**Mathematical sociology** is the area of sociology that uses 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.

In another model, ties have relative strengths. 'Acquaintanceship' can be viewed as a 'weak' tie and 'friendship' is represented as a strong tie. Like its uniform cousin discussed above, there is a concept of closure, called strong triadic closure. A graph satisfies strong triadic closure If A is strongly connected to B, and B is strongly connected to C, then A and C must have a tie (either weak or strong).

In these two developments we have mathematical models bearing upon the analysis of structure. Other early influential developments in mathematical sociology pertained to process. For instance, in 1952 Herbert A. Simon produced a mathematical formalization of a published theory of social groups by constructing a model consisting of a deterministic system of differential equations. A formal study of the system led to theorems about the dynamics and the implied equilibrium states of any group.

The model constructed by Simon raises 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.

Sociologist, James S. 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. In addition, 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 provided impetus for the emergence of a good deal of effort to link rational choice thinking to more traditional sociological concerns involving social structures.

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).

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 (Berger 2000).

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.

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 amongst 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.

Mathematical sociology textbooks cover a variety of models, usually explaining the required mathematical background before discussing important work in the literature (Fararo 1973, Leik and Meeker 1975). The *Journal of Mathematical Sociology* (started in 1971) has been open to papers covering a broad spectrum of topics employing a variety of types of mathematics, especially through frequent special issues. Articles in *Social Networks,* a journal devoted to social structural analysis, very often employ mathematical models and related structural data analyses. In addition – importantly indicating the penetration of mathematical model building into sociological research – the major comprehensive journals in sociology, especially *The American Journal of Sociology* and *The American Sociological Review,* regularly publish articles featuring mathematical formulations.