments of VII.6.3 also do not depend on those of Subsection VII.6.2, the reader can skip VII.6.2 without any problem. We have added VII.6.2 since it extends Mandelbrot's argument and since the result supports the one obtained in Subsection VII.6.3.

VII.6.2 Extension of the argument of Mandelbrot to 2-word phrases

We will use the same tools and notation as in Subsection 1.3.5. The results are from Egghe (1999a). We suppose we have an alphabet of N letters, which have an equal chance of occurrence. Words are formed with these letters and these words are separated by a blank. We put p as the probability for a letter to occur. Since there are also blanks, we have that (cf. 1.3.5) p < —. The probability for having a 2-word phrase consisting of k letters (in total) N hence is

P2(k) = p k ( l - N p ) 2

(VII.46)

(1 — Np is the probability to have a blank). Since this is decreasing in k we can estimate the rank r of this word as follows (cf. the similar argument given in Subsection 1.3.5)

N 2 + 2N3 +... + ( k - 2 ) N k " ' < r < N2 + 2N3 +... + ( k - 1 ) N k

(VII.47)

330

Power laws in the information production process: Lotkaian informetrics

(since, for every i, there are (i — l) N' possible 2-word phrases consisting of i letters in total the blank can indeed be at i —1 places). A long calculation, but only involving sums of geometric series yields

N + 2N 2 +... + iNi = — ^ - ^ ( i N i + 1 - ( i + l)N' + l) ; (N-l) V

(VII.48)

for all i. Applied in (VII.47) this yields

(k - 2) N k "' - (k - 1 ) Nk~2 < (^—H r - 1 < (k - 1 ) N k - kN k -'.

(VII.49)

To fix r, we will take the average of both sides:

p z i j r_iSiI[(k-2)Nk-1-(k-l)Nk-2+(k-l)Nk-kNk-1]

= lN"[(NJ-l)(k-l)-2N

2izlJ r - l « i N k - 2 ( N 2 - l ) ( k - l ) ;

(VII.50)

for large r (hence k). This is the (average) rank of 2-word phrases consisting of k letters in total. From (VII.46) it follows that

P 2 (r) = p k ( l - N p ) 2 = p k - 1 p ( l - N p ) 2 .

Hence

Further applications in Lotkaian informetrics

p(l-Np) V k-l = — ^ J—. lnp

331

(VII.51)

Formula (VII.51) in (VII.50) yields

lnNl

A

InN

P2(r) "P Pa(r) '" 2 (p(l-Np) ' p(l-Np)2 P

=

2NlnN fN-lf N2-l I N J

} 2(lnN)(N-l)r j N(N + 1)

for large r. So

InN

P Cri ,, 2\ ,2 p(l-Np)

lnp

=4(rr)

(VII.53)

with

2(lnN)(N-l) N(N + 1) '

a fixed number, and where \ is the inverse of the function x —> x lnx . We finally have

P2(r) =

7^V

where

M = p(l-Np) 2

(VIL54)

332

Power laws in the information production process: Lotkaian informetrics

and

P= - Y ^ lnN

(VH.55)

Note that we refind formula (1.58) for p being — where Ds is the fractal dimension of the text consisting of single words (see also formulae (V.I7) and (V.I8)). It is not easy to express the fractal dimension of the text, consisting of 2-word phrases since (VII.54) is not a power law: (3 acts on 4(yr), where \ is the inverse of the function x —> xlnx . This is left as an open problem.

The above argument has been given for historical reasons and since we will be able to compare the result with the ones in Subsection VII.6.3, but the argument above has many pitfalls:

(i)

it is only valid for N = 2 (i.e. 2-word phrases) although an extension to general N is feasible, but intricate,

(ii)

it only works for N-word phrases, not N-grams,

(iii)

the argument uses many approximations (for large ranks r),

(iv)

it supposes that all letters have the same probability,

(v)

it supposes that all N-word phrases occur and (because of (iv)) in equal quantities,

(vi)

it supposes that letters follow independently in words.

So the above cannot be considered as a definitive treatment of the rank-distribution of Ngrams and N-word phrases. In the next section we will give a general treatment where the above pitfalls do not occur anymore (except for (vi): independence will still be supposed and this is to be considered as a simplification of an otherwise too intricate problem).

Further applications in Lotkaian informetrics VII.6.3

333

The rank-frequency function of N-grams and N-word phrases based on ZipPs law for N = 1

In this subsection we will determine the rank-frequency function for general N-grams and Nword phrases (N = 2,3,4,...) given the rank-frequency function for letters (i.e. 1-grams) and for words (i.e. 1-word phrases). In this formulation the pitfalls (i), (ii) of the previous argument (Subsection VII.6.2) are removed. Also (iii) will not be a problem since we will derive exact results (no approximations). Of course, as is always done in the theories of this book, we will work with the continuous framework (Chapter II).

As to (iv) and for N-word phrases, it is clear that we can suppose Zipf s law for single-word occurrences. The reasons have been explained before: Zipf s law is a power law, a very common distribution in linguistics (Chapter I) and is fully a part of Lotkaian informetrics (Section II.4), because of Subsection II.4.2. Note that the Zipf function to use has the form (denoting it g, instead of g to express that it is the rank-frequency function for words (i.e. 1word phrases):

S W

' =(I77

-

re[O,T] and where p , E > 0 are constants (see Subsection II.4.2). By Proposition II.1.2.2 and the properties of the function U we have that

f T g,(r)dr = A.

(VII.57)

J0

So we have that

(r) =

^ (itr

-

is the rank-frequency density function for words. We will also substitute r' = 1 + r (and then replace r' by r) so that we have

334

Power laws in the information production process: Lotkaian informetrics

P,(r) = £

(VII.59)

with r 6 [1,T +1] (cf. formula (11.89)) and D = —.

As to (iv) for N-grams we would like to use (VII.59) also for the occurrence of letters. In Egghe (2000a) we checked 4 data sets of letter occurrence. Here we will present two of them. The first are data of Dewey (as presented in Heaps (1978), p. 200) on the occurrence of the 26 letters in English. The graph is given in Fig.VII.5 (see also Egghe (2000a)).

Although not perfect we can use (VII.59) as a first approximation. We have noted already the importance of N-grams for Asian languages and also in this context we have data: we used the website of Beckman (see Beckman (1999)) on the occurrence of more than 13,000 Chinese symbols. Fig.VII.6 gives the graph of the occurrence of the 100 most important Chinese symbols (see also Egghe (2000a)).

It is clear that the data are very close to a power law. In conclusion: also for the study of Ngrams we can and will use power law (VII.59).

Using (VII.59) also improves pitfall (v) a lot. The skew distribution of single words and of letters, as expressed by (VII.59) is also decisive for which N-word phrases or which N-grams are more popular (more occurring) and which are not.

Finally we can discuss (vi). To reduce the complexity of the problem we will suppose that letter occurrence in N-grams is independent of the place in the N-gram. Similarly, we will suppose that the word distribution in N-word phrases is independent of the place of the word in the N-word phrase. In symbols this means that, for every i = 1,...,N — 1 we have

P(r,|r i+l ,...,r N ) = P,(ri)

(VII.60)

(and we put P (rN) = P, (r N )) i.e. to have a letter (for N-grams) or a word (for N-word phrases) with rank r; on the ith place in the N-gram (N-word phrase), given that on the j t h place

Further applications in Lotkaian informetrics

335

(j = i + l,...,N) we have a letter or word with rank r^, is simply P,(r;) (as in (VII.59)), unconditional with respect to the letter or words occurring on places i + l,...,N. Note that (VII.60) also implies that the same rank-frequency distribution applies for each place i, which is a natural assumption for N-word phrases and which is clearly true for N-grams since we use redundant N-grams, hence on each place the same letter distribution applies, by construction.

The independence assumption is certainly not satisfied in practical examples but it is nevertheless worthwhile to develop a model in this case because it can serve as a first approximation of real examples. In fact, as indicated in Egghe (2000b), dropping the independence assumption leads to non-analytic formulae for the rank-frequency function of N-word phrases and N-grams. Of course, with the above described generalizations we reached a much more general model than the one used in Subsection VII.6.2.

Fig. VII.5

Letter occurrence, Dewey data (English).

Reprinted from Egghe (2000a), Fig. 1, p. 241. Reproduced with kind permission ofKluwer Academic Publishers.

336

Power laws in the information production process: Lotkaian informetrics

Fig. VII.6

Occurrence of the 100 mostly used Chinese characters.

Reprinted from Egghe (2000a), Fig. 3b, p. 244. Reproduced with kind permission ofKluwer Academic Publishers. We have the following theorem.

Theorem VII.6.3.1 (Egghe (2004f)): Let N e N be fixed and assume (VII.59) to be valid. Denote by PN (r) the rank-frequency probability density function of N-word phrases or N-grams. Then re[o,T N ] and

?».(*)=,

.

^

NlUp

CVH.61)

(^(r+ (-lD) where 5N' denotes the inverse of £,N and where ^N is the function

w.fjdZjaii

(VI,,2)

Further applications in Lotkaian informetrics

337

and In' y = ln(y)...ln(y), the ith power of ln(y). i times

Proof: Since ranks are determined by (decreasing) productivity we have that x = PN (r), where

r = vol{(r 1 ,...,r N )|P(r p ...,r N )>x}

(VII.63)

, where P(r,,...,rN) denotes the probability of occurrence of an N-gram or N-word phrase for which the ith letter (respectively word) has rank i; (in the single occurrence), i = 1,...,N . Here vol(S) denotes the volume of the N-dimensional set S. Now, by definition of conditional probability density (cf. Grimmett and Stirzaker (1985), p.61), repeatedly used:

P(r,,-,r N )

= P(r,|r 2 ,...,r N )P(r 2 ,...,r N )

= P(r 1 |r 2 ,...,r N )P(r 2 |r 3) ...,r N )P(r 3 ,...,r N )

= P(i; |r 2 ,...,r N )P(r 2 |r3,...,rN)...P(rN_, |rN)P(rN)

= P,(r1)P1(r2)...P1(rN_1)P,(rN)

by (VII.60). So, by (VII.63) we have

r = vol{(r,,...,rN) | P, (r,)P, (r2)...P, (rN) > x}

withx = P N (r), x e [ 0 , l ] .

(VII.64)

338

Power laws in the information production process: Lotkaian informetrics

Note that, because of (VII.56) and (VII.59), the real ranks (in the sense of the informetric theory of Chapter II) r; should be lowered with 1 but, in (VII.59), we can work with r( e [l,T + l] itself and the set S is only a translation of the rank N-tuples (r, -l,...,r N — l) over the vector (l,...,l) (N coordinates), so that the volume is the same. Hence we can use the T{S themselves in (VII.64). Note, however, that r itself denotes the real rank of N-grams or Nword phrases (in the sense of Chapter II). Indeed, let there be T letters (in case of N-grams) or T words (in case of N-word phrases (cf. (VII.56)), then r e [o,TN] and r = TN is obtained for x = 0, the set S being S = [l,T + l f x= —

which volume is TN and r = 0 is obtained for

since then vol(S) = 0 for the following reason: using (VII.64) we have

P 1 (r,)P 1 (r 2 )...P 1 (r N )>x = ( | ) N

But by (VII.58), each P , ^ ) ^ —. So A.

