Statistical Evaluation of Algebraic Constraints for Social Networks

A multirelational social network on a set of individuals may be represented as a collection of binary relations. Compound relations constructed from this collection represent various labeled paths linking individuals in the network. Since many models of interest for social networks can be formulated in terms of orderings among these labeled paths, we consider the problem of evaluating an hypothesized set of orderings, termed algebraic constraints. Each constraint takes the form of an hypothesized inclusion relation for a pair of labeled paths. In this paper, we establish conditions under which sets of such constraints may be regarded as partial algebras. We describe the structure of constraint sets and show that each corresponds to a subset of consistent relation bundles between pairs of individuals. We thereby construct measures of fit for a given constraint set. Then, we show how, in combination with the assumption of various conditional uniform multigraph distributions, these measures lead to a flexible approach to the evaluation of fit of an hypothesized constraint set. Several applications are presented and some possible extensions of the approach are briefly discussed. Copyright 2000 Academic Press.