Finite-graph concentration and consistency results for random graphs with complex topological structures

Statistical inference for exponential-family random graphs with complex topological structures is challenging. We stress the importance of additional structure and show that additional structure facilitates statistical inference. A simple and common form of additional structure, which is observed in multilevel networks, is a random graph with neighborhoods and local dependence within neighborhoods. We develop the first concentration results for maximum likelihood and $M$-estimators of a wide range of canonical and curved exponential-family random graphs with local dependence. All results are non-asymptotic and cover random graphs of fixed and finite size, provided the neighborhoods are small relative to the size of the random graph. We discuss extensions to larger random graphs with more neighborhoods along with concentration results for subgraph-to-graph estimators. As applications, we consider canonical and curved exponential-family random graphs, with local dependence induced by sensible forms of transitivity and parameter vectors whose dimensions depend on the number of nodes.