The Arithmetic of Social Relations: The Interplay of Category and Network

This paper develops a conceptual scheme that merges the qualitatively stated propositions of Blau's recent axiomatic theory of social structure with the quantitative approach of social network analysis. The conceptual scheme is used to describe a set of inescapable features of intergroup and intragroup relations. We examine, both qualitatively and formal equations, the tautologies that govern contact rates and network densities for any population that can be divided into two categories. We show how assumptions about the partitioning of populations into social categories can be translated into precise probabilities of contact within and between categories. We present several illustrations of the immodest implications of apparently modest assumptions. Following simulations of a high school within an adult community and an old boy network within a larger bureaucracy, we apply our conceptual scheme to actual data on social relations within a regional elite. Accompanying these examples is a discussion of new perspective on reference groups, as well as a development of a common exception to Blau's theory. We conclude with a formal statement of the substantive propositions that follow from our conceptual scheme.