Mathematical sociology

1800s: Martineau · Tocqueville · Marx · Spencer · Le Bon · Ward · Pareto · Tönnies · Veblen · Simmel · Durkheim · Addams · Mead · Weber · Du Bois · Mannheim · Elias Mathematical sociology is an interdisciplinary field of research concerned with the use of mathematics within sociological research.

[1] Starting in the early 1940s, Nicolas Rashevsky,[2][3] and subsequently in the late 1940s, Anatol Rapoport and others, developed a relational and probabilistic approach to the characterization of large social networks in which the nodes are persons and the links are acquaintanceship.

For instance, in 1952 Herbert A. Simon produced a mathematical formalization of a published theory[6] of social groups by constructing a model consisting of a deterministic system of differential equations.

The benefits of this approach include increased clarity and the ability to use mathematics to derive implications of a theory that cannot be arrived at intuitively.

This means making specified assumptions about some social phenomenon, expressing them in formal mathematics, and providing an empirical interpretation for the ideas.

The models typically used in mathematical sociology allow sociologists to understand how predictable local interactions are and they are often able to elicit global patterns of social structure.

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.

[12] 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[13]).

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.

[19] 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.

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.

The formal techniques employed remain many of the standard and well-known methods of mathematics: differential equations, stochastic processes and game theory.

This provides another way of taking note of recent contributions but with an emphasis on continuity with early work through the use of the idea of “research program,” which is a coherent series of theoretical and empirical studies based on some fundamental principle or approach.

[21] The Foundation book combined accessible examples of how rational choice theory could function in the analysis of such sociological topics as authority, trust, social capital and the norms (in particular, their emergence).

In this way, the book showed how rational choice theory could provide an effective basis for making the transition from micro to macro levels of sociological explanation.

[23] Nevertheless, many sociologists drew upon Coleman's formulation of a general template for micro-macro transition to gain leverage on the continuation of topics central to his and the discipline's explanatory focus on a variety of macrosocial phenomena in which rational choice simplified the micro level in the interest of combining individual actions to account for macro outcomes of social processes.

[24] (2) Structuralism (Formal) and Harrison C. White: In the decades since his earliest contributions, Harrison White has led the field in putting social structural analysis on a mathematical and empirical basis, including the 1970 publication of Chains of Opportunity: System Models of Mobility in Organizations which set out and applied to data a vacancy chain model for mobility in and across organizations.

[26] White's later contributions include a structuralist approach to markets[27] and, in 1992, a general theoretical framework,[28] later appearing in a revised edition.

He and his colleague and frequent collaborator Morris Zelditch Jr not only produced work of their own but created a doctoral program at Stanford University that led to an enormous outpouring of research by notable former students, including Murray Webster, David Wagner, and Hamit Fisek.

Collaboration with mathematician Robert Z. Norman led to the use of mathematical graph theory as a way of representing and analyzing social information processing in self-other interactions.

It was the origination of a research program that has included his further theoretical and empirical studies and those of other sociologists, such as Lynn Smith-Lovin, Dawn Robinson and Neil MacKinnon.

This research program comprises several of the key chapters in a 2006 volume[42] of contributions to control systems theory (in the sense of Powers 1975 [43]) in sociology.

Skvoretz, in addition to this other contributions, has been a frequent collaborator in a variety of theoretical research programs, often using mathematical expertise as well as skills in experimental design, statistical data analysis and simulation methods.