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

Sarah Ouadah (1) , Stéphane Robin (1) , Pierre Latouche (2)

1 MIA Paris-Saclay - Mathématiques et Informatique Appliquées
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

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 Erdös-Rényi 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.

Mots clés: goodness-of-fit; graphon; W-graph; degree variance; random graphs

Domaines: Statistiques [math.ST]

Sarah Ouadah, Stéphane Robin, Pierre Latouche. Degree-based goodness-of-fit tests for heterogeneous random graph models : independent and exchangeable cases. Scandinavian Journal of Statistics, 2020, 47 (1), pp.156-181. ⟨10.1111/sjos.12410⟩. ⟨hal-02454464⟩