Statistical inference for exponential-family models of random graphs with dependent edges is challenging. We stress the importance of additional structure and show that additional structure facilitates statistical inference. A simple example of a random graph with additional structure is a random graph with neighborhoods and local dependence within neighborhoods. We develop the first concentration and consistency results for maximum likelihood and M-estimators of a wide range of canonical and curved exponential-family models of random graphs with local dependence. All results are non-asymptotic and applicable to random graphs with finite populations of nodes, although asymptotic consistency results can be obtained as well. In addition, we show that additional structure can facilitate subgraph-to-graph estimation, and present concentration results for subgraph-to-graph estimators. As an application, we consider popular curved exponential-family models of random graphs, with local dependence induced by transitivity and parameter vectors whose dimensions depend on the number of nodes.
Keywords:curved exponential families, exponential families, exponential-family random graph models, M-estimators, multilevel networks, social networks.
Schweinberger, M., and Stewart, J. "Concentration and consistency results for canonical and curved exponential-family models of random graphs" Annals of Statistics, to appear (2019).