Degree-based goodness-of-fit tests for heterogeneous random graph models : independent and exchangeable cases

TL;DR

This paper develops and evaluates degree-based goodness-of-fit tests for heterogeneous independent and exchangeable random graph models, including the Erdős-Rényi and W-graph models, with applications to real-world networks.

Contribution

It introduces formal goodness-of-fit tests for both independent and exchangeable graph models, extending the use of degree statistics in network analysis.

Findings

Asymptotic normality of degree mean square established

Tests demonstrate good power in simulations

Applications to social and ecological networks show practical utility

Abstract

The degrees are a classical and relevant way to study the topology of a network. They can be used to assess the goodness-of-fit for a given random graph model. In this paper we introduce goodness-of-fit tests for two classes of models. First, we consider the case of independent graph models such as the heterogeneous Erd\"os-R\'enyi model in which the edges have different connection probabilities. Second, we consider a generic model for exchangeable random graphs called the W-graph. The stochastic block model and the expected degree distribution model fall within this framework. We prove the asymptotic normality of the degree mean square under these independent and exchangeable models and derive formal tests. We study the power of the proposed tests and we prove the asymptotic normality under specific sparsity regimes. The tests are illustrated on real networks from social sciences and ecology, and their performances are assessed via a simulation study.