# Mathematical Sociology

# MATHEMATICAL SOCIOLOGY

*Mathematical sociology* means the use of mathematics for formulating sociological theory more precisely than can be done by less formal methods. The term thus refers to an approach to theory construction rather than to a substantive field of research or a methodology of data collection or analysis; it is also not the same as statistical methods, although it is closely related. Mathematical sociology uses a variety of mathematical techniques and applies to a variety of different substantive research fields, both micro and macro.

Theory involves abstraction from and codification of reality; formulation of general principles describing what has been abstracted; and deduction of consequences of those formulations for the sake of understanding, predicting, and possibly controlling that reality. When social phenomena can be described in mathematical terms, the deductive power of mathematics enables more precise and more detailed derivations and predictions based on original premises.

Mathematical expression also enables sociologists to discover that the same abstract forms and processes sometimes describe what seem to be diverse social phenomena. If the same type of formulation describes both the spread of a disease and the adoption of an innovation, then a common type of process is involved and the theorist can search further for what generates that commonality. Ideally, therefore, mathematics provides the basis for very general and powerful integrative theory.

The vigor of mathematical sociology varies widely over the different subfields of sociology. Precise formulation requires precise observations and careful induction of general patterns from those observations. Some sociologists have rejected any attempts to quantify human behavior, either on the grounds that what is important is in principle not subject to precise measurement or from a philosophical unwillingness to consider human behavior to be in any way deterministic. That issue will not be addressed here, but clearly it is much easier to obtain precise information in some areas of inquiry than in others. For example, census data provide reasonably precise counts of many sociological interesting facts, with the consequence that mathematical demography has long flourished.

Mathematics can be thought of as an elegant logic machine. Application of mathematics to any substantive discipline involves careful translation of the substantive ideas into mathematical form, deriving the mathematical consequences, and translating the results back into a substantive interpretation. There are three key aspects to the process: (1) finding a satisfactory way of expressing the substantive ideas in mathematical terms, (2) being able to solve the mathematical puzzle, and (3) eventually being able to compare derivations from the model with data from the substantive application.

The mathematical expression of a substantive theory is called a *model*. Often, models are created with primary emphasis on their being tractable, or readily solved. Some types of models appear frequently in substantive literature because they are widely known, are relatively simple mathematically, and have easy solutions. Basic models and applications can be found in Coleman (1964), Fararo (1973), and Leik and Meeker (1975). Mathematics and sociological theory are discussed in Fararo (1984). A wealth of more complex models can be found in specialized volumes and in periodicals such as the *Journal of Mathematical Sociology*.

Simple models have the advantage of presenting an uncluttered view of the world, although derivations from such models often do not fit observed data very well. When the goal is heuristic, a simple model might be preferable to one that matches more closely the reality being modeled. However, when the goal is accurate prediction and possible control, then more complex models are typically needed.

There are two general questions to be raised in deciding whether developing a mathematical model has been useful. One is whether the mathematics of the model lead to new ideas about how the system being modeled operates. This is purely a theoretical question, concerned with understanding reality better by creating an abstraction of it that enables us to think more clearly about it. The second question concerns how well the model fits that reality, and is a statistical question.

Statistical models are concerned solely with fitting an underlying mathematical model to data from a sample of real-world cases. The underlying model may be complex or simple, but the statistical concerns are whether the sample can be assumed to represent adequately the population from which it was selected and whether the parameter (equation constant) estimates based on that sample can be considered accurate reflections of how the variables of the underlying model are related in that population.

## A HISTORY OF RECENT APPROACHES TO FORMAL THEORY

During much of this century, there has been concern in sociology over the relationship between theory and research. Whereas theory was abstract and typically discursive, research increasingly employed statistical methods. The gulf between verbal statements about theoretical relationships and statistical tests of empirical patterns was great. Beginning in midcentury, some sociologists suggested that one way to bridge this gulf is to translate theoretical ideas into mathematics. Proposals for ways that theoretical ideas could be represented mathematically included those by Simon (1957) and several summarized by Berger et al. (1962).