P1(r1)P1(r2)...P,(rN) = ( | ) N

c

But 0
p r

i( i) = f

( VIL65 )

for every i = 1,...,N . From (VII.58) and (VII.59) this implies

r{ = 1

(VII.66)

Further applications in Lotkaian informetrics

339

for every i = 1,...,N . Hence S = {(1,1,..-,1)}, a singleton in KN and hence vol(S) = 0.

The inequality

P1(r,)P1(r2)...P1(rN)>x

leads to, using (VII.59)

DN

-, j>* (r,r2...rN)

(VII.67)

hence

r,r2...rN<^=:a

(VII.68)

, by notation of a for reasons of simplicity. Formula (VII.68) implies

1 < r, < —^— r2...rN

(VII.69)

This gives us the number of possible r,s but dependent on the different r2s,...,rNs that are possible. This will be determined now. Formula (VII.69) yields

1 < r2 < — — r3...rN

(VII.70)

I
(VII.71)

Formula (VII.70) implies

r

4- r N

340

Power laws in the information production process: Lotkaian informetrics

and so on until

l
(VII.72)

N

and

l
(VII.73)

So vol(S) of (VII.64) is found when we remark that r, ranges in an interval of length — r2...rN

1 (by (VII.69)), where eachr2,...,rN range as indicated in (VII.70)-(VII.73). Hence

rN—a

r=a / ^ rN=l

r

rN

^

/

^ r N

'N-.-1

3-- r N

... /

N=a

r

^-/drN r

r2=l

- '

r

,

2

rN_l

r

N

/

3-- r N

drN_, ... /

rN_,-l

dr2

(VII.74)

r;=l

The evaluation of (VII.74) is tedious but easy. The first term in (VII.74) (called (I)) is calculated as follows: since

f

^ = -ln ^ 5 L

(VII.75)

(> 0 by (VII.70)), we have that

(l) =

3^5i

_ a J^J^... / ^ d r 3 r N =,

But

1

''=~!~ln

"N-I=—

r

N

rN _,_,

r

N-I

r rj=1

3

(VII.76)

Further applications in Lotkaian informetrics

r

i

3

341

2 I aJ

as is readily seen. This value goes in (VII.76) yielding

rN_,=JL

,4=^_ 1 , 2fr4...rNl

(l) = - a / ^ l J ^L ... J rN-l

^

r

rN_,=l

N-l

2 { a J^ T

r4=l

4

But

2

f

a

i

r

i

dr^lln-M. 4

4

3 ! 1 aJ

Note that each time the sign switches. This leads to

. , - . <--j(-l)"-'to»-"fSi=i.' rN=l

()

V1^

N rN_,=l

"aiV

V

^

^ W

J

(N-2)!

(N-l)!

=

/-rN-l

aln^a> (N-l)!~

aj

(vn77)

342

Power laws in the information production process: Lotkaian informetrics

since a = —— > r,...rN > 1, using (VII.68).

Now we calculate the second term in (VII.74), called (II).

r N-|—

a

r 2~

r

_

"

r

(ll) = -Jdr N /"dr,., ... / dr2 r

1N=1

N-i=l

r2=l

_ a

_

a

(ll) = -Jdr N /drN_, ... f 'N=l

%-,=!

_ a

'3=1

__

^—lW 3

"

N

-1

a

^ + lldr4

(ll) = -7drN / drN_, ... r _ l _ t o - 2 r N =l

r N _,-l

r 4 =l

1

4"-IN

M—'N

(ll) = -JdrN /NdrN_, ... J N I _ ? _ ^ _ 5 J

h

rN

^,

2r

rf=1

5-rN

r5..-rN

^ — ' N

^in fs-'N

r

ldr5 5-r N

i,...rN

j

a

(H) = - / d r N / rN=l

drN_, ...

rN_,=l

_

a

' T - l ^ - l n 3 ^ ^ -i—-5—ln 2 f^^-l J=]

3!r6...rN

a

In general we have (j + 3 = 2,...,N)

2 r6...rN

i, a

J

5—tofSi^.l r6...rN

I

a

J

5—+ lldr6 r6...rN

j

Further applications in Lotkaian informetrics 343 _

a

(ii)=-JdrN ... jr"lll(-iy^L_iii11'[ii^L+^L_(_iy+(_i)HL r N =l

r j + J =l

'j+S-'N

i=l

K

d

r

j+3- r N

Hence

(n)=-7 (-ir^E>( *)+(-irk + 7-(-irx rN=i

r

N i»i !

a )

I

fN=l

rN

(ll) = (-ira|^(-ln-(l))-(-ir(a-l)-(-iralna

(n) = (-irg a( ^' + ' a +(-l) N alna + (-ira + (-l)N

'

i=0

(n) = E ( " ° , i=0

» + (-!)"

Cvn.79)

1 !

Now (VII.78) and (VII.79) yield, by (VII.74)

aln N -a ^ ( - l ) N + M a l n - a (N-l)!+^ i!

r = E

(-l)

+(

alna+(_i)N

Using (VII.68) and the fact that x = PN (r), we have by (VII.80)

N l)

^

^

344

Power laws in the information production process: Lotkaian informetrics

r + (-\f-x=^—^-r,

(VII.81)

,(P N W)V where

!

i=0

, i.e. formula (VII.62). By (VII.68) the arguments of the logarithms, appearing in 4N

are

greater than or equal to 1, hence positive. Note that i;N is an injection on [l,+oo[. Indeed:

^NW

U

+

i!

i

^

i!

N—1

^N ( y ) = -^—I- > 0 NV ; (N-l)!

(VII.82)

ony e ]l,+co[. So ^N is a strictly increasing function on [l,+oo[ and hence an injection. But

a= —>1 xp

by (VII.68), hence we can take the inverse of ^N in (VII.81) yielding

P

"")=(c(r+(-.r)r

Further applications in Lotkaian informetrics

345

where ^N' denotes the inverse of the function ^ N .

The function PN (r) is not simple. We have the following corollary, proved in Egghe (1999a) and Egghe (2000a) as an approximate result:

Corollary VII.6.3.2 (Egghe (2000a)): If r is large, we have that

°W

PNO-K

„

(VII.83)

( XN '((N-l)!r)[

where xN' is the inverse of the function

XN(y) = yln N -'(y)

(VII.84)

(again lnN"' (y) denotes the (N - l)th power of ln(y)).

Proof: The number r large enough forces all the ranks rj,...,rN to be large by (VII.64). Since r, is large we have by (VII. 69) that

a rr..rN

^

a r2...rN '

In other words, in the proof of the above theorem we only calculate (I) for r and put (II) ~ 0 . By (VII.78), this yields the result.

Note that formula (VII.83) is, up to constants, the same as (VII.54) for N = 2 although both arguments are very different: the one leading to (VII.54) uses many artificial assumptions and approximations, as indicated in Subsection VII.6.2.

346

Power laws in the information production process: Lotkaian informetrics

This approximation was used in Egghe (1999a, 2000a) because evaluating (II) did not seem to lead to any useful result. Indeed, formulae (VII.61) and (VII.62) are much more complicated than (VII.83) and (VII.84) and if it were not for the results in the sequel we would not consider these intricate results as important. We are, however, lucky: in the next subsection we will derive the size-frequency function fN linked to the above rank-frequency distribution PN and we will show that the exact result (VII.61) leads to a very simple formula for fN, simpler than the one derived from the inexact (VII.83)!

The derivation of the size-frequency function fN is based on the general formulae of Chapter II on the link between the rank- and the size-frequency function. Therefore we first have to determine the rank-frequency function (called g in Chapter II and called gN here to show the N-dependence) derived from the rank-frequency density function PN in (VII.61). gN follows from PN simply by multiplying with the total number of items in the case of N-grams or Nword phrases, which we will denote by AN (in Chapter II this is denoted by A) - see Proposition II. 1.2.2 and the properties of U. Consequently, we have

g N (r)=

1

/ ^ ^ ^

fc'H- )"-))

( VII -85)

for r e [0,T N ], using Theorem VII.6.3.1.

In the proof of Theorem VII.6.3.1 we showed that 4N strictly increases, hence the same is true for 4N', so gN strictly decreases, using (VII.85). From (VII.82) it follows that £N (y) > 0 and ^" N (y)>0 on ]l,+oo[. This can be used in (VII.85) to show that gN is convexly decreasing, as it should (by the very definition of PN). We leave this as an exercise.

There are not many practical data on N-grams or N-word phrases. A convexly decreasing rank-frequency function for N-grams can be found in Cavnar and Trenkle (1994). These authors use the name "Zipfian" distribution which, visually, and probably also statistically, is a normal observation. In this Subsection VII.6.3 we only tried to show the mathematical link

Further applications in Lotkaian informetrics

347

between 1-gram (1-word phrase)-theory (i.e. Lotkaian, Zip flan informetrics) and N-gram (Nword phrase)-theory. In general, the above theory (and the one to follow on the size-frequency function) can be considered as the mathematical theory on how to describe informetrically the Cartesian product of N IPPs with the same Zipfian rank-frequency distribution.

The result (VII.85) on gN is intricate and not easy to work with. In the next subsection we will determine the size-frequency function fN that is equivalent with the rank-frequency function g N , using the model in Chapter II. The result on fN will be surprisingly simple (although its derivation is, once more, tedious).

VII.6.4

The size-frequency function of N-grams and N-word phrases derived from Subsection VII.6.3

We have the following theorem.

Theorem VII.6.4.1 (Egghe (2004f)): The size-frequency function fN that is equivalent with the rank-frequency function gN of (VII.85)isgivenby

fN(j) = - ^ l n N - ' 2JM j1+f J

(VII.86)

for je[l,p m (N)], where p m (N) is the maximal item density in the case if N-grams or Nword phrases, given by

P m (N)

= ANDN

(VII.87)

and where C is the constant

i

C= ^ p L P N (N-1)!

(VII.88)

348

Power laws in the information production process: Lotkaian informetrics

Proof: By the very definition of size-frequency function, we have (see formula (11.10)):

for je[l,p m (N)] with p m (N) the maximal item density in the case of N-grams or N-word phrases. Formula (VII.85) yields

gN(r)(#(r + (-l)N"))P=ANDN

Hence, taking derivatives

gN(r)(s"(r+ (-ir))T+gH(r)p(rH1(r + (-ir))r.%l(x = r + ( - i r ) = O where

%l(,. r+( -ir) means: the derivative of the function 4N' in the point r + (—l) ~ . S o

But, by (VII.82)

fa(r)^(r + (-ir)

=—

= ^

—

(VII.91)

Further applications in Lotkaian informetrics 349

Now we use (VII.85), yielding

;i

"Hnw;;;l.-,i

'

So

~PAND

a (A-

(VII 93)

Since j = gN (r) denotes the item density (by Definition (II.8) - see also (11.13)), we have by (VII.85) that

S'(r + (-ir) = (^ff

(VU.94)

in the point r = g"1 (j). So (VII.94) in (VII.93) yields

gN (g,' (J)) =

=^4^

lni j which yields, by (VII.89), the result

(M^f

(N-l)!

rr

(VIL95>

350

Power laws in the information production process: Lotkaian informetrics

fN(j) = - M L l n - ^ 1 N

p j'

+p

J

(N-l)!

