Random graphs with node and block effects: models, goodness-of-fit tests, and applications to biological networks

Many popular models from the networks literature can be viewed through a common lens. We describe it here and call the class of models log-linear ERGMs. It includes degree-based models, stochastic blockmodels, and combinations of these. Given the interest in combined node and block effects in network formation mechanisms, we introduce a general directed relative of the degree-corrected stochastic blockmodel: an exponential family model we call $p_1$-SBM. It is a generalization of several well-known variants of the blockmodel. We study the problem of testing model fit for the log-linear ERGM class. The model fitting approach we take, through the use of quick estimation algorithms borrowed from the contingency table literature and effective sampling methods rooted in graph theory and algebraic statistics, results in an exact test whose $p$-value can be approximated efficiently in networks of moderate sizes. We showcase the performance of the method on two data sets from biology: the connectome of \emph{C. elegans} and the interactome of \emph{Arabidopsis thaliana}. These two networks, a neuronal network and a protein-protein interaction network, have been popular examples in the network science literature, but a model-based approach to studying them has been missing thus far.