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

TL;DR

This paper develops Bayesian and frequentist Monte Carlo goodness-of-fit tests for various stochastic blockmodel variants in network data, utilizing algebraic statistics and Markov bases to improve model assessment.

Contribution

It introduces the first application of algebraic statistics machinery to latent-variable models, specifically for goodness-of-fit testing in stochastic blockmodels.

Findings

Tests effectively distinguish model fit in simulated data.

Markov bases facilitate understanding of model behavior.

Method extends to finite mixtures of log-linear models.

Abstract

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 \emph{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 behavior. 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.