(VII.96) '

,je[l,p n ] (N)], a remarkably simple result, taking into account from where we came! By definition of p m (N), see formula (II.8) we have

(N) fc(0)

-

by (VII.85). But £,N (l) = (—1)

-(#T

as follows readily from (VII.62). Hence, since we showed

in Theorem VII.6.3.1 that ^N is an injection on [l,+oo[, we have that 4 N ' ( ( ~ ^ f

j = ^ an<^

so, from (VII.97)

P m (N)

= ANDN

proving (VII.87). Now (VII.97) gives

fN(j) = ^ r m N - E E M " j1+p

J

with C as in (VII.88), hence we have proved (VII.86), for j € [l, pm (N)].

D

Note that, in terms of Lotka's a , see (11.87), we have that (VII.86) also reads as

fN(j) = -^lnN-> P B M

(VH.98)

Further applications in Lotkaian informetrics

351

hence a product of a power law and a power of a logarithm. It is easy to see that fN < 0 and fN > 0 hence fN is convexly decreasing on [l,pm (N)| = [l, A N D N 1.

Note also that gN and fN , for N = 1, reduce to the given laws of Zipf and Lotka (as it should). j_ ! Pm

F

Indeed, for f, this is clear (with C = - — as follows from (VII.88), agreeing with the results in Subsection II.4.2 (formula (11.91)), since we supposed Zipf s law for g,). For g,, we have by (VII.85)

g (r)=

'

(^(rTIjf=(^

since \x (y) = y by (VII.62), and hence

\ S i(r)=

'

E

(rTIf'

the same function as (VII.56), using that we denoted D = —. A In the next subsection we will use the size-frequency function fN to calculate the averages u. (here denoted as ji N ) of items per source and (x* (here denoted as n*N) being the Type/TokenTaken average as discussed in Chapter III. In terms of the present notations we could say that the Type/Token-Taken theory of Chapter III was based on f,; in the next subsection we will use fN (N > 2). Of course, the general defining formulae for u and \i (i.e. for general sizefrequency functions) of Chapter III also apply here.

352

Power laws in the information production process: Lotkaian informetrics

VII.6.5

Type/Token averages jxN and Type/Token-Taken averages \xK for N-grams and N-word phrases

As follows from Chapter III (formulae (III. 14) and (III. 15)), we have that the TT averages nN and the TTT averages \i'N are given by

^ = ^

cvn.99)

w ^N=T 1 L

(VII. 100)

where

Pm(N)

TN=JfN(j)dj

(VII.101)

i

P.(N)

AN=/jfN(j)dj

(VII.102)

i

Pn,(N)

W N = J ff N (j)dj

(VII.103)

i

and where fN is given by (VII.86). All these integrals are tedious to calculate but we can use the following formula found in Gradshteyn and Ryzhik (1965) (p.203 (2.722)):

J x - h T x dx = ^ — ^ ( - l ) k ( m + l)m(m-l)...(m-k + l ) —

valid forall n e » \ { - l } andmeN.

^ - (VII. 104)

Further applications in Lotkaian informetrics

353

For the calculation of TN (i.e. in function of pm (N), which will be our free parameter, just as it was the case with p m in Chapter III), we have two equivalent alternatives: or we can calculate (VII.101) directly or (which we will do here) use the following short argument. We note that j = g N (r) and hence l = g N (T N J (r = TN was the highest rank as proved in Theorem VII.6.3.1). Formula (VII.85) yields

ANDN

x

(s'(i*+(-ff so

T N +(-I) N = 4 N ( ( A N D 7

Using (VII.62) we have

N

> l ) N + 1 > N D 7 l n ' (ANDN)?

TN+(-ir = E

n

hence, by (VII.87)

N _ 1 (-l)

TN=(-1)N+^ i=o

N+ 1

" (p m (N))^n 1 f( Pm (N))F| : 1-

^

1

valid for all N e N and all p > 0.

We are left with the calculation of (VII. 102) and (VII. 103), using (VII.86). We have

(VII.105)

354

Power laws in the information production process: Lotkaian informetrics

AN = P"f -^ln1"-1 H s M dj. J 1

(VII.106)

1

~

J

/

Since

d^M=-P#dj j j

(VII.107)

we have that

lnK-I

f

PjN) ^dj^-p^N) —

1

^

f ^ J

;P

1

'

taN-.

P H Md P ^

1

i

J

J

(VH.108)

i

J

So, for p > 0, (3 ^ 1 we can apply (VII. 104) yielding

lnN-i

/

lnN_k_, p m (N)

Pn(N) 1 k

r ^ d J = -^g(- ) N(N-l)...(N-k)—

J^

(H jnN-k Pm(N)

= E^4(N-l)(N-2)...(N-k + l)—

^-

[H where we note that, for k = l, we have to take ( N - l ) ( N - 2 ) . . . ( N - k + l) = 1. (VII.106) now yields, using (VII.88)

Further applications in Lotkaian informetrics

(pm(N))p

355

^ ( - l ) > - l ) . . . ( N - k + l)ln N -> m (N))

(-!)>-!)!

valid for all N and p > 0 , p * 1 and noting that, for k = 1, (N-l)...(N-k + l) = l.

For P = 1, we have

\

J

J

p (N)ln N (p lHmV(N)) AN = ^ ^ "

(VII. 110)

as is easily calculated using (VII. 107) and (VII.88) for p = 1.

For WN we have

WN =

f ^ - l n N - ' H H I M dj l i J jP

(VII.Ill)

J

But, using (VII. 107), we have

lnN-,

J

PJM

_ ^ d j = -( Pm (N)) 2 -5 J P^l

P

m- ^ M

which can be calculated, using (VII. 104) for all p ^ —. This gives

d

PJM

(VII.112)

356

Power laws in the information production process: Lotkaian informetrics

taN.

J

PsM N _ ln— k -ji— T7^dj = -r-Lg(-l) N(N-l)...(N-k)—

f

fN"

9_M

1-2

JN-.C P m ( N ) ' j —

N C_i\ k

ln

where, for k = 1, we have to take (N — l)...(N — k +1) = 1. Hence we have, from (VII.Ill), using (VII.88)

WN =

( P - W

(-1)N(N-1)!

A ( - l ) k ( N - l ) . . . ( N - k + l)ln w - 1 '(p in (N))

valid for all N and P ^ — and where we have to take (N — l)...(N — k +1) = 1 for k = 1.

For p = - we have, using (VII. 107)

lnN->

Pm(N)

^dj =

f J So (VII. 111) and (VII.88) yield

^N Pm(N)'

j

L_ N

Further applications in Lotkaian informetrics

WN = 2 N ( P ;j N ) ) ln N (p m (N))

357

(VII.114)

for all N and (3 = - .

With these formulae for T N , AN and WN we are able to calculate \i.H and |i*N via (VII.99) and (VII. 100). Of course, formulae (111.33), (111.34), (III.38)-(III.4O) can be used in the calculation of (j, = \xx and \x = n*, i.e. TT and TTT averages in the case of 1-grams (single letters) or of 1-word phrases (single words), for comparison.

As examples we will take (3 = 1 (i.e. Lotka's a = 2) and (3 = — (i.e. Lotka's a = 3) and we will take N=l,2,3: the case of 2(3)-grams or 2(3)-word phrases in comparison with single letters or words will be informative enough for higher values of N. In addition the cases N = 2 and N = 3 are the most important cases for all applications.

Let us take P = 1 first. For N = 2 we have from (VII. 105), (VII. 110) and (VII. 113)

T 2 = l - P m ( 2 ) + Pm (2)ln(p m (2))

(VII.115)

A2=iPm(2)ln2(pm(2))

(VII.116)

W 2 =(p m (2)) 2 - P m (2)ln(p m (2))-p m (2)

(VII.117)

Hence

P m (2)ln 2 ( Pn ,(2)) ^-2(l-Pm(2)

+

p m (2)ln( Pm (2)))

( m U 8 )

358

Power laws in the information production process: Lotkaian informetrics

2fp (2)-ln(p ( 2 ) ) - l )

which yields Table VII.2.

Table VII.2

Values of \i2 and JJ.* for diverse values of pm (2), for P = 1.

Reprinted from Egghe (2004J), Table 1, with permission from Elsevier.

p (2)

I

^5

2

3

5

10

100

^

L140

L244

L397

T600

L890

2.933

^-

1.150

1.277

1.494

1.846

2.526

8.902

This can be compared with the values of \\.l and (i*, i.e. the non-composed case. For P = 1 (hence a = 2) we use the formulae (111.34) and (111.39)

H= h = - k £ a L . 1—Pm

(VII. 120)

H*=H;=H^i ln Pm

(VII.121)

and

yielding Table VII.3.

Further applications in Lotkaian informetrics Table VII.3

359

Values of p., and \i\ for diverse values of p m , for (3 = 1.

Reprinted from Egghe (2004/), Table 2, with permission from Elsevier.

1.5

2

3

5

10

100

1.216

1.386

1.648

2.012

2.558

4.652

1.233

1.443

1.820

2.485

3.909

21.498

Pm

h

We see that, for the same value of the input "seed " p m (2) or p m , we have that the values (j., and \i\ are larger than the values \x2 and \i\ respectively. We also see that n* — \\.2 < \i\ — \x, showing that the average screen lengths (e.g. in the case of the use of 2-grams by a cataloger - see Chapter III) are shorter than the ones given in the 1-gram case. Note further that n* > \xl and \i'2 > \x2 as it should, following Theorem III. 1.3.2.1.

Now we calculate the case N = 3, still with p = 1. We have from (VII. 105), (VII. 110) and (VII. 113)

T3 = - l + pm(3)-Pm(3)ln(pm(3)) + ipm(3)ln2(pm(3))

(VII.122)

A 3 =ip m (3)ln 3 ( Pm (3))

(VII.123)

W3 = (p m (3)) 2 -ip m (3)ln 2 (p nl (3))-p m (3)ln( Pm (3))-p m (3)

(VII.124)

Hence

„ ^

=

p m ( 3 ) l n 3 M 3 ))

- 6 + 6p m (3)-6 P m (3)ln( P m (3)) + 3 Pm (3)ln 2 (p m (3))

(VII 125) l

'

>

360

Power laws in the information production process: Lotkaian informetrics , 6p m (3)-31n 2 (p m (3))-61n(p m (3))-6 3 \x] = - ^ ^ / ( ])

(VII. 126)

which yields Table VII.4.

