A degree-based goodness-of-fit test for heterogeneous random graph models

The degree variance has been proposed for many years to study the topology of a network. It can be used to assess the goodness-of-fit of the Erd\"os-Renyi model. In this paper, we prove the asymptotic normality of the degree variance under this model which enables us to derive a formal test. We generalize this result to the heterogeneous Erd\"os-Renyi model in which the edges have different respective probabilities to exist. For both models we study the power of the proposed goodness-of-fit test. We also prove the asymptotic normality under specific sparsity regimes. Both tests are illustrated on real networks from social sciences and ecology. Their performances are assessed via a simulation study.