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Annu. Rev. Sociol. 2002. 28:197–220 doi: 10.1146/annurev.soc.28.110601.140942 c 2002 by Annual Reviews. All rights reserved Copyright °
MATHEMATICS IN SOCIOLOGY Christofer R. Edling Department of Sociology, Stockholm University, S-106 91 Stockholm, Sweden; e-mail:
[email protected]
Key Words mathematical models, rational choice, social mechanisms, social process, social structure ■ Abstract Since mathematical sociology was firmly established in the 1960s, it has grown tremendously. Today it has an impressive scope and deals with topical problems of social structure and social change. A distinctive feature of today’s use of mathematics in sociology is the movement toward a synthesis between process, structure, and action. In combination with an increased attention to social mechanisms and the problems of causality and temporality, this synthesis can add to its relevance for sociology in general. The article presents recent advances and major sociological research streams in contemporary sociology that involve the application of mathematics, logic, and computer modeling.
INTRODUCTION In this article I present the use of mathematics in contemporary sociology by pointing out recent advances and highlighting some major sociological research streams in which mathematics is used intensively. The overview is selective and by no means exhaustive. One restriction is important to mention. A special characteristic of the mathematical approach to sociological problems is that it involves many scholars from outside the sociology discipline. Such examples include John Harsanyi (1976), Anatol Rapoport (1983), and Herbert Simon (1957), three influential scholars who have conducted research over an extended period of time in areas highly relevant to sociology.1 However, I have decided to concentrate on work by sociologists only. Most of the references are to books and journals of sociology. The decision to keep the text free of equations has two rationales. First, I believe that a presentation of the use of mathematics in sociology deserves to be presented in a widely accessible format, and second, even if kept at minimum, a fruitful 1 Among these three, Harsanyi actually held a PhD in philosophy with minors in sociology from the University of Budapest. Before emigrating in 1950, he was employed at the University Institute of Sociology in the same city. According to his own account, the “conceptual and mathematical elegance” of economics made him switch when he had to study for a new degree in Australia in the early 1950s (Fr¨angsmyr 1995).
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technical discussion of the work presented here would require considerably more space than is available. I refer to the subject area under review interchangeably as mathematical sociology, mathematics in sociology, the mathematical approach to sociology, etc. A reason for this is that the label mathematical sociology is a bit problematic. In many sciences, physics for instance, the mathematical is a meta-theoretical activity that develops principles for modeling and analyzes consistency of theories. In sociology, however, theory is not widely associated with any sort of mathematics or formalization. Therefore, mathematical sociology has become a wide umbrella for a heterogeneous field using mathematics, logic, and computer simulation to illustrate and solve sociological problems (Feld 1997a). It is a heterogeneous field in the same sense that sociology at large is. It includes all possible types of subjects and areas, and it utilizes the whole range of possible methodological techniques for empirical testing. In addition, it is also heterogeneous in the type of mathematics, logic, and computational procedures applied to the various problems. This is not to say that sociology avails itself of an exceptionally broad range of mathematics, but the fact is that two sociologists approaching the social world by means of mathematics may have very little in common in terms of the approach and the models they use, even though they share a fundamental formal inclination (also, see the discussion in Freese 1980). The use of mathematics to solve and illuminate questions about society can be dated at least back to the eighteenth-century French philosopher de Condorcet, who did work on probability and decision-making. Modern mathematical sociology however, was born in the late 1940s to mid-1950s; classic texts include Karlsson (1958), Lazarsfeld (1954), and Rashevsky (1951). The approach really gained impetus in the 1960s, the classic being Coleman’s (1964) Introduction to Mathematical Sociology. Thus, since around the middle of the century just past, a number of sociologists have come to describe themselves as mathematical sociologists. As the label makes plain, these scholars are sociologists who in one way or another apply mathematics to sociology. Mathematical sociology has had some success, but counted in number of adherents, it has remained quite small: In July 2001, the mathematical sociology section of the American Sociological Association had 185 members. It remains an important and vital activity, however, and the main impression has to be that mathematical sociology has grown tremendously over the past 30 years. Lately, we have even witnessed a sort of revitalization of the field, much of it through the growth of social network analysis (Hummon & Carley 1993, Doreian & Stokman 1997), and the emergence of computational modeling (Hummon & Fararo 1995, Prietula et al. 1998, Gilbert & Troitzsch 1999). Even though the last major review was written in the mid-1970s (Sørensen 1978), both the journal Sociological Forum (1997, vol. 12, no. 1) and Sociological Theory (2000, vol. 18, no. 3) recently featured special issues in which prominent scholars reflected on and discussed the role of mathematics in sociology (see Abell 2000, Berger 2000, Fararo 1997, 2000, Feld 1997a,b, Jasso 1997, Heise 2000, Lieberson 1997, Skvoretz 2000, White 1997, 2000).