Table VII.4

Values of \i, and |a* for diverse values of pm (3), for p = 1.

Reprinted from Egghe (2004f), Table 3, with permission from Elsevier.

Pm(3) M-3

1.5

2

3

5

10

100

1.103

1.179

1.288

1.431

1.630

2.329

1.110

1.200

1.348

1.577

1.989

5.148

The same comments as for \i2, [i*2, given above, can be given here for (j.3, n*. Note again that the values of (x3, \x'} are smaller than the values of u.2, \i2 respectively.

We, finally, give formulae for (3 = — and N = 2, 3 and compare with the case N = 1. For N = 2 and P = — we have the following formulae, following from (VII. 105), (VII. 109) and (VII. 114)

Hence we have

T2=l-(Pm(2))2+2(Pm(2))2ln(Pm(2))

(VII. 127)

A2 = 4 P m (2) + 4(p m (2))2 ln( Pra (2)) - 4(p m (2)

(VII. 128)

W2=2(Pm(2))V(pm(2)

(VII. 129)

Further applications in Lotkaian informetrics

361

„ _ 4 P m ( 2 ) + 4(p m (2)) 2 ln(p m (2))-4(p m (2)) 2 "2=

(VIU30)

l-(p m (2)) 2 + 2(p m (2)) 2 ln( P m (2))

^ - 2

+

p.(2)h'(p a (2)) 2p m (2)ln( Pm (2))-2 Pra (2)

(VIU31)

yielding Table VII.5.

Table VII.5

Values of \x2 and u.* for diverse values of p m (2), for [5 = —.

Reprinted from Egghe (2004J), Table 4, with permission from Elsevier.

P-(2) P-2

1.5

2

3

5

10

100

1.130

1.214

1.321

1.433

1.552

1.761

1.140

1.244

1.397

1.600

1.890

2 .933

Compare now with the case N = 1, P = — (hence a = 3), using formulae (III.33) and (111.40):

H= h = ^ Pm+!

(VII.132)

H'=JI;=-!5PH_

(VII.133)

Pm yielding Table VII.6.

362

Power laws in the information production process: Lotkaian informetrics

Values of JI, and \i' for diverse values of p m , for (3 = —.

Table VII.6

Reprinted from Egghe (2004J), Table 5, with permission from Elsevier.

1.5

2

3

5

10

100

1.200

1.333

1.500

1.667

1.818

1.980

1.216

1.386

1.648

2.012

2.558

4.652

Pm Hi

h

For N = 3, p = - we have now, using (VII. 105), (VII. 109) and (VII. 114)

T3 = - 1 + (pm(3))2 - 2 ( p m (3))2 ln(p m (3)) + 2(p m (3)) 2 In2 (pm (3))

(VII.134)

A3=-8p n ,(3) + 4(p m (3)) 2 ln 2 (p m (3))-8(p m (3)) 2 ln(p n ,(3)) + 8( Pm (3)) 2 (VII.135)

W 3 =|(p m (3)) 2 ln 3 (p m (3))

(VII.136)

Hence

-8p m (3) + 4(p m (3)) 2 ln 2 (p m (3))-8(p n ,(3)) 2 ln(p ni (3)) + 8(pm(3))2 3

-l

+

( P r a (3)) 2 -2( P m (3)) 2 ln( P m (3)) + 2(p m (3)) 2 ln 2 (p m (3))

„ = Pm(3)'n3M3)) 2 * - 6 + 3 P m (3)ln ( P m (3))-6( P m (3))ln( P m (3)) + 6( Pra (3))

yielding Table VII.7.

( VII (

138) '

)

Further applications in Lotkaian informetrics

Table VII.7

363

Values of ]x3 and u^ for diverse values of p m (3), for |3 = —.

Reprinted from Egghe (2004J), Table 6, with permission from Elsevier.

Pn (3)

1.5

2

3

5

10

100

1.097

1.160

1.241

1.330

1.429

1.635

1.103

1.179

1.288

1.431

1.630

2 .329

We see again that the same tendencies of the comparison of n, ( (x*, |x2, (j,*, u3, uj are found as in the case (3 = 1.

We close with an open problem.

Open Problem: Describe the TT average and TTT average in case of N-grams where the number of items is limited to the number of documents in a database (e.g. an OP AC, used by a cataloger, as described in Chapter III). Since, here, the number of items (denoted A) is fixed and since there are TN N-grams (cf. Theorem VII.6.3.1), we might end up, for not even very large N with the relation TN > A , hence with more sources than items, which is out of the scope of the informetric theory, developed in this book.

This page is intentionally left blank

APPENDIX

APPENDIX I In this appendix we will give a self-contained proof of the characterization of scale-free functions as power functions (basic for Lotka's law) and on the characterization of functions, transforming products into sums, as logarithmic functions (basic for entropy). Both results are based on the characterization of functions, transforming sums into sums, as linear functions.

Theorem A.I.I: (i)

The following assertions are equivalent for a continuous function y : K —> K (a)

v|/(x + y) = v|/(x) + v)/(y)

(A.I.I)

for all x , y € R (b)

there is a real number c such that i|/(x) = cx

(A.I.2)

for all x e M.

(ii)

The following assertions are equivalent for a continuous function \\i: K+ —> K (a)

y(xy) = H/(x) + v|/(y)

(A.I.3)

for all x , y € K + (b)

there is a real number c such that i|;(x) = clnx

(A.I.4)

for all x e K+ (c)

there is a positive real number a such that \|/(x) = log a x

(A.I.5)

for all x e K + .

(iii)

The following assertions are equivalent for a continuous function \\i: M+ —> M+ (a)

\|/ is scale free (Definition 1.3.2.1)

366

Power laws in the information production process: Lotkaian informetrics (b)

there exist constants a E R + , c e t such that \|/(x) = ax c

(A.I.6)

for all x e R + .

Proof: It is clear that all (b)s imply all (a)s and that (ii)(b) is equivalent with (ii)(c). Therefore we

only have to prove the three implications (a) => (b)

(i)(a)=»(i)(b) Complete induction on (A.I.I) gives

vj/(nx) = nv|/(x)

(A.I.7)

for all x € R and n £ N. Let now m, n E N, t E R, x = — . Hence xn = mt and n hence, by (A.I.7), nvj/(x) = m\|/(t) and so

w

m . —t n

m

= n

V

u\ t .

Putting vj;(l) = c we see that (A.I.2) is valid for all positive rational numbers. Let x € R + arbitrarily. Since Q + is dense in R + , there exists a sequence (qn ) neN , qn E Q + for all n 6 N such that lim qn = x . Since \\i is continuous we have ll-*0O

\j/(x) = v|/(limqn)

= limv|/(qn) n—>oo

= lim cqn

= cx,

Appendix

367

hence (A.I.2) is proved for all x € R + . Let now x € W U {0}. If x = 0:

vj/(O) = O = cO

by (A.I.I) (take x = y = 0 there). If x < 0,

i|/(x) = \)/(0)-i|/(-x)

(by (A.I.I)). So

i|/(x) = -\|/(-x)

= -c(-x)

= cx,

completing the proof of (i).

ii)(a)=»(ii)(b) Define, for all x € R

f(x) = V (e").

Then

f(x + y) = v1/(e"+y)

= \|/(eV)

(A.I.8)

368

Power laws in the information production process: Lotkaian informetrics = v|/(ex) + v|/(ey)

(by (A.I.3))

= f(x) + f(y)

for all x, y G K , by (A.I.8). By (i) we have that there is a c e R such that

f (x) = ex

for all x G R. Hence, for all x G R +

\|/(x) = f (lnx) = clnx.

(iii)(a)=>(iii)(b) From (1.26), defining the scale-free property of \y, we have that: for every positive constant C, there is a positive constant D such that

V (Cx)

= Di|>(x)

for all x in the domain of \|/ i.e. R + here (and D only depends on C). Hence (since l€R+):

v|/(C) = D V (l)

hence V (Cx)

=^

W

(A.I.9)

Appendix 369 for all C,xeIR + . For all x e K + , define

cp(x) = l n ^ M

(A.I.10)

Hence cp : R+ —> R and we have

#) 2 (since (A.I.9) is valid for all C,x £ R+ and take C = y)

v(i)

v(i)

= cp(x) +
Also cp is continuous since \\i is. Now (ii) implies that there is a real number c such that

(p(x) = clnx

Since In is injective it follows that (by (A.I.10))

V( x ) = v(l)x c

370

Power laws in the information production process: Lotkaian informetrics

i|/(x) = ax

for all x e R + . Finally note that a = i]/ (l) > 0 .

D

The above theorem is a result on the characterization of functions satisfying certain so-called functional relations (such as the scale-free property or (A.I.I) or (A.I.3)). For more results in this area we refer the reader to Roberts (1979).

APPENDIX II Table of the experimental fractional frequency distribution of author scores in mathematics (Mathematical Reviews - 1990). Partially reprinted from Egghe and Rao (2002a), Table in Appendix, p. 801. Copyright John Wiley & Sons Limited. Reproduced with permission.

Fractional score 0.02 0.04 0.06 0.07 0.08 0.09 0.1 0.13 0.14 0.16 0.17 0.2 0.25 0.29 0.33 0.35 0.38 0.39 0.41 0.42 0.43 0.45 0.46 0.48

No. of authors 1 1 9 2 1 13 17 12 35 3 93 341 1,166 4 4,772 16 1 2 17 2 1 6 1 2

Fractional score 1.13 1.14 1.15 1.16 1.17 1.2 1.23 1.25 1.28 1.29 1.3 1.32 1.33 1.37 1.38 1.4 1.41 1.42 1.45 1.48 1.5 1.53 1.55 1.57

No. of authors 2 22 2 15 189 37 2 177 3 1 1 2 868 2 1 2 2 19 6 4 2,547 9 1 1

Fractional score 2.37 2.38 2.4 2.41 2.42 2.44 2.45 2.5 2.55 2.58 2.63 2.66 2.67 2.7 2.73 2.75 2.83 2.86 2.87 2.9 2.91 2.92 2.95 3

No. of authors 1 1 2 3 20 1 4 789 2 34 1 6 55 9 1 25 130 1 3 1 1 9 1 1,191

Appendix 0.5 0.53 0.56 0.58 0.59 0.62 0.63 0.64 0.65 0.66 0.69 0.72 0.74 0.76 0.78 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 0.99 1 1.02 1.04 1.06 1.08 1.09 1.1 1.11 1.12

11,673 36 1 136 3 2 1 5 2 17 474 23 1 170 3 2 919 7 1 6 2 48 6 1 1 14,507 9 2 2 66 3 2 2 2

1.58 1.6 1.62 1.64 1.65 1.66 1.67 1.7 1.73 1.75 1.79 1.83 1.9 1.91 1.92 1.95 1.98 2 2.03 2.06 2.07 2.08 2.09 2.12 2.14 2.15 2.16 2.17 2.2 2.25 2.28 2.29 2.33 2.36

27 1 1 1 1 1 149 11 1 65 1 328 2 1 9 1 6 3,255 6 1 1 47 1 2 2 1 2 70 26 50 2 1 288 1

371

3.03 3.08 3.1 3.11 3.17 3.2 3.23 3.25 3.33 3.36 3.42 3.45 3.48 3.5 3.53 3.58 3.6 3.66 3.67 3.7 3.72 3.75 3.78 3.8 3.83 3.91 3.92 3.95 3.98 4 4.08

3 10 1 1 42 9 1 29 126 1 4 2 1 33 1 5 2 2 33 2 1 13 1 1 80 1 1 1 1 386 9

17.2 17.58 18 Total

1 1 1 46,853

372

Power laws in the information production process: Lotkaian informetrics

Fig. A.I A fractional frequency curve (experimental data of previous table). The numbers in the abscissa refer to the number of fractional scores (their ranks), not to the fractional score itself. Reprinted from Egghe and Rao (2002a), Fig. 1, p. 790. Copyright John Wiley & Sons Limited. Reproduced with permission.

