Bayesian Models for Directed Graphs

Abstract Holland and Leinhardt (1981) proposed a simple exponential model called the p 1 model for analyzing digraphs that arise in studies of networks. A digraph consists of a set of g nodes and a g × g adjacency matrix (Xij ), where Xij = 1 if node i relates to node j and Xij = 0 otherwise. The underlying cell probabilities pij = Pr(Xij = 1) are to be estimated from these dichotomous responses. The p 1 model imposes an additive structure on a log-odds version of the pij . It provides information about the abilities of an individual node to attract and to produce relational ties, as well as the tendency of a pair of nodes to reciprocate ties. For digraphs of realistic sizes, the maximum likelihood estimates (MLE's) of the p 1 exponential parameters are often unsatisfactory, particularly when some of the row and column marginal totals of the adjacency matrix are small. A Bayesian approach, using an exchangeable normal prior on the parameters representing the attractiveness and expansiveness characteristics of the nodes, is proposed. The Bayesian p 1 model explicitly recognizes the association between these two characteristics of a node, an important feature ignored by its fixed effects counterpart. An algorithm for finding the MLE's of the covariance components based on a marginal likelihood is presented. An approximate posterior estimation procedure for the exponential parameters is proposed. Using an empirical example, it is shown that the Bayesian p 1 model can yield answers quite different from those of the fixed effects model.