An exact test for the mixed membership stochastic block model

TL;DR

This paper introduces an exact goodness-of-fit test for the mixed membership stochastic block model (MMSBM) that does not rely on asymptotic approximations, using algebraic statistics and Markov bases.

Contribution

It develops the first finite-sample test for MMSBM, enabling precise evaluation of model fit with a novel Markov chain approach.

Findings

Test maintains nominal size in simulations

Exhibits strong power against misspecification

Operates without asymptotic assumptions

Abstract

We present the first finite-sample goodness-of-fit test for the mixed membership stochastic block model (MMSBM). Using algebraic statistics theory, we derive a Markov basis that lets a Metropolis-Hastings sampler explore exactly the set of networks compatible with a fitted MMSBM. The resulting exact $p$-value, based on a partial conjunction statistic, requires no asymptotic approximations. Simulations show nominal size and strong power against misspecified block numbers and connection patterns.