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THE USE OF MATHEMATICS IN SOCIOLOGY The presentation of mathematical tools is not part of the presentation of theory or methods in sociological textbooks, and it has been a long time since a textbook on mathematical sociology has appeared. The last broad introductory text written by sociologists appeared more than 25 years ago (Leik & Meeker 1975). Today it might not even be feasible to put together an encyclopedic textbook of the sort that Fararo (1973) wrote in the early 1970s because of the sheer amount of sociology and mathematics that would have to go into it. Even though there are recent introductory texts on selected areas (Bradley & Meek 1986), and introductory texts aimed at mathematics undergraduates (Beltrami 1993), the student who wants an up-to-date introduction to the use of mathematics in sociology has to turn to research papers in selected areas. Several useful collections exist. For example, the volume by Szmatka et al. (1997) can be read as a nice, not very technical introduction to the use of mathematics in sociology. The book contains papers on the role of mathematics in sociological research and its relation to theory, as well as applied mathematical models in the areas of status and networks. The book is also interesting because it brings together a vast sociological research stream, the accumulated results of more than 30 years of work. Contributors would probably claim that this unity in advanced research could not have been accomplished without the language of mathematics (Berger 2000). Other, slightly more demanding collections on topical problems such as solidarity (Doreian & Fararo 1998) and the evolution of social networks (Doreian & Stokman 1997) have recently been compiled. A general text that concentrates on theory is Sociological Theories in Progress (Berger et al. 1989). This is the third volume of a series of three (Berger et al. 1966, 1972) that gives a nice illustration of the development of formal theory in sociology over several decades. As I hope to show, mathematics may be fruitfully used to address sociological problems. Of course, the discussion about the utility of formalization in sociology is on-going (see Wilson 1984). Ever since the birth of the discipline, sociology has been haunted by the chasm between science and literature (Lepenies 1988). There is no reason why people who believe that sociology belongs to the arts should embrace the use of mathematics in sociology (though mathematics has aesthetic qualities). However, also among sociologists who subscribe to the idea that sociology belong to the sciences, there is a diffused skepticism toward the use of mathematics. I think this is partly because mathematics is falsely associated with a vulgar version of the natural sciences. It is true that the ideas of a unified science have inspired some sociologists to use mathematics, but even these efforts have always been very sensitive to the particularities of sociology and social science (e.g., Fararo 1989). But for most scholars who use mathematics to do sociology, this use is only for clarity and precision. Indeed, Herbert Simon (1957, p. 89), commenting on his own work on the theories of Festinger (Simon 1957, ch. 7) and Homans (Simon 1957, ch. 6), claims “that the mathematical translation is itself a substantive contribution to the theory,” and that “[m]athematics has become
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the dominant language of the natural sciences not because it is quantitative—a common delusion—but primarily because it permits clear and rigorous reasoning about phenomena too complex to be handled in words.” And Patrick Doreian, long-time editor of the Journal of Mathematical Sociology, puts it in the following way. “Mathematics is a language, and all that mathematical sociology means to me is that sociological ideas are expressed in mathematical terms, and that we try and take advantage of using the mathematics. I don’t see mathematics as being that special, at least in my own work”.∗ If one takes this view, that mathematics is a language, it is reasonable to claim that there is nothing about mathematics in itself that would make it a hindrance to success. Still, one could ask whether it is a language that is appropriate in all of sociology. According to Philip Bonacich, there may be areas where it is more difficult to formalize, fields “where the mathematics is not well developed or no one has thought about what kind of mathematics to use. Mathematical sociology is still a very open area, in which there is still a lot of room for discovery in terms of what kinds of mathematics can be used. So, I wouldn’t say about anything, for example culture, that mathematics can’t be used. It just hasn’t been done yet” (Interview). This idea—that if anything meaningful at all can be said about society, there are no grounds for claiming that it cannot be done with mathematics—we find already in Lazarsfeld (1954). However, everyday language and mathematics are not the same, and in the translation the risk of losing touch is always present, because, in the words of Thomas Fararo: We always know more than we can say. And we always can say more than we can really formally put down in more exacting terms. So as you go further and further from the fundamental intuitions in the interest of being logical and mathematical, you can potentially lose contact with the governing intuitions. But the main gain would be to try to bring the mathematics back into, and as close as possible to the basic intuitions of the field. Trying to represent those intuitions in some way. [. . .] You know, you think sociologically, and then you think mathematically. But these are often hard to fit together. The mathematics enforces a discipline that the other discipline doesn’t really value in the same way. It has its own forms of rigor but they’re not the same. To bring those two into conjunction has always been the sort of thing that I thought of as important.∗ When reflecting upon this “translation problem,” it is important to note, as did Bonacich in the quotation above, that mathematics, and sociology for that matter, is an evolving discipline. In consequence, we have to realize first that there is an immensely large number of combinations to be tried out between sociological intuition and the mathematics at hand. Without more scholars investigating these combinations, we will never know the limits of formalization. Second, there will be new mathematics tomorrow that can and will be put in the service of sociology. ∗
This is quoted from an interview conducted by the author. See acknowledgments.
