Monte Carlo goodness-of-fit tests for degree corrected and related stochastic blockmodels

We construct Bayesian and frequentist finite-sample goodness-of-fit tests 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 latent block model versions combine a block membership estimator with the algebraic statistics machinery for testing goodness-of-fit in log-linear models. We describe Markov bases and marginal polytopes of the variants of the stochastic blockmodel and discuss how both facilitate the development of goodness-of-fit tests and understanding of model behaviour. The general testing methodology developed here extends to any finite mixture of log-linear models on discrete data, and as such is the first application of the algebraic statistics machinery for latent-variable models.

Automated summary

We propose Bayesian and frequentist finite-sample goodness-of-fit tests for three variants of the stochastic blockmodel for network data. By leveraging the algebraic statistics machinery, our tests combine a block membership estimator with the log-linear models to test the goodness-of-fit for the latent block model versions. We discuss the Markov bases and marginal polytopes of the stochastic blockmodel variants, which help facilitate the development of goodness-of-fit tests and enhance our understanding of model behavior.