APPENDIX III Statistical determination of the parameters in the law of Lotka

A.III.l Statement of the problem

The purpose of this book was to investigate how far we can go (i.e. which results can be proved) when we only use the size-frequency fucntion

f

0) = ^

(A.m.i)

j £ [l,p m ], C, a > 0, i.e. a decreasing power law, i.e. Lotka's function. In this sense, formula (A.III.l) is given (and, in fact, it is the only thing that is given) and hence we suppose C, pm and a to be known parameters.

Appendix

373

However, in practise, one has a table of data (i.e. n e N (of course finite) versus f(n) = the number of sources with n items) and one is interested to determine the "best fitting" function of the type (A.III.l) ("best fitting" to the data or, rather, the graph of the data (n,f(n)V n = 1,2,3,...). This is important to know since the values of C, pm and a (especially a ) determine certain crucial properties of concentration (Chapter IV), of fractal complexity (Chapter V) or even the form of citation graphs (Subsection VII.5).

There are three problems to tackle: (i)

the fact that our practical data are never complete and hence that we have a sample on which, possibly, another size-frequency function than the one of the (unknown) complete data set applies,

(ii)

Clearing out the difference between the continuous Lotka function (A.III.l) and the discrete one

f(n) = 4

( A - IIL2 )

n n = l,2,...,n mx . (iii)

Considering the data set as such (cf. (i)) how can we determine a, a (most importantly) and C, K, pm and n ^

in the most accurate way. This is a three-

parameter problem. Often, of course, n ^ is high and hence, nraax and pm can be taken as +oo so that the problem is reduced to a two-parameter determination (C and a for (A.III.l); K and a for (A.III.2)).

The next section deals with (i) and Section A.III.3 with (ii). The rest of this Appendix III is devoted to possible solutions of problem (iii).

A.III.2 The problem of incomplete data (samples) and Lotkaian informetrics

In this section we will reveal a unique property of Lotkaian informetrics in connection with the determination of the size-frequency function f based on an incomplete set of data. Let us consider a complete IPP (unknown) with size-frequency function f and a "sample" IPP (known) which is a subset of the unknown one with size-frequency function f *.

374

Power laws in the information production process: Lotkaian informetrics

We will consider the following sampling types, which we define, using the continuous sizefrequency function f (Chapter II) of which the domain is extended to K+ (as is also the case in Subsection 1.3.2), and verify that the names reflect their intuitive meaning (see also Dierick (1992)).

Definition A.III.2.1 A sample is a systematic sample in the items (or an item systematic sample), with sample fraction 6 (i.e. the sample size is 9 A, a fraction 6 of all the items) if, for every j e R + :

f(9j) = f(j).

(A.III.3)

Definition A.III.2.2 A sample is a systematic sample in the sources (or a source systematic sample), with sample fraction r| (i.e. the sample size is r|T , a fraction r| of all the sources) if, for every j € K + :

f ( j ) = rif(j).

(A.III.4)

We have the following result, characterizing Lotkaian informetrics. Theorem A.III.2.2 (Egghe (2004h)): The following assertions (i) and (ii) are equivalent: (i)

Item and source systematic samples are the same, i.e.: (a)

For every 9 € ]0,l] there exists a r\ e ]0,l] (only dependent (injectively) on 8) such that every item systematic sample with fraction 9 is a source systematic sample with fraction r). Reversely:

(b)

For every r\ e ]0,l] there exists a 9 e ]0,l] (only dependent (injectively) on r|) such that every source systematic sample with fraction r| is an item systematic sample with fraction 9.

(ii)

The function f is scale-free and hence, equivalently (Corollary 1.3.2.3), f is a decreasing power law (i.e. Lotka's law).

Appendix

375

If the above assertions are true, the relation between r\ and 0 is given by

r| = 9 a

(A.III.5)

where a is Lotka's exponent, see (A.III.l).

Proof:

Suppose (i)(a). We have given formula (A.III.3) and hence, by (i)(a), we also have formula (A.III.4) with r| = r|(9), i.e. t| is an injective function of 0 . So, for all j £ R + :

f(ej) = f(j)

and

f(9j) = Tlf(©j).

Hence

f(6j) = - f ( j )

(A.III.6)

for all j € R + , where r\ = r\(6) for all 6 e ]0,l].

Suppose (i)(b). We have given formula (A.III.4) and hence, by (i)(b), we also have formula (A.III.3) with 0 = 0(r|), i.e. 0 is an injective function of r|. So, for all j e K + :

f ( j ) = Tlf(j)

376

Power laws in the information production process: Lotkaian informetrics

Hence

f ^ j ) = nf(j)

(A.m.7)

for all j e M+.

Note that the fact that 6 = 0(r|) is an injection implies that T| is a function of —, for all 6 0e]0,l]. Hence, by (A.III.6) and (A.m.7), f is scale-free (since 6e]0,l] and — £ [l,+oo[ and hence all values in M+ are covered) and the result follows.

(ii)=>(i)

Let f be given as a decreasing power law

f(j) = f j e R+ Define, given 0 e ]0,l]

f(9j) = f(j)

(i.e. given an item systematic sample) for all j £ K + . Hence, for all j 6 K + :

Appendix

111

f ( j ) = e°f(j)

Putting r| = Qa we see that we have a source systematic sample (with fraction r| = 9a note that indeed 6e]0,l] implies r|e]0,l] and that r| is an injective function of 0). Hence we proved (i)(a).

Let us now have a given source systematic sample (i.e. given r| e ]0,l])

f(j) = -nf(j)

for all j e E + . Hence, for all j e K + :

f* r, a j =T|f Tfj

n f T)°j = f ( j )

Hence we have an item systematic sample with fraction 6 = ria (note again that t| 6 ]0,l] implies 9 6 ]0,l] and that 9 is an injective function of r|). Hence we proved (i)(b). This completes the proof of the theorem.

For results on concentration aspects of systematically sampled (in items or sources) IPPs we refer the reader to Egghe (2902c).

378

Power laws in the information production process: Lotkaian informetrics

Note: In the above results as well as in Subsection 1.3.2 we used a size-frequency function f : R + —> R + . In Chapter II, for f, we limited ourselves to a domain [p(0),p(A)l and we even supposed (for the sake of simplicity) that p (0) = 1. Note that, in general (by the definition in formula (11.(5)), p takes values in K+ so that the above interval can be replaced by K + . This broader domain is needed here in order to be able to define size-frequency functions for sampled IPPs and to prove the scale-free property of f.

The above result shows that Lotkaian informetrics (and only Lotkaian informetrics) allows for a sample size-frequency function f * which is (up to constants) the same as the population size-frequency function f, a remarkable conclusion. We hence can delete (i) and give full attention to the solution of the problem itself, i.e. to (ii) and (iii). One can also do this in nonLotkaian informetrics but, as explained above, one only finds the appropriate size-frequency function for the sample (incomplete IPP) at hand which is then different from the one of the complete IPP. Luckily Lotka himself was not in this situation when he checked his power law on the volume of the Chemical Abstracts (1907-1916), where he limited himself to author names starting with the letter A or B, a clear sample! From the above, it is expected that, if Lotka had not applied this severe restriction he would have ended up, by (A.III.4), with (more or less) the same power law (i.e. with the same a ) as he has found in his article Lotka (1926).

A.I11.3 The difference between the continuous Lotka function and the discrete Lotka function

So we consider the functions

f(j)=

7

j £ [l,p m ], i.e. the continuous function that was the topic of study in this book and, denoting fd (d for discrete) to indicate the difference with f,

Appendix

379

£ n = l,2,...,nmax. For easy reasoning we will take n ^ = pm = +00. Suppose our data yield that there are T sources and A items in total. Then we repeat the following results from Chapter II (existence off) ((11.28) and (11.29) in Proposition II.2.1.1.1):

a= 2Azl =

A-T

AT

izl

(AJH.S)

(j. — 1

A

C = - ^ — = -— A-T n - 1

(A.III.9)

From this it is clear that the functions f and fd above are different. Indeed, (A.III.9) implies that C = f (l) > T while, obviously, K = fd(l) < T (fd(l) being the number of sources with 1 item). This difference can be explained by the fact that f represents a density function so that f (l) has no actual interpretation in terms of our data. The link between f and fd can be clarified by the "discretization" of the continuous f. The very definition of a density function (as f) gives that, for every n = 1,2,3,...

I(f)(n)=:/f(j)dj=/^dj.

For a ^ 1 this gives

I(f)

w=^r(^-(^r)

(AJIL10)

Note that l(f) is not a power law (and hence l ( f ) ^ f d ) but we will study the difference between l(f) and fd. It will turn out that both functions are very similar which will make our problem of determining fd easier.

380

Power laws in the information production process: Lotkaian informetrics

Note that

I f

( )w = ^r( 1 -^)

(AJIL11)

From (A.III.8) and (A.III.9) (or from Subsection II.2.1.1) it follows that

— =T a-1 so that (A.III.l 1) implies

l(f)(l) = T ( l - ^ i r ) < T

(A.III.12)

which explains the continuous model and C = f (l) > T . In fact we see that l(f )(n) < T for every n e N. Let us now make the following approximative calculation of (A.III.10):

C_ 1 a - 1 (n + l)*"1

,-1-ifl] a-ldnK-'J (A.III.13) now gives

I(f)(n)-^

1_ n"-1

(A.III.13)

Appendix

381

showing that l(f) has the same shape as f. Hence for f and fd to have the same shape we conclude that

a^a

(A.III.14)

if a and a are not too high (otherwise f (n) RS fd (n) RS 0 even if a and a are very different). Whether (A.III.14) is true can be checked as follows.