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This is further stressed by Harrison White, who “cannot imagine that the successes we want to have will ever be achieved without any mathematics. It’s bound to be there. But it’s a subtle matter because the mathematics we need might not be the mathematics we have at the moment. I’ve always been interested in quite different kinds of mathematics. Most people don’t realize that mathematicians are always throwing up new kinds of maths” (Interview). White (1997) has advised sociologists to scan the mathematical literature for good ideas and to take advantage of the great mathematical advances made during the twentieth century (see Casti 1997, 2000 for a popular account). There is much to pick from, and so far sociologists have used only a little of it. Novel sociological insight can be gained simply by applying existing mathematics in a new way (Heise 2000, Skvoretz 2000). Much of the recent substantive and methodological progress in economic sociology (Burt 1992) and organizational sociology (Hannan & Freeman 1989), for example, is due to the application of mathematics to the goal of increasing our understanding of social phenomena. And if one needs examples of enduring contributions from mathematical sociology, lasting and increasing interest in structural analyses of social networks, innovation diffusion, and debates over the concept of rational man are convincing. Various approaches can be taken to classify the use of mathematics in sociology (see Coleman 1964, Berger et al. 1962, Allen 1981). In the most recent major review of mathematical models in sociology, Sørensen (1978) made the distinction between models of structure and models of process. Structure and process involve different sorts of mathematics as well as different sorts of substantive questions (but see Hernes 1976 for an early discussion of combining the analysis of structure and process). This distinction follows a more extensive survey by Sørensen & Sørensen (1977) in which they distinguished between four different classes of models: stochastic models for social processes, deterministic models for social processes, models of structure, and purposive actor models. I follow Sørensen & Sørensen’s outline and keep the discussion under three headings: structure, process, and actor. But, as becomes evident, a distinctive feature of today’s use of mathematics in sociology is that it is increasingly difficult to keep process, structure, and action separated. A second important change since the late 1970s is the growing use of computer simulations as an alternative to mathematical models (Gilbert & Troitzsch 1999). Although the distinction is not always clear-cut, it brings to the fore the difference between experimentation and analytical solutions. Traditionally you construct a mathematical model for a problem and then solve the model analytically. This means that every problem has an exact solution. The more complicated the problem, and the more equations are involved, the harder to solve the model analytically. Eventually, the model becomes so complicated it cannot be solved analytically, in which case the modeler opts for the second best, which is to find a numerical solution by testing a large number of different initial conditions and calculating the answers. The next logical step is to construct a computer program that has all the parts that the modeler believes are important, and then to run the
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program, again with a broad variation in initial conditions. The outcome from the computer program is analyzed with the preferred technique, as would be any other sociological data. The analytically solvable model and the computer simulation model are the two endpoints on the model spectrum, but they should be regarded as complementary. However, as Kathleen Carley notes, computer simulations may prove to be a new entry into the field of mathematical sociology. “The mathematical part is in a funny way harder for most people than computer modeling, and I think that that’s going to be even more true five years from now. In part because our high schools are lousy teaching maths, they’re much better teaching computer science: and the kids love it. They think this is cool. [. . .] So we may ironically be better off in a math perspective, when we get people to kind of backtrack it through computer science” (Interview). Most classifications of mathematical sociology, including that of Sørensen & Sørensen (1977), points to the use of mathematics in constructing theoretical models of social phenomena. Even though many mathematical sociologists are using and sometimes even develop quantitative methods, they often point out explicitly that the use of mathematics in sociology should not be equated with statistics (Meeker & Leik 2000). In practice, the distinction between using mathematics in theoretical and in statistical model construction is fuzzy. For example, a linear regression model can very well be used as a theoretical model, although this is seldom done. Rather it is used because linear regression makes for a straightforward way to obtain parameter estimates from statistical data (see Sørensen 1998 for a critique). However, an important point is that the use of mathematics in sociology is not about giving a quantitative approach to data preference over a qualitative approach. Because arithmetic and statistics involve mathematics, many believe this to be the case. In formal theory there need not be any use of quantification or even testing, and some of the classic (Lorrain & White 1971, White 1963) and recent (Abell 1987) works of mathematical sociology involve no statistics and little quantification. At the same time it would be wrong not to acknowledge that statistical modeling is the area within sociology in which mathematics has the strongest impact on the field as a whole. The kind of statistical tools now available for doing network analysis (Wasserman & Faust 1994), event-history analysis (Blossfeld & Rohwer 1995), and hierarchical linear modeling (Snijders & Bosker 1999), for example, are pushing contemporary sociology to new levels of sophistication. This is partly because easy-to-use software packages are available that allow even the most mathematically inept sociologist to define and estimate statistical models. No such software packages are yet available for formal theory development (although there are several intuitive software packages for basic dynamic simulation and agent-based modeling, and modern mathematics packages do much to simplify mathematical analysis). Theoretical model building is an act of balancing realism, generality, and precision, in which one will have to stand back in favor of the others (Levins 1966). Mathematical sociology is often accused of sacrificing realism for precision. To
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some extent this critique is justified, but as has already been implied, it is not surprising because precision is what attracts many of the proponents of the field. Moreover, the way to manage this balancing act is not agreed upon. Fararo (1989) argues that realism is the driving force in model construction: We build the model because there is something there to be modeled, and models are deliberately constructed as representations of the real world. Obviously, model building is about making idealizations of a complex reality by using simplifying, and sometimes false, assumptions. However, few sociologists would base their models on obviously false assumptions if it meant distorting the essential feature of the problem. As a consequence, we also find among mathematical sociologists skepticism toward parts of economics. When I discussed this with Peter Abell, he said: If I look at contemporary economics, for instance, I’m sometimes worried about the extent to which technical facility is so highly rated that people can spend their time immersed in the technical problems and lose sight of the fact that we’re really trying to understand a complex world. They have to simplify the world that they want to look at to such a degree that one sometimes wonders whether it’s worthwhile. I would not like sociology to take that direction. I think the great strength of sociology, if it has any strength, is that it has tried to take empirical complexity seriously, and hasn’t done what some parts of economics have done, and I think we should preserve that (Interview). Still, the issue is delicate. I would prefer a sociological model to be general enough to explain social phenomena across time and space. In addition sociological models should be precise, or else they cannot serve as hypothesis generators and consistency checkers in any substantial way; and those are two important functions for theoretical models (Carley 1997). Consequently it can be argued that realism will have to give way to generality and precision. Such a modeling paradigm is characterized by the expectation “that many of the unrealistic assumptions will cancel each other, that small deviations from realism result in small deviations in the conclusions, and that, in any case, the way in which nature departs from theory will suggest where future complications will be useful” (Levins 1966, p. 422). All the work presented in the following three sections is motivated by the urge to understand social phenomena, and the authors try to reach this understanding through precise and general models.
