A Stein Goodness of fit Test for Exponential Random Graph Models

TL;DR

This paper introduces a new nonparametric goodness of fit test for exchangeable ERGMs using kernel Stein discrepancy, enabling assessment of how well a network fits a specified ERGM model.

Contribution

It develops a novel test statistic based on Stein's method and kernel functions for ERGMs, with theoretical analysis and practical validation.

Findings

The test accurately detects model misspecification in simulations.

Application to real networks demonstrates practical utility.

Theoretical properties ensure validity under certain conditions.

Abstract

We propose and analyse a novel nonparametric goodness of fit testing procedure for exchangeable exponential random graph models (ERGMs) when a single network realisation is observed. The test determines how likely it is that the observation is generated from a target unnormalised ERGM density. Our test statistics are derived from a kernel Stein discrepancy, a divergence constructed via Steins method using functions in a reproducing kernel Hilbert space, combined with a discrete Stein operator for ERGMs. The test is a Monte Carlo test based on simulated networks from the target ERGM. We show theoretical properties for the testing procedure for a class of ERGMs. Simulation studies and real network applications are presented.