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The degrees are a classical and relevant way to study the topology of anetwork. They can be used to assess the goodness-of-fit for a given randomgraph model. In this paper we introduce goodness-of-fit tests for two classesof models. First, we consider the case of independent graph models such as theheterogeneous Erd\"os-R\'enyi model in which the edges have differentconnection probabilities. Second, we consider a generic model for exchangeablerandom graphs called the W-graph. The stochastic block model and the expecteddegree distribution model fall within this framework. We prove the asymptoticnormality of the degree mean square under these independent and exchangeablemodels and derive formal tests. We study the power of the proposed tests and weprove the asymptotic normality under specific sparsity regimes. The tests areillustrated on real networks from social sciences and ecology, and theirperformances are assessed via a simulation study.

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