PROCESS The mathematics of social processes can be broadly divided into stochastic models and deterministic models. For examples of the former see Bartholomew (1982), for the latter see Epstein (1997). In a deterministic process we can fully determine its future state if we know the current state of the process. If we are dealing with a stochastic process, on the other hand, its future state can only be predicted from the present with some probability. Deterministic processes are described by differential
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(or difference) equations. The main tool for describing stochastic processes is the stationary Markov process, of which the Poison process and Brownian motion are variants (differential equations are used in constructing stochastic models as well as to model change in probability distributions). The use of process models in sociology derives from Coleman’s (1964) Introduction to Mathematical Sociology. For example, that book had a major impact on the development of event-history techniques in sociology (Blossfeld & Rohwer 1995, Tuma & Hannan 1984). Variants of Markov processes have been extensively used to model the social mobility of individuals (Stewman 1976). Examples of recent models in that direction are a structural model of employment (Montgomery 1994) and an empirically oriented analysis of the effect of divorce on personal efficacy (Yamaguchi 1996). Other models involving Markov processes include models of voluntary association (McPherson 1981) and the emergence of network ties (Fararo & Skvoretz 1986). Another fascinating application is Padgett’s (1981) analysis of the budget process. The vacancy chain model (Chase 1991), pioneered by White (1970), is an interesting example of the clever use of simple stochastic models for mobility where structural positions move rather than people. Sørensen (1977) formulated a model for status attainment based on the idea of vacancy competition. For a related application to organizational attainment, see Hedstr¨om (1992). In the case of vacancy chains, it is the novel use of old mathematics (by turning the process upside down) rather than new or refined mathematics that is striking. Historically, stochastic models have been used more frequently than deterministic ones in sociology. There are several reasons for this, one being that sociologists in general regard deterministic models as suspect, another that Coleman’s (1964) textbook dealt with stochastic process models. Third, with stochastic process models, change in discrete variables can be modeled directly. As most sociological data are discrete, this property seems very attractive (see, e.g., Tuma & Hannan 1984). As far as the basic mathematics goes, the distinction is quite appropriate. A deterministic model deals with change in variables, and a stochastic model deals with change in probability distributions. But it is unclear whether the distinction matters that much in sociological application. Techniques for estimating continuous deterministic models with discrete data are easily derived (Huckfeldt et al. 1982, Tuma & Hannan 1984), and stochastic components can be incorporated into differential equations. Many of the deterministic dynamic models that utilize differential equations of interest to sociologists are found in epidemiology and biology (Murray 1993), and in the literature on innovation diffusion (Mahajan & Peterson 1985). An interesting analysis of diffusion models and their relationship with event-history models is given by Diekmann (1989), and Granovetter & Soong (1983) provide a central discussion of threshold models of diffusion. Examples of sociological analyses that translate models from epidemiology and biology more directly include models of membership competition (McPherson 1983), street gang growth (Crane et al. 2000), and church growth (Hayward 1999). Some limitations of such
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models are analyzed in an evolutionary model of group formation (Dittrich et al. 2000). Deterministic models are also the building blocks for models of chaos and catastrophe (Brown 1995). These have been enthusiastically reviewed, but few interesting applications have been suggested, possibly because they are hard to test without a very large number of data points (Williams 1997), which is seldom available in sociological data-sets. One of the most successful process-oriented research programs in contemporary sociology has been organizational ecology (Hannan & Freeman 1989). This approach borrows from evolutionary biology and ecology, and draws largely upon the idea of selection. It is at least partly mathematically based. Most of the basic theory has been developed using deterministic models of population dynamics that are translated into an organizational context. The general idea is that organizations compete for resources in niche space and that the survival of an organization is contingent upon the organizational environment. Although organizational ecology originated from an interest in organizational dynamics, it quickly grew into a statistically inclined organizational demography (see Carroll & Hannan 2000). Given the empirical nature of sociology, it is not surprising that the greatest impact of organizational ecology on sociology as a whole is the event-history models derived by Hannan & Freeman (1989) to analyze organizational founding and death rates based on empirical data. There has been some work on the theoretical backbone of organizational ecology (Hannan 1991), including a microsimulation on the niche concept (Hannan & Ranger-Moore 1990). In addition, the theoretical apparatus has been scrutinized by means of first-order logic (Peli et al. 1994) to derive novel implications (Vermeulen & Bruggeman 2001) and to check for logical consistency (Bruggeman 1997, Hannan 1998). Related approaches to organizational dynamics include elaborations on the niche concept (McPherson 1983), individual level analysis of membership selection (McPherson & Ranger-Moore 1991) influenced by complexity theory (Butts 2001, Kauffman 1995), and computer simulations of organizational adaptation (Carley & Svoboda 1996). Interestingly, the research on organizational dynamics is one of the few modern sociological research programs that has opened up a debate with the biological and ecological sciences, a debate much more in evidence in the early days of sociology (but see Boorman & Levitt 1980). Although the issue here is the ecology of types of organizations (Hannan & Freeman 1989) or organizational members (McPherson & Ranger-Moore 1991), which can hardly be equated with animal or plant species, it is interesting to note the leverage gained in organizational analysis by incorporating and modifying a few simple ideas from ecology. In parallel, we should make note of the renewed interest in evolutionary theory that has revitalized current economics (Weibull 1995). Such cross-disciplinary exchanges are healthy, and it can be argued that the common language of mathematics facilitates them. A weakness of the models discussed in this section is that they do not allow for much individual heterogeneity. This is mainly due to the way the mathematics works. Modeling true heterogeneity means adding a new equation for each
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individual. Even with moderately large social systems, this quickly becomes cumbersome. This approach to modeling processes is therefore best left for macroprocesses and for the analysis of aggregate data. Stemming partly from this critique of homogeneous models, we have witnessed a growing interest in microlevel process models. The broad research tradition of group processes (Szmatka et al. 1997), discussed in the structure section, and the application of agent-based models, discussed in the section on purposive actors, can also be regarded as process-oriented activities. In these models, however, the process itself is represented not directly by one or few equations, but as an outcome of social interaction over time.