The exact relation between a and a can, numerically, be calculated as follows: given T and A as above (A > T) we have, from (A.III.8) and (A.III.9) (or from Subsection II.2.1.1, formula (11.30)) that

^ =T oc-1

^ a-2

= A

hence

^=^=^4 T

a —2

From (A.III.2) we have

n=l

n

n=l "

where C, denotes the classical Riemann-zeta function (see Apostol (1957)). So

(AJILI5)

382

Power laws in the information production process: Lotkaian informetrics

A = ^(a-1)

£(a)

T

So for a given data set, A and T, hence \x can be determined. From tables of ^(a) one can construct tables of

, .

versus a (provided in Egghe (2004g) and hence a follows from

(A.III.16). Also a follows from (A.III.15) (or see (11.30)):

a = ^—H-l

(A.III. 17)

and the comparison can be made. This is done in Egghe (2004g) where more details can be found. The relation (A.III. 14) is confirmed there.

It furthermore follows from (A.III.15) and (A.III.16) that a > 2 iff a > 2 (hence also a < 2 iff a < 2) even if a and a are different (for high, uncommon, values). This is an important conclusion because many - from Lotka's function f derived properties (e.g. shape of the cumulative first-citation distribution (Chapter VII), existence of the Groos droop (Chapter II), concentration (Chapter IV) and fractal properties (Chapter V)) - are common for different as above 2 and for different as below 2 but are different for values of a below and above 2.

So, from the above, the practical calculation of the exponent a (based on the data) (see Subsection A.III.4) yields a clear estimate of a which determines the Lotkaian informetrics theory, based on f, as developed in this book. The relationship between l(f )(l) and fd (l) = K can also be described as follows (in this way also indicating the closeness of l(f)(n) and fd(n) for all n e N ) . From (A.III. 12) and the fact that from (A.III.2) it follows that

n=l

n

Appendix

we conclude that we have to compare 1

383

— with (using (A.III.14))

;- 1 (a) = - J —

(A.III. 18)

Tables of ^"' (a) are available and we reproduce one here (Table A.I) since we will also need it further on (the table is reproduced from Egghe and Rousseau (1990a), with permission from the publisher - see also Nicholls (1987)).

Note that only the values C,{l) and C,(4) are exactly known since

?( 2 ) = E ^ T = ^ n—1

(A.III.19)

^

(cf. formula (I.I 1)) and

as can e.g. be found in Gradshteyn and Ryzhik (1965). Some examples are found in Table A.2.

We also see that, the higher a, the closer the results, i.e. I (f) (l) PS fd (l). This is due to the fact that

y(a>=

< i1 i — = i = * d i - £ \

An analogous argument shows this to be true for every n = 1,2,3,... .

(A-IIL2I>

384

Power laws in the information production process: Lotkaian informetrics

Table A.I Table of — - for a <= [1.11,3.49] with increments of 0.01. Reprinted from Egghe and Rousseau (1990a), Table IV.6.6, p. 357, with permission from Elsevier. 1 a 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49

^( a ) 0.1033 0.1121 0.1208 0.1294 0.1378 0.1462 0.1545 0.1627 0.1708 0.1788 0.1868 0.1946 0.2024 0.2100 0.2176 0.2251 0.2325 0.2399 0.2471 0.2543 0.2614 0.2685 0.2754 0.2823 0.2891 0.2958 0.3025 0.3090 0.3156 0.3220 0.3284 0.3347 0.3409 0.3471 0.3532 0.3592 0.3652 0.3711 0.3770

1 a 1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89

W)

0.3828 0.3885 0.3942 0.3998 0.4054 0.4109 0.4163 0.4217 0.4270 0.4323 0.4375 0.4427 0.4478 0.4528 0.4578 0.4628 0.4677 0.4725 0.4773 0.4821 0.4868 0.4914 0.4961 0.5006 0.5051 0.5096 0.5140 0.5184 0.5227 0.5270 0.5313 0.5355 0.5397 0.5438 0.5479 0.5519 0.5559 0.5599 0.5638 0.5677

1 a 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29

<^(a) 0.5715 0.5753 0.5791 0.5828 0.5865 0.5902 0.5938 0.5974 0.6009 0.6044 0.6079 0.6114 0.6148 0.6182 0.6215 0.6249 0.6281 0.6314 0.6346 0.6378 0.6409 0.6441 0.6472 0.6502 0.6533 0.6563 0.6593 0.6622 0.6651 0.6680 0.6709 0.6737 0.6766 0.6793 0.6821 0.6848 0.6875 0.6902 0.6929 0.6955

1 a 2.30 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 2.40 2.41 2.42 2.43 2.44 2.45 2.46 2.47 2.48 2.49 2.50 2.51 2.52 2.53 2.54 2.55 2.56 2.57 2.58 2.59 2.60 2.61 2.62 2.63 2.64 2.65 2.66 2.67 2.68 2.69

^( a ) 0.6981 0.7007 0.7033 0.7058 0.7083 0.7108 0.7133 0.7157 0.7181 0.7205 0.7229 0.7252 0.7276 0.7299 0.7322 0.7344 0.7367 0.7389 0.7411 0.7433 0.7454 0.7476 0.7497 0.7518 0.7539 0.7560 0.7580 0.7600 0.7620 0.7640 0.7660 0.7680 0.7699 0.7718 0.7737 0.7756 0.7775 0.7793 0.7811 0.7830

1 a 2.70 2.71 2.72 2.73 2.74 2.75 2.76 2.77 2.78 2.79 2.80 2.81 2.82 2.83 2.84 2.85 2.86 2.87 2.88 2.89 2.90 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99 3.00 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09

1 a

0.7848 0.7866 0.7883 0.7901 0.7918 0.7935 0.7952 0.7969 0.7986 0.8003 0.8019 0.8035 0.8052 0.8068 0.8083 0.8099 0.8115 0.8130 0.8145 0.8161 0.8176 0.8191 0.8205 0.8220 0.8235 0.8249 0.8263 0.8277 0.8291 0.8305 0.8319 0.8333 0.8346 0.8360 0.8373 0.8386 0.8399 0.8412 0.8425 0.8438

3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29 3.30 3.31 3.32 3.33 3.34 3.35 3.36 3.37 3.38 3.39 3.40 3.41 3.42 3.43 3.44 3.45 3.46 3.47 3.48 3.49

^( a ) 0.8450 0.8463 0.8475 0.8488 0.8500 0.8512 0.8524 0.8536 0.8547 0.8559 0.8571 0.8582 0.8593 0.8605 0.8616 0.8627 0.8638 0.8649 0.8660 0.8670 0.8681 0.8691 0.8702 0.8712 0.8723 0.8733 0.8743 0.8753 0.8763 0.8772 0.8782 0.8792 0.8801 0.8811 0.8820 0.8830 0.8839 0.8848 0.8857 0.8866

Appendix

385

Table A.2 Comparison of l(f)(l) and K (both divided by T). Reprinted from Egghe (2004g), Table 2. Copyright John Wiley & Sons Limited. Reproduced with permission.

C'(a) a = 1.5

0.2929

0.3828

0.5

4- = 0.6079

a=2.5

0.6464

0.7454

a=3

0.75

0.8319

a=3.5

0.8232

0.8875

a=4

0.875

* J = 0.9239

a=2

Note also that l(f)(l)< K = f d (l) in all cases (assuming that a = cc and using (A.III.12), fd (l) = K = T C (a) and the fact that

1 L

\

!

oo

1

\2—J-\ra — I k=l

K

as is readily seen; see also Table A.2 as an illustration) so that we have the following order in n = 1 (by the above discussions)

I(f )(1) < K = fd (1) < T < C = f (1)

(A.III.22)

In Egghe (2004g) it is also shown that fd (n) < f (n) for all n e N, if we assume that a = a .

This clearly shows the relation between the functions f, fd and l(f). The graph in Fig.A.I shows the relation between the curves off, fd and l(f).

386

Power laws in the information production process: Lotkaian informetrics

Fig. A.I Qualitative illustration of the relation between f, fd and l(f). Reprinted from Egghe (2004g), Fig. 1. Copyright John Wiley & Sons Limited. Reproduced with permission.

A.III.4 Statistical determination of the parameters K, a, n _ in the discrete Lotka function — ,n = l,...,nmax n

We note that, finding the values of C, a and pm in the continuous function

f(j) = |

(A.IH.1)

je[l,p r a ] is explained in Subsection II.2.1.1 (for pm = o o ) and in Subsection II.2.1.2 (for pm < oo ), which is the mathematical equivalent of the fitting model of

f

dM = £

(A-m.2)

Appendix

387

n = l , . . . , ! ^ that we will explain here. See also Tague and Nicholls (1987) for similar results on the determination of the parameters of (A.III.l). For reasons of simplicity, we will denote fd by f since, from now on, only this function (A.III.2) will be considered.

A.III. 4.1 Quick and Dirty methods

Here we suppose n ^ = oo. Given A, the total number of items and T, the total number of sources (hence A > T) we can use (A.III.8) for estimating a (by (A.III.14)):

OA

y

a= ——-

(A.III.23)

K = f(l)

(A.III.24)

K can be estimated as

, the number of sources with 1 item (can be read from the data). This method was used in Egghe and Rousseau (1990a) and in Rousseau (1990b). Tague (1988) and Tague and Nicholls (1987) suggest the use of the formula

a=

f 2 — In 2

(A.III.25)

where f (i)(i = 1,2) denotes the number of sources with 1 or 2 items (which can be read from the data) which follows from (A.III.2) noting that (remember we use the notation f for fd now) f (1) = K, f (2) = ^ . So (A.III.23) together with (A.III.24) or (A.III.25) together with (A.III.24) give an easy estimate of the parameters K and a.

The above methods are "Quick and Dirty" since they are indeed quick but do not recognize that, on the values of A, T, f (l) and f (2) one has measurement errors. Criticism on the use

388

Power laws in the information production process: Lotkaian informetrics

of (A.III.24) can be read in Griffith (1988). In favor of these methods is the fact that all values T, A, f (l), f (2) are rather stable since they deal with total number of sources and items and the largest group of sources f (l), f (2), namely the ones with 1 or 2 items.

More exact methods now follow.

A.M. 4.2 Linear Least Squares method

The Linear Least Squares (LLS) method has been advocated by Pao (1982b, 1985, 1986), but already used in the historic paper Lotka (1926), see also Tague and Nicholls (1987), and goes as follows. Formula (A.III.2) (we denote f for fd now) yields, when taking logarithms, for n = l,2)...,nII]1I:

ln(f(n)) = l n K - a l n n

(A.III.26)

which is the equation of a straight line in a log-log scale. So, when we transform all our data in a log-log scale, we have solved the problem of determining a and K by applying linear regression analysis on these data. This gives (see any book on statistics or Egghe and Rousseau (1990a) for a derivation) the following formulae for a and K

n_£(tan)ln(f(n))-£mn£ln(f(n)) a=

^

0=1

^

(A.III.27)

2

n_£(lnn) -£lnnl n=l

n=l

lnK = ln(f(n)) + ahin

)

(A.III.28)

where ln(f (n)) denotes the average of the data values lnf (n) and Inn denotes the average of the values Inn, n = l,...,nmax. Here we used the notation (n,f(n)) for the data points in our data set as well.