Another approach to more formal presentation of theories that seemed to offer a bridge over the chasm between verbal statements and statistical tests was Zetterberg's concept of axiomatic theory (1965). The popularized result was *axioms* in the form of monotonic propositions ("The greater the *X*, the greater the *Y*") and hypotheses derived solely from concatenating (multiplying) the signs of the relationships. For example, the axioms "As *A* increases, *B* increases" and "As *B* increases, *C* decreases" lead to the hypothesis that "As *A* increases, *C* decreases." Standard statistical tests of the deduced hypotheses were presumed to be proper tests of the theory that generated the axioms. The approach was quick, convenient, and readily understood, and did not require expertise in mathematics or statistics. Consequently, it quickly became popular.

Numerous inadequacies with such theories soon became apparent, however (see for example critiques in Hage 1994), and interest turned to *path models* based on earlier work by Wright (1934). Path models assume that empirical measurements coincide exactly with theoretical concepts; that all variables are continuous (or reasonably close to continuous); that all relationships (paths) are bivariate, linear, and causal; that there is no feedback in the system (the recursive assumption); and that all relevant variables have been included in the model (the "closed-system" assumption). The underlying theoretical model is therefore very simplistic but does allow for the introduction of assumptions about multiple causal paths, and for different types of causal relationships including intervening, spurious, modifying, and counteracting.

If the assumptions are reasonable, then the causal effects, or path coefficients, are equal to ordinary multiple regression coefficients. That is, the underlying mathematical model feeds directly into a well-known statistical model, and the tie between theory and research seems well established. Are the assumptions reasonable?

The closed-system assumption, for example, implies that there is no correlation between the prediction errors across the various equations. If error correlations appear, then path coefficient estimates based on the statistical model will be in error, or biased, so the theory will not be tested properly by the statistics. Only two solutions are possible: (1) add more variables or paths to the model or (2) develop a statistical model that can accommodate correlated errors.

The assumption that measurement equals concept poses a different problem whenever various scales or multiple indicators are used to represent a theoretical concept not readily assessed in a simple measure. Traditionally, the statistical model called *factor analysis* has been used to handle this measurement-concept problem, but factor analysis was not traditionally linked to the analysis of theoretical systems.

Recent years have seen very extensive development and elaboration of statistical models for linear systems, and these models address both the correlated error and the measurement-concept problems while allowing departure from recursiveness. They are called *linear structural models* (Joreskog 1970; Hayduk 1987). The underlying mathematical model still said that all variables are continuous and all relationships bivariate, linear, and causal. This development focused entirely on technical statistical questions about bias in parameter estimates. Furthermore, although the theory generally supposed that one variable affects another over time, the data are normally from only one or a very few points of time. From a general theory point of view, the underlying linear model's assumptions, which were imposed for tractability, are highly restrictive.

## MOVING BEYOND LINEAR MODELS

Consider the assumption of linearity. If a dynamic process is being modeled, linear relationships will almost always prove faulty. How change in some causal factor induces change in some consequent system property is typically constrained by system limits and is likely to be altered over time through feedback from other system factors.

As a disease like acquired immune deficiency syndrome (AIDS) spreads, for example, the rate at which it spreads depends on how many people are already infected, how many have yet to be infected, and what conditions allow interaction between the not-yet-infecteds and the infecteds. With very few infected the rate of spread is very small because so few cannot quickly infect a very large number of others. As the number of infecteds grows, so does the rate of spread of the disease, because there are more to spread it. On the other hand, if nearly everyone had already been infected, the rate of spread would be small because there would be few left to spread it to. To complicate theoretical matters further, as the disease has generated widespread concern, norms governing sexual contact have begun to change, influencing the probabilities of transmission of the virus. In mathematical terms, the implication is that the rate of change of the proportion infected (i.e., the rate of spread of the disease) is not constant over time, nor is it a constant proportion of change in any variable in the system. In short, the process in inherently nonlinear.