STRUCTURE Put crudely, modern mathematical sociology has two fathers, James Coleman and Harrison White. Coleman (1964) pioneered process models, and later rational choice theory (Coleman 1990); White (1963, 1970) pioneered models of structure. Of course, White has also studied actors and interests (e.g., White 1992), but these have never been given the same explicit mathematical treatment as have his analyses of social structure. Both of these authors’ imprint on contemporary work is still clearly recognizable, but at the present, White’s network models exercise the more dramatic influence on mathematical sociology. When Sørensen (1978) surveyed the field, models of social process dominated mathematical sociology, but structural analysis, or social network analysis, was just about to burst forth. Since 1978, social network analysis has had a dedicated journal called Social Networks, edited by sociologist Linton Freeman, that hosts contributions mostly from sociologists, anthropologists, and mathematicians. At the beginning of the twenty-first century, this is undoubtedly the theoretical area in which mathematics is most forcefully put to work. In contrast to the mathematics of processes, the mathematics of structure has been partly developed as an answer to problems in the social sciences (Harary et al. 1965). There are few such examples in mathematics, but this also holds true for utility theory and game theory discussed in the next section. Large bodies of social network analysis have been criticized for being theoretically underdeveloped. And indeed, much of the work that is reported in the journal Social Networks can be categorized as methods and statistics. I concentrate here on the parts of social network research that have theoretical ambitions. Social network analysis (Burt 1980, Wasserman & Faust 1994) always entails the representation of actors and/or objects linked together either by social connections (e.g., two persons are friends or enemies) or shared experience (e.g., they go to the same school). Graph theory and matrix algebra are typically used. Modern classics of the first kind include White’s (1963) analysis of kinship structure. An example of the second is the work on structural roles (Boorman & White 1976, Lorrain & White 1971, White et al. 1976). This work initiated a continuing stream of further work on block modeling (Borgatti 1992, Robins et al. 2001) and structural equivalence (Batagelj et al. 1992, Doreian 1988). What White and collaborators
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proposed is a way of identifying structure by a systematic search for the absence of social ties. The same logic underlies Burt’s (1992) influential idea of strategic use of the “structural holes” of a network. Of related interest is Butts’s recent theoretical work on network complexity (Butts 2000, 2001). Network analysts have taken on classic sociological ideas, such as Simmel’s on the duality of group and individual (Breiger 1974, 1990), as well as everyday puzzles such as the “class size paradox” (Feld 1991). A contemporary idea that has been more extensively elaborated is Granovetter’s (1973) thesis about tiestrength. Granovetter proposed that infrequent social contacts, called weak ties, such as those to old schoolmates and brief acquaintances, can be very important transmitters of crucial information on job vacancies and that weak ties are as important as the strong ties we have to partners, close friends, and relatives. For example, job seekers’ minimum acceptable wage has been included in a formal model (Montgomery 1992), and implications from strong- and weak-tie interaction on unemployment have been analyzed (Montgomery 1994). In an ongoing effort toward theoretical unification, Fararo & Skvoretz (1987) suggested an embedding of the weak-tie hypothesis into biased net theory (see Skvoretz 1990). Expectation-states theory (Zelditch & Berger 1985) and closely related Group Processes (Szmatka et al. 1997), both in themselves formal and mathematical approaches with many ramifications (Berger 2000), constitute a long-standing research program that has gained vigor from formal structural analysis. The core of this tradition of small group research deals with the idea that inequality in face-to-face interaction is determined by the relative status of group members. Integrated with social network analysis under the name E-state Structuralism (Fararo & Skvoretz 1986, Skvoretz & Fararo 1996, Skvoretz et al. 1996), the emergence of network structure can now be investigated from the point of view of individuals’ expectations. This approach to social network evolution, and several others, are included in a volume by Doreian & Stokman (1997) that testifies to the increased attention paid to the emergence and evolution of social networks by network analysts. Thus, the critique that social network analysis is too static is beginning to attract serious attention from social network analysts. A sustained line of research in structural analysis is Friedkin & Johnsen’s theory of social influence (Friedkin & Johnsen 1990, 1997), recently compiled into an exemplary book containing parts on the theory, measurement, and analysis of social influence (Friedkin 1998). In a sense this project is the opposite of E-state Structuralism because it looks at the effect of social structure on interpersonal (dis)agreement through interpersonal social influence. This work investigates the classic problem of social differentiation from a social-psychological perspective by utilizing many ideas, including structural role analysis (White et al. 1976) and spatially structured social influence (Marsden & Friedkin 1993). Although perhaps primarily concerned with experimentation, Network Exchange theory (Willer 1999) is another body of research on social structure that examines interpersonal relations and provides a synthesis between network analysis and exchange theory, with an almost exclusive focus on power relations (Cook & Yamagishi 1992). Other
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analyses of power in networks incorporate individual decision-making (Bonacich 1999, Yamaguchi 2000) and interests (Whitmeyer 1997). With a few exceptions—work applying random and biased-net theory being one (see Skvoretz 1983, 1990), sociologists have mainly concentrated their efforts on rather small social structures. Normally, network size does not exceed 150– 200 actors and is often considerably smaller. This can be explained partly by the strong impact from small groups research. And until quite recently, computational limitations posed a genuine obstacle. Contemporary analysis of the small-world phenomenon (Kochen 1989) demonstrates that theoretical analysis can be undertaken with large graphs as well, due largely to powerful computers. Watts (1999) concentrated on the dynamic implications of network structure and demonstrated that even small variations in local network structure are utterly important for global dynamics. This work brings together ideas of tie-strength and previous work on random nets, and these hold much promise for future development of structural analysis.