Appendix

389

Note, however, that when applying the above method, one is not guaranteed that

n=l

"

hence one lacks the property that — is a distribution, which is unacceptable since we do not have a model anymore for the fraction of sources with n items for each n = 1 , . . . ^ ^ . See Sutter and Kocher (2001) and criticism on this work in Rousseau (2002a).

Therefore, one only uses (A.III.26) to determine a and one derives K, once a is known via (A.III.24) or by the following (better) method. Since we must have

"mix

V

£i7 n=l

=T

(A.III.29)

n

we have that —, hence K, is determined by

— = —*—

(A.III.30)

For n ^ < +oo (and not too high!) this can be calculated directly and for n ^ too high (or n

max = + ° ° ) O n e

Can a

PPty

— = —!— = - ) —

(A.III.31)

with i^(a) as in Table A.I. i^(a) is not calculable directly but one can use the following sharp approximating formula, proved by Pao and Singer, see Pao (1985) (a shorter proof being available in Berg and Wagner-Dobler (1996))

390

Power laws in the information production process: Lotkaian informetrics oo

1

V

P-l

1

-V

1

1

l

i +

i +

[

„

a

|

+

^-ir^ (^T)^ ^ 24(p-ir

rAnn2i 1

(AJIL32)

In fact, the values in Table A.I were calculated using (A.III.32), see Nicholls (1987), but (A.III.32) can also be used if one needs other values of a than the ones covered in Table A.I.

The above method was used in Alvarado (1999), Gupta and Karisiddippa (1999), Gupta, Sharma and Kumar (1998), Pao (1982b, 1985, 1986) Nicholls (1986, 1987), Tague and Nicholls (1987) and Newby, Greenberg and Jones (2003), Pulgarin and Gil-Leiva (2004) and mentioned in Wilson (1999).

The most classical criticism on the LLS method is that, by using the logarithms of the data one does not, actually, fit the data themselves and that deviations at one end of the graph are several orders of magnitude greater than at the other end, cf. Griffith (1988). A methodology could be applying non-linear regression techniques to fit (A.III.2) but we have only the reference Nelson (1989). But also the non-linear least squares technique suffers from the fact that — is not necessarily summing up to 1 anymore.

The following technique, replacing LLS (but also using (A.III.31) or (A.III.32)) seems to be the best one to find the exponent a.

A.III.4.3 Maximum Likelihood Estimating method

The Maximum Likelihood Estimating (MLE) method has been advocated by Nicholls (1986, 1987, 1989) although the technique was already developed in 1952, see Seal (1952). The technique, used to determine the exponent a (in replacement of (A.III.27)) gives the following formula:

"max

y^f(n)lnn

„ , , >.

C (a)

' v i

n=l

—

f n

E( ) n=l

^ \ I

K

'

(\

TTT

v

Appendix

391

where, again, C, denotes the Riemann-zeta function as defined above and where we again used the notation (n,f (n)) for the data points in our data set. Formula (A.III.33) can only be solved numerically e.g. using Table A.3 composed by Rousseau, see Rousseau (1993) (reproduced with permission). Using Table A.3 one can determine the value of the exponent —C a, given the value of —-—, determined by (A.III.33).

The value of K is still determined by (A.III.30) and, for high nmax (or nmax = + o o ) , by (A.III.31)and(A.III.32).

This combined technique was programmed by Rousseau and Rousseau - see Rousseau and Rousseau (2000), the program being freely useable on the Web; since it incorporates the most advanced technique of fitting Lotka's law we encourage the reader to make use of this very user-friendly package. We remark that this program was used to recalculate the parameters K and a of the data of Lotka himself in Lotka (1926), on which we reported in Chapter I.

This technique was used in Rousseau and Rousseau (2000), Rousseau (1997), Gupta and Karisiddippa (1999), Kinnucan and Wolfram (1990), Rousseau (2002a), Nicholls (1986, 1987, 1989), Tague and Nicholls (1987) and mentioned in Tague (1988) and, of course, in Wilson (1999) which is a very good review article, not only on the fitting of Lotka's law but on the complete "state-of-the-art" of informetrics at the year 1999.

392

Power laws in the information production process: Lotkaian informetrics

-C(a) . Table A.3 Values of—T\^- in function of a.

CM Reprinted from Rousseau (1993), Table 1, p. 411-412. Reproduced with permission from Emerald Group Publishing Limited.

a 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59

-C'(a) C(a)

a

4.4583 4.2219 4.0071 3.8112 3.6317 3.4666 3.3144 3.1736 1.0430 2.9214 2.8081 2.7022 2.6029 2.5098 2.4223 2.3398 2.2620 2.1885 2.1189 2.0529 1.9904 1.9309 1.8744 1.8205 1.7691 1.7201 1.6733 1.6285 1.5857 1.5446 1.5052 1.4675 1.4312 1.3963 1.3628 1.3306 1.2995 1.2696 1.2408 1.2129

1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99

a 1.1861 1.1601 1.1351 1.1108 1.0874 1.0647 1.0427 1.0215 1.0008 0.9809 0.9615 0.9427 0.9244 0.9067 0.8895 0.8728 0.8565 0.8407 0.8253 0.8104 0.7958 0.7816 0.7678 0.7544 0.7413 0.7285 0.7161 0.7039 0.6921 0.6805 0.6693 0.6583 0.6475 0.6370 0.6268 0.6168 0.6070 0.5974 0.5880 0.5789

2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.30 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39

0.5700 0.5612 0.5527 0.5443 0.5361 0.5281 0.5202 0.5125 0.5050 0.4976 0.4904 0.4833 0.4763 0.4695 0.4629 0.4563 0.4499 0.4436 0.4374 0.4314 0.4254 0.4196 0.4139 0.4083 0.4028 0.3974 0.3920 0.3868 0.3817 0.3767 0.3717 0.3669 0.3621 0.3574 0.3528 0.3483 0.3438 0.3395 0.3352 0.3309

Appendix

2.40 2.41 2.42 2.43 2.44 2.45 2.46 2.47 2.48 2.49 2.50 2.51 2.52 2.53 2.54 2.55 2.56 2.57 2.58 2.59 2.60 2.61 2.62 2.63 2.64 2.65 2.66 2.67 2.68 2.69

0.3268 0.3227 0.3187 0.3147 0.3108 0.3070 0.3032 0.2995 0.2959 0.2923 0.2887 0.2853 0.2818 0.2785 0.2752 0.2719 0.2687 0.2655 0.2624 0.2593 0.2563 0.2533 0.2504 0.2475 0.2446 0.2418 0.2391 0.2363 0.2336 0.2310

2.70 2.71 2.72 2.73 2.74 2.75 2.76 2.77 2.78 2.79 2.80 2.81 2.82 2.83 2.84 2.85 2.86 2.87 2.88 2.89 2.90 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99

0.2284 0.2258 0.2233 0.2208 0.2183 0.2159 0.2135 0.2111 0.2088 0.2065 0.2042 0.2020 0.1998 0.1976 0.1955 0.1934 0.1913 0.1892 0.1872 0.1852 0.1832 0.1813 0.1793 0.1774 0.1756 0.1737 0.1719 0.1701 0.1683 0.1666

3.00 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29

393

0.1648 0.1631 0.1614 0.1598 0.1581 0.1565 0.1549 0.1533 0.1517 0.1502 0.1486 0.1471 0.1456 0.1442 0.1427 0.1413 0.1399 0.1385 0.1371 0.1357 0.1344 0.1330 0.1317 0.1304 0.1291 0.1278 0.1266 0.1253 0.1241 0.1229

A.III.5 General remarks

A.III. 5.1 Fitting Zipf s function

Since the function of Zipf is of the same type as the one of Lotka one can use the same techniques, as described above, to fit Zipf s function. Of course, this fitting limits itself to the graph of Zipf s function and does not take into account the fact that the arguments in Zipf s function are ranks. For a study of this topic we refer to Tague and Nicholls (1987).

394

Power laws in the information production process: Lotkaian informetrics

A.III. 5.2 The estimation of pm and nnlax

So far we did not go into the issue of estimating the maximal item density pm (in the continuous model (A.III. 1)) or the maximal number of items per source, nmax (in the discrete model (A.III.2)). For pm this is completely solved (mathematically) in Subsection II.2.1.2, given A > T . For n lmx , the statistical estimate is discussed in Tague and Nicholls (1987). It turns out that, if the sample size is high enough, we can use the "sample" nmax for the population n ^

(to use in the model) and that the maximum likelihood estimate for the

population n ^ is its sample value (Tague (1988), Tague and Nicholls (1987)).

One can also put pm or n ^ equal to +oo, hereby simplifying the 3-parameter model to a 2parameter one. Both type of models are compared in Kinnucan and Wolfram (1990).

A.III.5.3 Fitting derived functions such as Price's law

The fitting techniques described here allow us to use the results developed in this book. In this way we can check for the exact form of derived results (e.g. the type of law of Price that is valid, given parameters C, a and pm - see Chapter IV). There is, therefore, no need to perform separate tests for these derived results as e.g. was done in Berg and Wagner-Dobler (1996), Gupta, Sharma and Kumar (1998) and Nicholls (1988). That Price's law fails (in the latter article) because of its relation to "Lotka's law, which is considered as an inverse square law rather than as a generalized model taking variable parameter values" is, in view of Corollary IV.3.3.2, a correct statement: Price's law is equivalent with Lotka's law with a = 2 and pm = C, so a very special case of Lotka's law (allowing no parameter freedom). In Geisler (2000) p. 173, note 22, this is rephrased in a wrong way: Geisler only refers to "an inverse relationship" which is meaningless.

If one considers Zipf s law as a derived function of Lotka's law (cf. Chapter II) one could, in view of the remarks made above on Price's law, also state that a fit of Lotka's law suffices and that, from the obtained parameters, Zipf s law can be reconstructed. This argument is also given in Urzua (2000) (but notice that what they call "Pareto's law" is the (continuous) informetric definition of Lotka's law, the functional relation being given by formula (11.13)).

Appendix

395

A.III. 5.4 Goodness-of-fit tests

We do not go into the diverse techniques to test the quality of the fit, i.e. how close is the calculated model to the data. Techniques such as x2

or

Kolmogorov-Smirnov can be read in

books on general statistics. We even dare to say that, for applying the results of this book on Lotkaian informetrics, we assume that a power law applies for the size-frequency function or the rank-frequency function and we just want to know the "best" parameters C, a (and, if pm cannot be taken as +oo, pm ). It is very well possible, due to the stability of the mathematical results in this book, that a Lotkaian model with the calculated parameters does not fit in the statistical sense but that the model, nevertheless, can be used to see what are the derived properties. Also, in many cases, it even suffices to see whether a < 2 or a > 2 in order to know the form of the derived property (e.g. concave or S-shaped first-citation distribution see Subsection VII.5, Groos droop or not - see Fig. II. 1, and so on). Even in the case of a nondecreasing size-frequency function (such as in some cases of number of authors per paper) we can use a Lotkaian function as a first approximation and use it for the derived properties. Also in this case a statistical fit does not help us, which is also the case in which a non-power law decreasing model can be fitted to the data but where a power law still can be used as a mathematical model due to general considerations.

