Exponential family models for sampled and census network data

Much progress has been made on the development of statistical methods for network analysis in the past ten years, building on the general class of exponential family random graph (ERG) network models first introduced by Holland and Leinhardt (1981). Recent examples include models for Markov graphs, "p*" models, and actor-oriented models. For empirical application, these ERG models take a logistic form, and require the equivalent of a network census: data on all dyads within the network. In a largely separate stream of research, conditional log-linear (CLL) models have been adapted for analyzing locally sampled ("egocentric") network data. While the general relation between log-linear and logistic models is well known and has been exploited in the case of a priori blockmodels for networks, the relation for the CLL models is different due to the treatment of absent ties. For fully saturated tie independence models, CLL and ERG are equivalent and related via Bayes' rule. For other tie independence models, the two do not yield equivalent predicted values, but we show that in practice the differences are unlikely to be large. The alternate conditioning in the two models sheds light on the relationship between local and complete network data, and the role that models can play in bridging the gap between them.