PURPOSIVE ACTOR MODELS AND BEYOND As previously mentioned, the various uses of mathematics in sociology have grown tremendously during the past several decades. This is also true with respect to purposive actor theory, or rational choice theory, as it is more often called. For a long time, sociologists regarded rational choice theory strictly as an economics activity. In their extensive review of mathematical sociology, Sørensen & Sørensen (1977) listed Coleman’s (1973) early work on collective action as the only sociological contribution. Fararo (1973) dedicated one part of his textbook on mathematical sociology to game theory but remained rather uncertain about its significance. Sørensen (1978) speculated that such models would become more widely used in the future, but he did not discuss purposive actor applications at all in his review of mathematical models in sociology because of their insignificance at the time.2 Of course, this is no longer the case. In the late 1980s Coleman founded the journal Rationality and Society, which has continued to publish rational choice sociology. Rational choice theory starts from the simplifying and universal assumption that social actors strive in all situations to optimize the outcome of their actions, as they see it. For a review of sociological applications of this simple idea, see Voss & Abraham (2000). If formulated in terms of utility theory, the idea of rational choice gains a tremendous deductive power. Coleman’s (1990) Foundations of Social Theory is the most thorough introduction to sociological rational choice analysis, and Coleman & Fararo (1992) provides further discussion of pros and cons. The second part of Foundations contains a mathematical presentation of the theory. 2
When the Journal of Mathematical Sociology published some of its first papers on game theory in 1977, the editor felt obliged to reassure his readers that it was not his intention to turn the journal into one dedicated to game theory (Hinich M, Laing J, Lieberman B. 1977). Editor’s comment. J. Mathematical Sociology 5:149–50.
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Among other things, the book has spurred a fruitful contemporary debate on trust and social capital. Sociologists working within a rational-choice framework have conducted formal analyses of problems such as addiction (Skog 1997), collective action (Heckathorn 1998, Oliver 1993), power (Yamaguchi 2000), and educational choice (Breen & Goldthorpe 1997). Sociologists have also been interested in trying to model formally the way in which people acquire the beliefs upon which they act. Examples include applying Bayesian updating to models of learning (Breen 1999) and the spread of panic (Butts 1998). Related examples include the volunteer’s dilemma (Diekmann 1993), threshold models of collective behavior (Braun 1995, Granovetter & Soong 1983), and Critical Mass Theory (Marwell & Oliver 1993, Oliver et al. 1985). When actors are assumed to pursue actions based on rational choice without the analysis taking into account the potential actions of other actors, the analysis is performed within the domain of decision theory. When the outcome of an actor’s action is also affected by the actions of one or several other actors, the analysis comes under game theory (Fudenberg & Tirole 1991; for a classic introduction, see Luce & Raiffa 1957). Strictly speaking, game theory and decision theory are not that distinct; a decision is also said to be a game against nature, i.e., against an unintentional actor. The use of game theory is now a major part of modern economics. Indeed, the only Nobel Prize ever awarded for a contribution to pure mathematics was the 1994 prize in economics given to John Harsanyi, John Nash, and Reinhardt Selten for their contributions to game theory. Although the use of game theory among sociologists has increased over the past 20 years (see Swedberg 2001), it is still not widely used outside of mathematical sociology. This is rather surprising because the core idea of game theory is that social actors interact, and that each actor is equally affected, albeit in different ways, by that interaction. The only way in which a game theoretical analysis differs from Weber’s analysis of strategic interaction is that it is carried out in a more systematic fashion. This has been pointed out by Abell (2000), who argued strongly that game theory ought to have greater influence in sociology. Even if sociologists are not contributing to the development of game theory per se, the application of game theory is growing in sociology. Macy & Skvoretz’s (1998) game theoretical analysis for explaining the emergence of trust in a population of strangers is one in a line of studies using formal game theory to tackle problems of social dilemmas (Heckathorn 1998, Raub & Snijders 1997, Weesie & Raub 1996), and the free rider problem (Diekmann 1993). In the continuing tradition of mathematical sociology, scholars also strive to integrate game theory with other modeling approaches, such as network theory (Markovsky 1997, Raub & Weesie 1990) and exchange theory (Bienenstock & Bonacich 1997, Braun 1997, Bonacich & Bienenstock 1993). Sociological variations of game theory include Burns’s work on social rule complexes (Burns & Gomolinska 2000) and Montgomery’s (1998) work on roles. Many sociologists are skeptical about rational choice theory and parts of game theory because of the assumption of utility maximization (see Petersen 1994).
