  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.