Nonlinear models in mathematics take many forms. If the variables are conceived as continuous over time, and the primary theoretical focus is on how variables change as a consequence of changes in other variables, then the most likely mathematical form is differential equations. The substantive theory is translated into statements about rates of change (Doreian and Hummon 1976). Most diffusion and epidemiology models, like the AIDS problem just noted, use differential equations. So do a number of demography models. Recently some of the numerous differential equations models dealing with topics such as conflict and arms control have been applied to theoretical questions of cooperation and competition in social interaction (e. g., Meeker and Leik 1997).

For relatively simple differential equations models, once the model is developed it is possible to determine the trajectory over time of any of the properties of the system to ascertain under what conditions the covariation of system properties will shift or remain stable, and to ask whether that system will tend toward equilibrium or some other theoretical limit, oscillate in regular patterns, or even "explode."

None of these questions could be asked of a linear model because the mathematics of the linear model leaves nothing about the model itself to be deduced. Only statistical questions can be asked: estimates of the regression coefficients that fit the model to the data and the closeness of that fit. To the extent that sociology addresses questions of process, appropriate theory requires nonlinear models.

What about the assumption of continuous variables? For cross-sectional data, or data from only two or three time points, new techniques of statistical analysis suitable for categorical data have been developed. However, these focus once again on technical statistical problems of bias in estimation of parameters and generally rely on assumptions of linearity. Within mathematical sociology, other approaches exist. If time or time-related variables were to be treated in discrete units, there are at least three different approaches available. For handling dynamic systems without the calculus of differential equations, difference equations are the appropriate form. Huckfeldt and colleagues (1982) provide a convenient overview of this approach.

For extensive time series with relatively few variables, there are Box-Jenkins and related types of models, although these have seen relatively little use in sociology. Because they are closer to statistical models than general theoretical models, they are only mentioned in passing.

Many theories treat systems as represented by discrete states. For example, over a lifetime, an individual is likely to move into and out of several different occupational statuses. Are certain status transitions more likely than others? In simplest form, the implied theoretical model involves a matrix algebra formulation that specifies the probability of moving from each of the states (occupational statuses for this example) to each of the other states. Then the mathematics of matrix algebra allows deduction of a number of system consequences from this "transition matrix." Such a treatment is called a *Markov chain*. An early application by Blumen and colleagues (1955) demonstrated that certain modifications of a Markov chain were needed for the theory to fit the data they had available on occupational transitions. Their work was the initial inspiration for a distinguished string of mathematical models of social mobility. Another classic is by White (1970), who conceptualized mobility of vacant positions as well as of individuals into and out of positions.

Another type of substantive problem that deals with discrete data is the analysis of social networks such as friendship structures. These can be modeled using the mathematics of *graph theory*, the basic concepts of which include *nodes* (or *points*) and *relationships between pairs of nodes* (or *lines*). One of the most vigorous modeling areas in sociology, network analysis has produced a rich and elaborate literature addressing a wealth of substantive issues. Typically, network data consists of whether or not any two cases (nodes in the network) are linked in one or more ways. The resulting data set, then, usually consists of presence or absence of a link of a given type over all pairs of nodes.

Early network analyses concerned friendships, cliques, and rudimentary concepts of structurally based social power. With the introduction of directed graph theory (Harary et al. 1965), random or probabilistic net theory, and block modeling, powerful tools have been developed to approach social structure and its consequences from a network point of view. As those tools emerged, the range of questions addressed via network analysis has greatly expanded. Over the past twenty years, numerous articles in the *Journal of Mathematical Sociology* have dealt with networks. The journal *Social Networks* also publishes work in this area. Overviews and examples can be found in Burt and Minor (1983), and in Wellman and Berkowitz (1988).

