The Erd\"os Renyi graph is a popular choice to model network data as it is parsimoniously parametrized, straightforward to interprete and easy to estimate. However, it has limited suitability in practice, since it often fails to capture crucial characteristics of real-world networks. To check the adequacy of this model, we propose a novel class of goodness-of-fit tests for homogeneous Erd\"os Renyi models against heterogeneous alternatives that allow for nonconstant edge probabilities. We allow for asymptotically dense and sparse networks. The tests are based on graph functionals that cover a broad class of network statistics for which we derive limiting distributions in a unified manner. The resulting class of asymptotic tests includes several existing tests as special cases. Further, we propose a parametric bootstrap and prove its consistency, which allows for performance improvements particularly for small network sizes and avoids the often tedious variance estimation for asymptotic tests. Moreover, we analyse the sensitivity of different goodness-of-fit test statistics that rely on popular choices of subgraphs. We evaluate the proposed class of tests and illustrate our theoretical findings by extensive simulations.

翻译：厄尔多斯—雷尼图因参数化简洁、解释直观且易于估计，成为网络数据建模的常用选择。然而，该模型在实际应用中适用性有限，因其常无法捕捉真实世界网络的关键特征。为检验该模型的适用性，我们针对同质厄尔多斯—雷尼模型提出一类新型拟合优度检验，以对抗允许非恒定边概率的异质性备择模型。我们考虑了渐近稠密与稀疏网络场景。该检验基于覆盖广泛网络统计量的图泛函，并以统一方式推导其极限分布。所得渐近检验类包含若干现有检验作为特例。此外，我们提出参数自举法并证明其相合性，该方法尤其能提升小规模网络的检验效能，同时避免渐近检验中繁琐的方差估计。进一步地，我们分析了基于常见子图选择的拟合优度检验统计量的敏感性。通过大量仿真实验评估了所提检验类并验证了理论结论。