Abstract

Network data may be represented as binary graphs, either directed or undirected, and have a long history of being used to model and describe interaction between people and other entities, with formal approaches dating back to at least the start of the twentieth century.  For a graph, the potential tie between a pair of nodes is represented by a binary indicator variable that we may call a tie-variable. These tie-variables are indexed by the labels of the nodes and can be organised in a so-called node by node adjacency matrix. Since the entries of the adjacency matrix are cross-classified by both the row node, and the column node, the tie-variables are highly interdependent. Exponential (family) random graph models (ERGMs) constitute a class of log-linear models with natural parameters that have as statistics a subset of graph statistics derived out of principled dependence assumptions. Due to these dependencies, the ERGM for a network does not marginalise and subgraphs of the network do not follow models of the same form. Here we discuss inference approaches for the parameters of the ERGM when some tie-variables are missing. The treatment of missing data in ERGM also applies to cases where data are missing by design, for example when the network data have been obtained through a link-tracing designs, such as snowball sampling. We describe a Bayesian approach for estimation and provide examples of applications to networks of young men who have sex with men, rebels in the Democratic Republic of the Congo, as well as the use of the Bayesian estimation scheme for imputing initial conditions in the analysis of network panel data. The latter case is illustrated with an application to social support networks in bushfire-affected communities in Australia. The use of the proposed approach is contingent of a number of fairly heroic assumptions, some of which will be brought up for discussion.