Graph theory may be applied to other theoretical issues; for example, a graph-theoretic model developed by Berger and colleagues (1977) helps explain the processes by which people combine information about the various characteristics of themselves and others to form expectations for task performance.

Small group processes have also generated a variety of mathematical formulations. Because observations of groups often generate counts of various types of acts (for example amount of talking by members of a discussion group) that display remarkable empirical regularity, these processes have intrigued model builders since the early 1950's. Recent developments, combining network analyses with Markov chains, include Robinson and Balkwell (1995) and Skvoretz and Fararo (1996).

At the most micro level of sociology, the analysis of individual behavior in social contexts has a long tradition of mathematical models of individual decision making. Recent developments include the satisfaction-balance decision-making models of Gray and Tallman (Gray et al. 1998).

An exciting aspect of these different levels of development is that, increasingly, inquiries into microdynamics based on social exchange theory are working toward formulations compatible with the more general network structural analyses. These joint developments, therefore, promise a much more powerful linking of micro-system dynamics with macro-system structural modeling (Cook 1987; Willer 1987). Recent work on power as a function of the linkages that define the exchange system is an example. The use of mathematics to express formal definitions has enabled researchers to pinpoint where there are theoretical differences, leading to productive debate (Markovsky et al. 1988; Cook and Yamagishi 1992; Friedkin 1993; Bonacich 1998).

There are other examples of work in mathematical sociology involving attempts to develop appropriate mathematical functions to describe a theoretically important concept. One is Jasso's innovative work on models of distributive justice (e.g., 1999). Similarly, affect control theory (e.g., Smith-Lovin and Heise 1988) represents mathematical formulation in an area (symbolic interaction and sociology of emotions) typically considered not subject to such treatment.

One other area of vigorous development deserves attention; the treatment of strings of events that constitute the history of a particular case, process, or situation (Allison 1984; Tuma and Hannan 1984; Heise 1989). Event history analysis has some of its origins in traditional demographers' life tables, but methods and models have experienced a great deal of attention and growth in recent years. If one had lifetime data on job placements, advancements, demotions, and firings (i.e., employment event histories) for a sample of individuals, then event history methods could be used for examining what contributes to differential risks of one of those events occurring, how long someone is likely to be in a given situation ("waiting time" between events), and so forth.

An important recent development is the use of complex computer simulations for developing and exploring mathematically expressed theories. We have noted above the tension between simple models that are mathematically tractable and more complex models that may be more realistic as for example including feedback loops and random processes. Computer simulation is a way of showing what can be derived from the assumptions of a model without an analytic mathematical solution; examples include Macy and Skvoretz (1998), Carley (1997), and Hanneman (1995). Like many other areas of social and behavioral science, mathematical sociology has been influenced by developments in game theory, from early work by Rapoport (1960) to the more recent idea of evolutionary games introduced by Axelrod (1984) and extensive interest in problems of collective action (Marwell and Oliver 1993). These consider how actions of individuals may produce unintended outcomes because of the logic of their interdependence with actions of others. The related theoretical area of rational choice theory also has a strong mathematical component (Coleman 1990); see also recent issues of the journal *Rationality and Society*.

A notable feature of current work in mathematical sociology is that the development, testing, and refinement of mathematical models is located within substantive research programs. Mathematical formulations appear in mainstream sociology journals and are becoming accepted as one component of continuing programs of research along with development and refinement of theory and collection and analysis of empirical data (several examples can be found in Berger and Zelditch 1993).

One indication of continuing interest in mathematical sociology is the recent formation of the Mathematical Sociology Section of the American Sociological Association. There is also a large amount of work internationally, including in Japan (Kosaka 1995) and in England and Europe (Hegselmann et al. 1996). Mathematical work in sociology is alive and vigorous. It truly does promise a higher level of theoretical precision and integration across the discipline.

(see also: *Paradigms and Models*; *Scientific Explanation*)

## references

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Barbara F. Meeker

Robert K. Leik

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