Maximum entropy based testing in network models: ERGMs and constrained optimization

TL;DR

This paper introduces a maximum entropy-based framework for testing goodness-of-fit and two-sample hypotheses in network models, including ERGMs and Erdős-Rényi graphs, across different graph density regimes.

Contribution

It develops a novel Lagrange multiplier-based test statistic for network models, unifying various regimes and connecting to classical score tests.

Findings

Consistent testing procedures for fixed and growing network sizes.

Applicable to dense ERGMs and sparse Erdős-Rényi graphs.

Leverages nonlinear large deviation theory for analysis.

Abstract

Stochastic network models play a central role across a wide range of scientific disciplines, and questions of statistical inference arise naturally in this context. In this paper we investigate goodness-of-fit and two-sample testing procedures for statistical networks based on the principle of maximum entropy (MaxEnt). Our approach formulates a constrained entropy-maximization problem on the space of networks, subject to prescribed structural constraints. The resulting test statistics are defined through the Lagrange multipliers associated with the constrained optimization problem, which, to our knowledge, is novel in the statistical networks literature. We establish consistency in the classical regime where the number of vertices is fixed. We then consider asymptotic regimes in which the graph size grows with the sample size, developing tests for both dense and sparse settings. In the dense case, we analyze exponential random graph models (ERGM) (including the Erd\"os-R\`enyi models), while in the sparse regime our theory applies to Erd{\"o}s-R{\`e}nyi graphs. Our analysis leverages recent advances in nonlinear large deviation theory for random graphs. We further show that the proposed Lagrange-multiplier framework connects naturally to classical score tests for constrained maximum likelihood estimation. The results provide a unified entropy-based framework for network model assessment across diverse growth regimes.