Mathematical Sociology

Mathematical sociology John Skvoretz University of South Florida, USA Thomas J Fararo University of Pittsburgh, USA abstract The history, presuppositions, current developments, challenges and prospects for mathemat - ical sociology are described. Common research threads in the field from its earliest post-Second World War days to the present are identified. A case for the importance of mathematical sociology to the future of sociological theory is made. keywords mathematical sociology ◆ model building ◆ process ◆ sociological theory ◆ structure Mathematical sociology weds mathematics and sociol - general perspectives in sociology is what is usually ogy to advance the scientific understanding of social meant by ‘sociological theory’. At the level of structures and social processes. Within the broad macrosocial analysis, one perspective (functionalism) arena of sociology, it stands in that corner defined by treats values in terms of their social integrative signif - a generalizing orientation, by the belief that a science icance while another (conflict theory) treats values as of social orders is possible, by a commitment to a log - weapons in struggles among groups. At the level of ical derivation of empirical regularities from formally microsocial analysis, one perspective (exchange theo - stated axioms or assumptions, and by a concern for ry) treats social interaction in terms of interchange of the integration and unification of sociological theory. resources of any kind, while another perspective (sym - Mathematics in this context is used to directly formu - bolic interactionism) treats social interaction as com - late theory and derive testable hypotheses. munication of social meanings. These perspectives Mathematics is also widely used in data analysis in the and others have arisen in pursuit of the goal to formu - social sciences but such use is not intended to be in late fundamental sociological problems – what sociolo - the service of theory formulation. Nevertheless, prac - gy should explain and how. In turn, the use of titioners of quantitative data analysis and practitioners mathematics in sociological theory has arisen in the of mathematical sociology can overlap substantially context of attempting to find more effective means of since the requisite skill set, facility with mathematical the pursuit of explanatory goals, drawing upon but formulations and the training to reason logically from not limited to existing theoretical perspectives. premises, is common to both enterprises. This article surveys the field of mathematical sociology, its histo - ry, its presuppositions and current areas of develop - History ment. It then outlines some of the basic challenges faced by mathematical sociology, closing with a com - Interest in mathematical sociology took off in the mentary on its future directions. post-Second World War period. The pre-war work of As an intended scientific discipline, sociology is Rashevsky (1939a, 1939b, 1940a, 1940b, 1941, relatively new and mathematical sociology even 1942) was emblematic of early efforts in mathemati - newer. The canonical problems of the discipline are cal sociology – problems were approached in a ‘grand not agreed upon and a variety of theoretical perspec - theory’ fashion with little attention to relevant data. tives exist as competing paradigms. The array of such The post-war period, the 1950s and 1960s, was a Sociopedia.isa © 2011 The Author(s) © 2011 ISA (Editorial Arrangement of Sociopedia.isa ) John Skvoretz and Thomas J Fararo, 2011, ‘Mathematical sociology’, Sociopedia.isa , DOI: 10.1177/2056846011102 1 Skvoretz and Fararo Mathematical sociology fruitful one for mathematical sociology in part task; in the internal system, the three variables char - because theorists began by paying attention to signif - acterize interpersonal relations such as friendships. icant empirical regularities, the precision of which The general idea is that the internal relationships virtually demanded formal analysis. Many of these emerge out of the social activity and interaction regularities derived from laboratory studies of small required for adapting to the group’s environment but group processes. Specifically, there were the studies then feed back to alter or reinforce that adaptation. of participation in task groups by Bales and col - For instance, a work group’s strong internal ties may leagues (Bales, 1950; Bales et al., 1951), the studies enable it to resist some new practice initiated by of the effect of the structure of a communication higher management (environment). network on group performance and individual out - Homans made several claims about how in gener - come by Bavelas and his colleagues (Bavelas, 1948, al activity is related to sentiment, sentiment to inter - 1950), Leavitt (1949, 1951), Smith (1950) and action, and so on from his analysis of five groups that Bavelas and Barrett (1951) and the studies of con - had been subjects of detailed field studies by social formity by Asch (1951, 1952, 1956). scientists. Simon formalized these assertions by spec - The importance of small group research was ifying mathematically the functional relationships clearly recognized in the title of one of the early between the variables of activity, sentiment and books to organize mathematical thinking in sociolo - interaction and connecting them via differential gy, Types of Formalization in Small Group Research equations into a dynamic system characterizing the (Berger et al., 1962). Useful to this day is their cate - group. The system of equations was analyzed for the gorization of formal models by three types of pri - conditions under which equilibrium points existed mary goal: explication, representation and and were stable. The hope was that such conditions theoretical construct. In the first type mathematics is could then be interpreted in meaningful sociological used in the ‘explication, or rendering of precise terms that expanded the theoretical understanding of meaning, to one or more basic concepts’ (Berger et the establishment and maintenance of social groups. al., 1962: 7). Their illustration is the use of graph Other foundational work appeared not in sociol - theory by Cartwright and Harary (1956) to render ogy journals but in the Bulletin of Mathematical more precise the idea of balance, the primary driver Biophysics. In a series of papers, Anatol Rapoport and in Heider’s theory of interpersonal relations (1944, colleagues set out and explored formal models for 1946, 1958). In the second type, ‘the theorist potentially extremely large groups (Landau, 1952; attempts to represent in as precise and formally sim - Rapoport, 1951a, 1951b, 1953a, 1953b, 1953c, ple a manner as possible a recurrent but specific 1957, 1958, 1963; Rapoport and Horvath, 1961; instance of an observed social phenomenon’ (Berger Rapoport and Solomonoff, 1951). In particular et al., 1962: 7). The paradigmatic illustration is Rapoport’s random and biased net theory focused on Cohen’s (1958, 1963) Markov chain model of the modeling social networks – the structure of social Asch conformity process. In the third type, the the - ties of a particular type in a bounded population. orist aims ‘to provide a direct means of developing a The models used probabilistic formalisms first to general explanatory theory which formally accounts describe properties of a random net and then to for a variety of observed processes’ (Berger et al., model departures from randomness by biases that 1962: 67; italics in original). The exemplar they dis - made certain connections more likely than chance cuss is the ‘stimulus sampling’ learning model of when the relational context was favorable. For Estes and Burke (1955), a model proposed to explain instance, the idea that a person, ego, is more likely to outcomes in one type of learning experiment (dis - chose another as a friend if the other has also chosen crimination learning) whose conceptual machinery ego as a friend was formally expressed by a ‘reciproc - was sufficiently general that it was elaborated to ity’ bias. The basic problem concerning Rapoport apply to multi-person situations (Atkinson and was the ‘tracing’ problem: how biases and the ran - Suppes, 1958). dom chance of connection affected reachability, the Another common route to mathematical models proportion of nodes in a population that could be was to recast verbally stated theoretical claims in for - reached from a randomly selected starter node. The mal terms. Simon’s (1952) formalization of Homans’ average proportions of newly reached nodes in each (1950) theory is the classic example. The conceptual generation, as links were traced out from an arbitrary foundation of Homans’ theory is a social system starting set, were termed the biased net ‘structure sta - model in which there are two subsystems, external tistics’ (Fararo and Sunshine, 1964). Biases were and internal. Each subsystem is described by three defined in the context of the tracing procedure. variables: activity, interaction and sentiment. In the Rapoport derived formal expressions for the struc - external system, the variables are specified as task- ture statistics but only under some strong approxi - related and consisting of interactions needed for the mation assumptions that later research showed 2 Skvoretz and Fararo Mathematical sociology needed correction to improve fit to available data Fararo was equally aware of the importance of (Skvoretz, 1990). systematic measurement to scientific advance yet the By the 1960s and 1970s, a sufficient

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