---
title: "Mathematical sociology"
chunk: 2/5
source: "https://en.wikipedia.org/wiki/Mathematical_sociology"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T04:35:37.275238+00:00"
instance: "kb-cron"
---

== Further developments ==
In 1954, a critical expository analysis of Rashevsky's social behavior models was written by sociologist James S. Coleman. Rashevsky's models and as well as the model constructed by Simon raise a question: how can one connect such theoretical models to the data of sociology, which often take the form of surveys in which the results are expressed in the form of proportions of people believing or doing something.  This suggests deriving the equations from assumptions about the chances of an individual changing state in a small interval of time, a procedure well known in the mathematics of stochastic processes.
Coleman embodied this idea in his 1964 book Introduction to Mathematical Sociology, which showed how stochastic processes in social networks could be analyzed in such a way as to enable testing of the constructed model by comparison with the relevant data. The same idea can and has been applied to processes of change in social relations, an active research theme in the study of social networks, illustrated by an empirical study appearing in the journal Science.
In other work, Coleman employed mathematical ideas drawn from economics, such as general equilibrium theory, to argue that general social theory should begin with a concept of purposive action and, for analytical reasons, approximate such action by the use of rational choice models (Coleman, 1990).  This argument is similar to viewpoints expressed by other sociologists in their efforts to use rational choice theory in sociological analysis although such efforts have met with substantive and philosophical criticisms.
Meanwhile, structural analysis of the type indicated earlier received a further extension to social networks based on institutionalized social relations, notably those of kinship.  The linkage of mathematics and sociology here involved abstract algebra, in particular, group theory.  This, in turn, led to a focus on a data-analytical version of homomorphic reduction of a complex social network (which along with many other techniques is presented in Wasserman and Faust 1994).
In regard to Rapoport's random and biased net theory, his 1961 study of a large sociogram, co-authored with Horvath turned out to become a very influential paper. There was early evidence of this influence. In 1964, Thomas Fararo and a co-author analyzed another large friendship sociogram using a biased net model. Later in the 1960s, Stanley Milgram described the small world problem and undertook a field experiment dealing with it. A highly fertile idea was suggested and applied by Mark Granovetter in which he drew upon Rapoport's 1961 paper to suggest and apply a distinction between weak and strong ties. The key idea was that there was "strength" in weak ties.
Some programs of research in sociology employ experimental methods to study social interaction processes.  Joseph Berger and his colleagues initiated such a program in which the central idea is the use of the theoretical concept "expectation state" to construct theoretical models to explain interpersonal processes, e.g., those linking external status in society to differential influence in local group decision-making. Much of this theoretical work is linked to mathematical model building, especially after the late 1970s adoption of a graph theoretic representation of social information processing, as Berger (2000) describes in looking back upon the development of his program of research. In 1962 he and his collaborators explained model building by reference to the goal of the model builder, which could be explication of a concept in a theory, representation of a single recurrent social process, or a broad theory based on a theoretical construct, such as, respectively, the concept of balance in psychological and social structures, the process of conformity in an experimental situation, and stimulus sampling theory.
The generations of mathematical sociologists that followed Rapoport, Simon, Harary, Coleman, White and Berger, including those entering the field in the 1960s such as Thomas Fararo, Philip Bonacich, and Tom Mayer, among others, drew upon their work in a variety of ways.

== Present research ==
Mathematical sociology remains a small subfield within the discipline, but it has succeeded in spawning a number of other subfields which share its goals of formally modeling social life.  The foremost of these fields is social network analysis, which has become among the fastest growing areas of sociology in the 21st century.  The other major development in the field is the rise of computational sociology, which expands the mathematical toolkit with the use of computer simulations, artificial intelligence and advanced statistical methods.  The latter subfield also makes use of the vast new data sets on social activity generated by social interaction on the internet.
One important indicator of the significance of mathematical sociology is that the general interest journals in the field, including such central journals as The American Journal of Sociology and The American Sociological Review, have published mathematical models that became influential in the field at large.
More recent trends in mathematical sociology are evident in contributions to The Journal of Mathematical Sociology (JMS).  Several trends stand out: the further development of formal theories that explain experimental data dealing with small group processes, the continuing interest in structural balance as a major mathematical and theoretical idea, the interpenetration of mathematical models oriented to theory and innovative quantitative techniques relating to methodology, the use of computer simulations to study problems in social complexity, interest in micro–macro linkage and the problem of emergence, and ever-increasing research on networks of social relations.  
Thus, topics from the earliest days, like balance and network models, continue to be of contemporary interest. The formal techniques employed remain many of the standard and well-known methods of mathematics: differential equations, stochastic processes and game theory. Newer tools like agent-based models used in computer simulation studies are prominently represented. Perennial substantive problems still drive research: social diffusion, social influence, social status origins and consequences, segregation, cooperation, collective action, power, and much more.