Mathematical Sociology

Mathematical sociology is the use of mathematics to construct social theories. Mathematical sociology aims to take sociological theory, which is strong in intuitive content but weak from a formal point of view, and to express it in formal terms. 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. In mathematical sociology, the preferred style is encapsulated in the phrase "constructing a mathematical model." This means making specified assumptions about some social phenomenon, expressing them in formal mathematics, and providing an empirical interpretation for the ideas. It also means deducing properties of the model and comparing these with relevant empirical data. Social network analysis is the best-known contribution of this subfield to sociology as a whole and to the scientific community at large. 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.

Starting in the early 1940s, Nicolas Rashevsky, 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. During the late 1940s, formulas were derived that connected local parameters such as closure of contacts – if A is linked to both B and C, then there is a greater than chance probability that B and C are linked to each other – to the global network property of connectivity.

Moreover, acquaintanceship is a positive tie, but what about negative ties such as animosity among persons? To tackle this problem, graph theory, which is the mathematical study of abstract representations of networks of points and lines, can be extended to include these two types of links and thereby to create models that represent both positive and negative sentiment relations, which are represented as signed graphs. A signed graph is called balanced if the product of the signs of all relations in every cycle (links in every graph cycle) is positive. This effort led to Harary's Structure Theorem (1953), which says that if a network of interrelated positive and negative ties is balanced, e.g. as illustrated by the psychological principle that "my friend's enemy is my enemy", then it consists of two subnetworks such that each has positive ties among its nodes and there are only negative ties between nodes in distinct subnetworks. The imagery here is of a social system that splits into two cliques. There is, however, a special case where one of the two subnetworks is empty, which might occur in very small networks.

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