Goodness of fit testing based on graph functionals for homogenous Erd\"os Renyi graphs

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