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Many sociological theories are strong in substantive content but do not employ any formal language that would enable the deduction of testable predictions about the phenomena of interest. Aiming to improve this situation, since the middle of the twentieth century some sociologists have engaged in the construction of mathematical models, stating sociological assumptions in mathematical terms so that derived consequences can be empirically tested by comparison with appropriate empirical data. Such mathematical models can deal with social structures and/or social processes.
For instance, a social network can be represented as a matrix in which rows and columns refer to social units and the entries pertain to the social relationship of each pair. Theoretical interest in social structure in this sense has led to the extensive use of mathematical methods with new work regularly published in the journal Social Networks. Various social processes, such as social influence and social mobility, have been treated in terms of the construction of mathematical models. Various fields of mathematics have proved useful, such as differential and difference equations, abstract algebra, probability theory and stochastic processes, and linear algebra. Examples may be found in the Journal of Mathematical Sociology, published since 1971. Where the analytical method tends to break down because of complexity, especially nonlinearity, more and more analysts have turned to computer simulation with the objective of deriving complex outcomes of processes described in terms of simple rules of interaction among agents.
Fundamental problems relating to social emergence, social cooperation and social order are being studied with such computational models. Some of this work involves applications of concepts from game theory. See the special issue of the American Journal of Sociology (110 (4) 2005).
- Edling, R. (2002) Mathematics in sociology. Annual Review of Sociology 28: 197-220.
- Fararo, T. J. (1973) Mathematical Sociology. Wiley, New York.
- Wasserman, S. & Faust, K. (1994) Social Network Analysis: Methods and Applications. Cambridge University Press, New York.