Goodness‐of‐fit testing based on graph functionals for homogeneous Erdös–Rényi graphs

The Erdös–Rényi graph is a popular choice to model network data as it is parsimoniously parameterized, straightforward to interpret 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 its adequacy, we propose a novel class of goodness‐of‐fit tests for homogeneous Erdös–Rényi models against heterogeneous alternatives that permit nonconstant edge probabilities. We allow for both 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 avoids the often tedious variance estimation for asymptotic tests and enables performance improvements for small network sizes. Moreover, under certain fixed and local alternatives, we provide a power analysis for some popular choices of subgraph counts as goodness‐of‐fit test statistics. We evaluate the proposed class of tests and illustrate our theoretical findings by simulations.