Higher-order graphon theory: Fluctuations, degeneracies and inference

Exchangeable random graphs, which include some of the most widely studied network models, have emerged as the mainstay of statistical network analysis in recent years. Graphons, which are the central objects in graph limit theory, provide a natural way to sample exchangeable random graphs. It is well known that network moments (motif/subgraph counts) identify a graphon (up to an isomorphism); hence, understanding the sampling distribution of subgraph counts in random graphs sampled from a graphon is pivotal for nonparametric network inference. Although there are a few results regarding the asymptotic normality of subgraph counts in graphon models, for many commonly appearing graphons this distribution is degenerate. This degeneracy phenomenon was overlooked until very recently and its consequences in network inference have remained unexplored. Toward this, we obtain the following results: We derive the joint asymptotic distribution of any finite collection of network moments in random graphs sampled from a graphon, which includes both the nondegenerate case (where the distribution is Gaussian) as well as the degenerate case (where the distribution has both Gaussian or non-Gaussian components). This provides the higher-order fluctuation theory for subgraph counts in the graphon model. Furthermore, we develop a novel multiplier bootstrap for graphons that consistently approximates the limiting distribution of the network moments (both in the Gaussian and non-Gaussian regimes). Using this and a procedure for testing degeneracy, we construct joint confidence sets for any finite collection of motif densities. This provides a general framework for statistical inference based on network moments in the graphon model. Examples and simulations are provided to validate the general theory. To illustrate the broad scope of our results, we also consider the problem of detecting global structure (i.e., testing whether the graphon is a constant function) based on small subgraphs. We propose a consistent test for this problem, invoking celebrated results on quasirandom graphs, and derive its limiting distribution both under the null and the alternative.