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The analysis of network data has gained considerable interest in recentyears. This also includes the analysis of large, high-dimensional networks withhundreds and thousands of nodes. While exponential random graph models serve asworkhorse for network data analyses, their applicability to very large networksis problematic via classical inference such as maximum likelihood or exactBayesian estimation owing to scaling and instability issues. The latter tracefrom the fact that classical network statistics consider nodes as exchangeable,i.e., actors in the network are assumed to be homogeneous. This is oftenquestionable. One way to circumvent the restrictive assumption is to includeactor-specific random effects, which account for unobservable heterogeneity.However, this increases the number of unknowns considerably, thus making themodel highly-parameterized. As a solution even for very large networks, wepropose a scalable approach based on variational approximations, which not onlyleads to numerically stable estimation but is also applicable tohigh-dimensional directed as well as undirected networks. We furthermoredemonstrate that including node-specific covariates can reduce nodeheterogeneity, which we facilitate through versatile prior formulations and anew measure that we call posterior explained variance. We illustrate ourapproach in three diverse examples, covering network data from the ItalianParliament, international arms trading, and Facebook; and conduct detailedsimulation studies.

Actor Heterogeneity and Explained Variance in Network Models -- A Scalable Approach through Variational Approximations