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However, evolutionary game theory (Weibull 1995) is based on the idea that games are played repeatedly over time, and that the best strategy in a game is determined not by forward-looking rational anticipation of actions and consequences, but by consideration of historical traits. Developed in mutual exchange between economics and biology, evolutionary game theory offers the same mathematical and deductive power as classic game theory, but without assuming rational actors. Evolutionary game models have only recently been incorporated into sociology. Macy (1996) provides a critical discussion and comparison between evolutionary games and neural networks, and recently a special section of Sociological Methods & Research (Pollock 2000) was devoted to evolutionary game theory. Agent-based modeling (e.g., Axelrod 1997) brings new vigor to models of agents. In the colorful words of Epstein & Axtell, the basic logic in agent-based modeling is to “grow artificial societies” from the bottom up (Epstein & Axtell 1996). Schelling’s (1978) classic model of segregation, and Axelrod’s (1984) analysis of the evolution of cooperation, can serve as templates for agent-based models. This means starting with a set of agents that use very simple and local behavioral rules, and then studying the effects of social interaction at a global level. Thus, it shares with rational-choice models a preference for grounding theoretical models in the actions of individual actors (Zeggelink et al. 1996b). Hummon (2000) simulates network dynamics with rational actors embedded in a social network, and Macy & Skvoretz (1998) use computer simulation to explore their game theoretical model on the emergence of trust (also see Burt 1999). Other models move beyond rationality assumptions and start with either very simple or very complicated actors. In Mark’s (1998) model of differentiation, the simple assumption made about social actors is that they choose to interact with people who resemble themselves. The model demonstrates that social differentiation can emerge even if we assume almost no individual differences. An opposite approach is to build computer models of social actors in much greater detail, using artificial intelligence (Carley 1996) and neural networks (Macy 1996). A very useful recent introduction to various simulation techniques for the social sciences is Gilbert & Troitzsch (1999). Although the use of mathematics in developing empirical methods is not covered in this article, the method of comparative narratives (Abell 1987) is a special case. It is a way of modeling that allows for formal comparison of two or more narratives. With regard to the comparison, this is undoubtedly a sort of method, and as such it is not without alternatives. Systematic and formalized analysis of qualitative data is a small but growing field embracing sequence analysis (Abbott 1992, Abbott & Tsay 2000), comparative method (Ragin 1987), and models of event structure (Heise 1989). All of these are mathematical approaches to qualitative analysis that provide powerful tools for dealing with historical and ethnographic data. The inclusion of this line of research in this section may or may not be appropriate. The reason is twofold. First, most event structure, sequence, and comparative narratives analysis deal with actions, and thus are related to other models of actors. Second, whereas most of these approaches (naturally) deal with methodological issues, Abell (1993) has suggested that a closer connection between the narrative method
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and game theory is fruitful.3 Abell’s (1987) project has been to formulate semantics of action that facilitates narrative analysis. The analytical strategy is based on the formulation of rules that connect sequences of action, and the idea is to provide a formal language that can be translated into a computer language to facilitate qualitative analysis. In several ways the approach is related to formal versions of interpretive sociology presented by Fararo (1989).
DISCUSSION As has been the case in several previous reviews, I have discussed the use of mathematics under the rubrics of process, structure, and action. But, as may have become evident, one salient characteristic of contemporary work is that this distinction is no longer clear-cut. Social network analysis has brought the concern about structure into nearly all types of models, and the study of networks is turning toward the emergence and dissolution of social ties, as well as to the dynamics of network structure. In these analyses, it is always actors who provide the motion. In agent-based modeling, process, structure, and action are most clearly brought into one inseparable representation of society. As the scope of this review reveals, the mathematical approach to sociology is topical. Still, the use of mathematics for solving sociological problems is not yet widespread. Nonetheless, several signs of movement in that direction appeared in the 1990s. At least three new journals that specialize in mathematical applications to sociological problems have been created within the past six or seven years. One, a conventional printed journal (also available online), is called Computational and Mathematical Organizations Theory and is edited by sociologist Kathleen Carley at Carnegie Mellon University; the other two are electronic journals: the Journal of Social Structure, edited by sociologist David Krackhart, also at Carnegie Mellon, and the Journal of Artificial Societies and Social Simulation, edited by sociologist Nigel Gilbert at the Centre for Research on Simulation in the Social Sciences at the University of Surrey. The first section for mathematical sociology within the American Sociological Association (ASA) was founded in the mid-1990s. Today the section counts 185 members, 30% of whom are students.4 It is hard to point to particularly important 3
This is rather controversial, as we would normally prefer to keep the method free from substantial sociological theory, and then let theory bear on the outcome of our comparisons. Abell received repeated criticism on this point and others in a special issue of Journal of Mathematical Sociology (1993, vol. 18, no. 2–3). 4 As of 2001, the mean size of the 42 different member sections in ASA is 427 members (std. dev. 214), the largest having 967 members and the smallest (in formation) only 96 members. The section for mathematical sociology is among the smallest five in the association. Section members are often members of other sections as well; the most popular overlaps are memberships in the sections for Social Psychology (62 persons), Rational Choice (61 persons), Methodology (50 persons), and Theory (40 persons).
