Non-asymptotic model selection for models of network data with parameter vectors of increasing dimension

Model selection for network data is an open area of research. Using the $\beta$-model as a convenient starting point, we propose a simple and non-asymptotic approach to model selection of $\beta$-models with and without constraints. Simulations indicate that the proposed model selection approach selects the data-generating model with high probability, in contrast to classical and extended Bayesian Information Criteria. We conclude with an application to the Enron email network, which has 181,831 connections among 36,692 employees.