Consistency of empirical distributions of sequences of graph statistics in networks with dependent edges

One of the first steps in applications of statistical network analysis is frequently to produce summary charts of important features of the network. Many of these features take the form of sequences of graph statistics counting the number of realized events in the network, examples of which include the degree distribution, as well as the edgewise shared partner distribution, and more. We provide conditions under which the empirical distributions of sequences of graph statistics are consistent in the $\ell_{\infty}$-norm in settings where edges in the network are dependent. We accomplish this by elaborating a weak dependence condition which ensures that we can obtain exponential inequalities which bound probabilities of deviations of graph statistics from the expected value. We apply this concentration inequality to empirical distributions of sequences of graph statistics and derive non-asymptotic bounds on the $\ell_{\infty}$-error which hold with high probability. Our non-asymptotic results are then extended to demonstrate uniform convergence almost surely in selected examples. We illustrate theoretical results through examples, simulation studies, and an application.