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centers or institutional settings. Most sociologists using mathematics are working in the United States, and they are indeed found in small numbers although in many universities across the country. Some of the most important people are at Carnegie Mellon, Chicago, Columbia, Cornell, Pittsburgh, Harvard, Santa Barbara, South Carolina, Stanford, and UCLA. Much of the work that goes on at these universities is represented in this article. Japanese sociologists are also availing themselves of mathematics (Kosaka 1989), but unfortunately most of this work has to date only been published in Japanese (see the journal Sociological Theory and Methods). In 2000, the first joint conference was organized between the Mathematical Sociology section of the ASA and the Japanese Association for Mathematical Sociology, so an increased exchange across the Pacific might be expected in the future. In Europe the use of mathematics is widespread at the InterUniversity Center for Social Science Theory and Methodology (ICS) in the Netherlands. In particular, scholars at ICS have contributed to game theory (Raub 1988, Raub & Snijders 1997, Weesie & Raub 1996), network analysis (Snijders 1996, Zeggelink et al. 1996a), and diffusion (Buskens & Yamaguchi 1999). Other mathematical sociologists who publish in English are scattered all over Europe, in Germany, the Netherlands, Norway, Sweden, and the United Kingdom, and cross-Atlantic exchanges in mathematical sociology are well established. Despite the fact that mathematical sociologists are quite few in number, the use of mathematics is now an increasingly important aspect of the empirical and theoretical analysis of social structure and change, as is evident from a quick glance through recent issues of leading journals such as the American Journal of Sociology and the American Sociological Review. Sociology will continue to benefit from this development as it enables a closer exchange with other social and physical sciences, and with mathematics. Within network analysis this synergistic relationship between sociology and other sciences is a well-established part of the tradition, and numerous examples of cross-disciplinary cooperation can be cited (Freeman 1984). I have mostly made reference to work done by sociologists, published mainly for a sociological audience. But mathematics is also an important way of communicating sociological knowledge to other sciences. If this is not done, there is a risk that much of sociology will be reinvented by computer scientists, economists, and physicists. So far, impressive proposals for a social physics (e.g., Helbing 1995, Weidlich & Haag 1983) have not had any impact on sociology, perhaps because they offer very little sociological insight and very few empirical examples. But there is some excellent and interesting research being done where one sometimes gets the feeling that sociology really should be able to contribute more. In economics, for instance, social norms (Lindbeck et al. 1999) and social interaction (Durlauf & Young 2001) are now taken very seriously by leading scholars. The authors in the volume edited by Durlauf & Young (2001) take a fresh look at genuinely sociological problems. But without the sociologists! Very recently physicists have begun to study social networks, and this line of work is likely to
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have a great impact on social network analysis. The interest was triggered by a revisit to the small-world phenomena (Watts 1999, Watts & Strogatz 1998) that showed that a random network needs very little rewiring to be transferred into a small-world network, with fundamental consequences for global dynamics. This work has been continued by physicists doing both theoretical (Amaral et al. 2000, Barabasi & Albert 1999) and empirical analysis (Newman 2001). At present the mathematical approach offers only limited hope for unifying sociological thinking. There are explicit attempts to use mathematics as a means to unify theory, most notably in Thomas Fararo’s ambitious project to unify sociological theory through formal and mathematical thinking (Fararo 1989, 2001, Fararo & Butts 1999). In addition, computer simulations are being used to investigate theoretical implications that are hidden in verbally formulated theories (Feld 1997b, Hanneman et al. 1995). Mathematics is often used in sociology to bring together different theoretical approaches. Montgomery (1998) proposed a marriage between the idea of embeddedness and role theory by utilizing a version of game theory in which “players” consist of roles instead of actors. Skvoretz (1983, 1991) applied biased net theory (Skvoretz 1990) to rephrase Peter Blau’s macrooriented theory of social structure and inequality, and in so doing Skvoretz added coherence and derived new theoretical implications [further models that draw on this and other works of Blau are Hedstr¨om (1991), McPherson & Ranger-Moore (1991), and Montgomery (1996)]. Despite these efforts, mathematical sociology will very likely continue to mirror the rest of sociology and to remain a heterogeneous field for a long time to come. Even though the movement toward integration of research and theoretical reasoning has been a major trend for some time (Costner 1988), one major critique that can still be directed against mathematical and formal sociology is that the gap between models and empirical analysis is too wide. Reducing this gap would certainly increase the attractiveness of applying mathematics to sociological problems (Skvoretz 2000), and it would bring theory closer to empirical analysis. Research debates have recently approached this problem. This is not meant to belittle the status of theoretical models. Some theories cannot be tested directly. For example, it is interesting to note that game theoretical analyses of the prisoner’s dilemma have become common place in the social sciences. Indeed, Axelrod’s (1984) fame is due to a computer tournament between rather abstract decision algorithms. What these models do is to propose exact mechanisms that account for social process. If the explanation proposed by such a model provides insight into an important phenomenon, then the model is useful despite the fact that some models cannot be subjected to empirical testing. Still, we have to be aware that testing provides the only feedback to theory. From a theoretical perspective, there is a discussion that proposes explicit social mechanisms (Hedstr¨om & Swedberg 1998, Skvoretz 1998) to bring predictive and deductive power into sociology. Social mechanisms are a long-standing interest in mathematical sociology (Karlsson 1958), and this invitation should be taken seriously. Finally, a call for the use of formal theory to strengthen statistical analysis (Blossfeld & Prein 1998, B¨ackman & Edling 1999,
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Goldthorpe 2000), and a renewed interest in the problems of temporality (Abbott 2001) and causality (Doreian 2001, Goldthorpe 2000, Winship & Morgan 1999) open up a broad discussion of the utility of mathematical models. This literature clearly points toward a direction for future work that would not only maintain the impressive scope of mathematical sociology, but would also further add to its relevance for sociologists in general. ACKNOWLEDGMENTS In 1998–1999 I conducted short interviews with Peter Abell, Philip Bonacich, Kathleen Carley, Patrick Doreian, Thomas Fararo, and Harrison White, of which extracts are quoted in the text. I am most grateful to them all for taking the time, and for being so forthcoming. Peter Hedstr¨om, Thomas Fararo, Fredrik Liljeros, Werner Raub, and one anonymous referee delivered extremely valuable suggestions and comments on an earlier version. David Bachman provided membership figures for ASA sections, and Kenneth Kronenberg edited the English of a previous version. I appreciate that the Sociology Department at Harvard University granted me library access and desk space during two hot summer months in 2001. Financial support from The Bank of Sweden Tercentenary Foundation and the Swedish Foundation for International Cooperation in Research and Higher Education is gratefully acknowledged. I dedicate this article to the vivid memory of Aage B. Sørensen. The Annual Review of Sociology is online at http://soc.annualreviews.org
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