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.

中文翻译：

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异构随机图模型的基于度的拟合优度检验：独立和可交换的案例

度数是研究网络拓扑的经典且相关的方法。它们可用于评估给定随机图模型的拟合优度。在本文中，我们介绍了两类模型的拟合优度检验。首先，我们考虑独立图模型的情况，例如异构 Erd\"os-R\'enyi 模型，其中边具有不同的连接概率。其次，我们考虑称为 W-graph 的可交换随机图的通用模型。随机块模型和期望度分布模型都属于这个框架。我们证明了这些独立可交换模型下度均方的渐近正态性并推导出形式检验。我们研究了所提出检验的功效，并证明了渐近正态性在特定的稀疏机制下。