Advances in biased net theory: definitions,derivations, and estimations

Random and biased net theory, introduced by Rapoport and others in the 1950s, is one of the earliest approaches to the formal modeling of social networks. In this theory, intended as a theory of large-scale networks, ties between nodes derive both from random and non-random events of connection. The non-random connections are postulated to arise through “bias” events that incorporate known or suspected systematic tendencies in tie formation, such as, mutuality or reciprocity, transitivity or closure in triads, and homophily—the overrepresentation of ties between persons who share important socio-demographic attributes like race/ethnicity or level of educational attainment. A key problem for biased net theory has been analytical intractability of the models. Formal derivations require approximation assumptions and model parameters have been difficult to estimate. The accuracy of the derived formulas and the estimated parameters has been difficult to assess. In this paper, we attempt to address long-standing issues in biased net models stemming from their analytical intractability. We first reformulate and clarify the definitions of basic biases. Second, we derive from first principles the triad distribution in a biased net, using two different analytical strategies to check our derivations. Third, we set out a pseudo-likelihood method for parameter estimation of key bias parameters and then check the accuracy of this relatively simple but approximate scheme against the results obtained from the triad distribution derivation.