Goodness of fit for log-linear ERGMs

We construct finite-sample tests of goodness of fit for three different variants of the stochastic blockmodel for network data. Since all of the stochastic blockmodel variants are log-linear in form when block assignments are known, the tests for the \emph{latent} block model versions combine a block membership estimator with the algebraic statistics method for log-linear models. We describe Markov bases and marginal polytopes, and discuss how both facilitate the development of the algorithms and understanding of model behavior. The general testing methodology extends to any finite mixture of log-linear models on discrete data, and as such is the first application of algebraic statistics sampling for latent-variable models.