A Network Formation Model Based on Subgraphs

We develop a new class of random-graph models for the statistical estimation of network formation that allow for substantial correlation in links. Various subgraphs (e.g., links, triangles, cliques, stars) are generated and their union results in a network. The challenge in estimating the frequencies with which subgraphs 'truly' form is that subgraphs can overlap and may also incidentally generate new subgraphs, and so the true rate of formation of the subgraphs cannot generally be inferred just by counting their presence in the resulting network. We provide estimation techniques for recovering the rates at which the underlying subgraphs were formed from the observation of a single (large) network. We provide results on identification of the true underlying rates of subgraph formation from various statistics, as well as a new Central Limit Theorem for correlated random variables that establishes asymptotic normality for our estimators. We also show that if the network is sparse enough then direct counts of subgraphs are consistent and asymptotically normal estimators. We illustrate the models with applications.