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SUBJECT INDEX

80/20-rule 188, 199, 200 - generalized 188

Brookes' law 153 Buckland's problem 40

A

C

adjusted counting 255 aging distribution - cumulative 323 - exponential 304, 316 - lognormal 304 alphabetical ranking 310 arc 297 ASCII coding 67 Asian languages 328 author count - pure geometric 255 - senior 254 - total 254 author rank 304 - distribution 308 authors - number per paper 90 authorship determination 328 average 110 average effort 70 average information content 69 average screen length 359

cardinality 11 career duration 93 Cartesian product of IPPs 326 Central Limit Theorem 302 Chinese symbols 334 Chinese words 87 citation 8 citation age 300 citation network 87 classification of subjects 328 CLT 302 coastline of Norway 241 coefficient of variation: see variation coefficient collaboration network 87 complexity 231 compression of texts 328 concave 193, 197, 315, 319, 320 concentration measure 189, 190, 194, 198, 205 concentration theory 187 - continuous 196 -discrete 192 -Lotkaian 187, 199 - of linear three-dimensional informetrics 218 - of Type/Token-Taken informetrics 226 conditional expectation 56, 270 conferences 7 continuous concentration theory 196 continuous Lotka function: see Lotka's law convolution 251, 277, 280, 281, 284 counting - adjusted 255 - fractional 247, 258 - normal 254 -proportional 258 - standard 254 - straight 254 -total 18,247,257 counting procedure 253 crediting system 253 cumulative advantage 45

B Benford's law 20 Bernoulli trial 309 bibliometrics 7 binary digit 67 binomial distribution 272, 309 bit 67 book 8 Booth's law 47 borrowing 202 box-counting dimension 239 Bradford formalism 105 Bradford multiplicator 22, 23, 126 Bradford's function (see also Bradford's law) 122, 129, 143 Bradford's law 22, 126, 154 - graphical formulation 24, 105, 127, 154, 156 - group-free version 23 - verbal formulation 24 branch 55

424

Power laws in the information production process: Lotkaian informetrics

cumulative aging distribution 323 cumulative citation distribution 325 cumulative first-citation distribution 314 cumulative nth citation distribution 323, 325 cumulative rank-frequency function 13, 105 D De Solla Price's law: see Price's law decimal digit 68 decreasing exponential function 15 decreasing power law 14 demography 9 density 26 diachronous 325 - obsolescence 40 dictionary 67 dimensional complexity 239 discrete concentration theory 192 discrete Lotka function: see Lotka's law dit 68 diversity 214 domain name 90 double logarithmic scale 98 dual function 252 dualIPP 13, 102 dual Lotka law 276 dual size-frequency function 144, 276 dual, duality 12, 102, 159, 325 E econometrics 9, 21 entropy 65, 69, 213 Erdos-Renyi model 33 Erdos-Renyi network 88 error correction 328 error detection 328 evolution in time 28, 31, 32 existence theorem 111, 116 exponential aging distribution 304, 316 exponential function 32 exponential growth 34 exponential obsolescence 34 exponentially decreasing rank-frequency function 135 F Failure Breeds Failure 48 FBF48 first-citation distribution 313 - cumulative 314

forgetting 33 fractal 232 - self-similar 232, 234, 243 fractal dimension 232, 236, 243 fraction of multinational publications 274 fractional 92 fractional author count 255 fractional count(ing) 18, 247, 258 fractional counting system 146 fractional frequency curve 372 fractional frequency distribution 288, 292, 295 fractional frequency score 146 fractional score 146 fractional scoring system 251 fractional size-frequency function 276 fractionated score 275 fractionated scoring system 256 Fubini's theorem 308 functional relation 370 G gambling theory 56 Gaussian distribution 302 generalized 80/20-rule 188 generalized bibliography 8, 101 geometric mean 269 Gini index 194, 198, 205 Goodness-of-fit 395 Gozinto theorem 175 graph 87 Groos droop 139, 298 growth 55 growth process 61 growth rate 34 H Harmonic Mean Response Times 314 Hausdorff-Besicovitch dimension 236 hyperlink 87 I identification of languages 328 immediacy index 314 incomplete data 373 indexation of music 328 indexing 327 inequality 187 inflection point 140 information content 66 information production process 8, 101 information retrieval 326

Subject index Information Science Abstracts 86 informetrics 7, 86 in-link 54, 89 Institute for Scientific Information 314 integral test 307 internet 87 intranet 87 IPP 8, 101 - complexity 231 - dual 13, 102 IR 326, 327 ISA 86 ISI314, 325 ISSI7 item 8 item systematic sample 374 J JCR314 Journal Citation Reports 314 journal productivity 300 L law of anomalous numbers 20 law of Bradford: see Bradford's law law of Leimkuhker: see Leimkuhler's law law of Lotka: see Lotka's law law of Mandelbrot: see Mandelbrot's law law of Pareto: see Pareto's law law of Price: see Price's law law of succession of Laplace 48 law of surfing 89 law of total probability: see theorem of total probability law of Weber-Fechner: see WeberFechner's law law of Zipf: see Zipf s law Leimkuhler curve 297 - arc 297 Leimkuhler's function: see Leimkuhler's law Leimkuhler's law 22, 122, 128, 153 lexicographical ordering 310 Library and Information Science Abstracts 86 line segment 233 Linear Least Squares method 388 linear regression 15, 241, 388 linear three-dimensional informetrics 161 linear three-dimensional Lotkaian informetrics 175 linguistics 20, 86 LISA 86

425

LLS 388 loan 8, 223 logarithmic effort function 75 logarithmic function 66 lognormal aging (distribution) 33, 304 Lorenz curve 190, 193, 196, 200 -construction 223 Lotka exponent 15 Lotka's function: see Lotka's law Lotka's general power function: see Lotka's law Lotka's law (see also Lotkaian informetrics) 14, 122, 128, 135 - continuous 378 - discrete 378 Lotka's power law: see Lotka's law Lotkaian concentration theory: see Lotkaian informetrics Lotkaian continuous concentration theory: see Lotkaian informetrics Lotkaian function existence theory 114 Lotkaian informetrics -basic theory 101 - examples 85 - introductory aspects 7 - linear three-dimensional Lotkaian informetrics 175 - Lotkaian concentration theory 187 - Lotkaian continuous concentration theory 199 - Lotkaian fractal complexity theory 231 - Lotkaian Type/Token-Taken informetrics 177 Lotkaian Type/Token-Taken informetrics 177 M Mandelbrot's argument 329 Mandelbrot's function: see Mandelbrot's law Mandelbrot's law 19, 42, 122, 128, 143, 245 martingale 60 Maximum Entropy Principle 76 Maximum Likelihood Estimating 390 mean citation age 301 mean reference age 304 measure of dispersion 214 measure of the concentration: see concentration measure median citation age 300 median reference age 304 memories 33

426

Power laws in the information production process: Lotkaian informetrics

MEP76 MLE 390 most-recent-reference distribution 325 multi-authored paper 304 multiple authorship 247 multiple sources 247 multiplicator of Lagrange 71, 77

publication game 46

N Naranan model 34 Neperian logarithm 68 network 87 - citation network 87 - collaboration network 87 N-gram 326 - non-redundant 327 - redundant 327 noblesse oblige 256 node 55 nonlinear regression 320 non-redundant N-gram 327 normal counting 254 N-word phrase 326

random network 33,88 random text 42, 245 rank-frequency function 12, 105, 333 ranking 11 ratio scale 28, 99 rectangle 233 redundant N-gram 327 reference 8 reference age - mean 304 - median 304 regression -linear 241, 388 - nonlinear 320 retrieval of music 328 Riemann-zeta function 17, 381, 391

O

S

Obsolescence 40 - diachronous 40 - synchronous 40 O-notation 306 out-link 89

sample 373 SBS45 scale-free (property) 27, 85, 86, 98, 246 - characterization of power laws 29 scale-free function 365 scale-invariant property 189 scattering 40 scientometrics 7 search key 328 seed 308, 310 self-similar, self-similarity 233, 246 self-similar fractal 232, 234, 243 sensation 68, 94 sensation law of Weber-Fechner: see Weber-Fechner's law significant-digit 20 similarity dimension 234, 236 similarity measure 327 sitation 90 size-frequency function 10, 105, 333 slope of the regression line 262 social network 88 source 8 source systematic sample 374 source-item identity 298 speech recognition 328 S-shaped (growth) 33, 315, 319, 320 standard counting 254

P parallelepiped 233 Pareto's law 21 PLE70 PME78 positive reinforcement 163, 175, 223 - concentration 219 post-coordinative retrieval 326 power law: see Lotka's law, Zipf s law Pratt's measure 196,207 pre-coordinative indexing 326 Price index 304 Price's law, Price's law of concentration 94,97,188,214,394 Price's square root law: see Price's law Principle of Least Effort 70 Principle of Most Effort 78 principle of nominal increase 190, 223 product property 29, 30, 251 proportional author count 255 proportional counting 258

Q quantitative linguistics 8 R

Subject index standard normal distribution 302 standards, standardization 10, 89 stimulus 68, 94, 98, 218 stochastic process 55 straight counting 254 stretched exponential 33 submartingale 60 success 309 Success Breeds Success 45 super source 253, 256 supermartingale 60 synchronous 325 - obsolescence 40 systematic sample in the items 374 systematic sample in the sources 374 T Theil's measure 195, 211 theorem of total probability 281, 305, 308, 310 three-dimensional informetrics 157 - linear three-dimensional informetrics 161 -linear three-dimensional Lotkaian informetrics 175 time dependence 92 total count(ing) 18, 247, 257 total influence 172 total scoring system 249 transfer principle 190 tree 55 triadic von Koch curve 234, 242 truncation 327 - generalization 327-328 TT (informetrics): see Type/Token TTT (informetrics): see Type/TokenTaken two-dimensional informetrics 9, 157 Type/Token 8, 168 - average 352 Type/Token-Taken 168, 226, 329 - average 352 - concentration 226 - Lotkaian Type/Token-Taken informetrics 177 Type-Token identity 298 U universal Bradford multiplicator: see Bradford multiplicator urn model 46

V variation coefficient 195, 207 W Weber-Fechner' s law 68, 75, 94, 218 WWW 87 Z Zipf s function: see Zipf s law Zipf s law 19, 150, 328, 393

All

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Informed Consent: Information Production and Ideology

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Power to the Edge: Command and Control in the Information Age (Information Age Transformation Series)

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Dictionary of Information and Library Management:

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