Consistent $M$-estimation of curved exponential-family random graph models with local dependence and growing neighborhoods

In general, statistical inference for exponential-family random graph models of dependent random graphs given a single observation of a random graph is problematic. We show that statistical inference for exponential-family random graph models holds promise as long as models are endowed with a suitable form of additional structure. We consider a simple and common form of additional structure called multilevel structure. To demonstrate that exponential-family random graph models with multilevel structure are amenable to statistical inference, we develop the first concentration and consistency results covering $M$-estimators of a wide range of full and non-full, curved exponential-family random graph models with local dependence and natural parameter vectors of increasing dimension. In addition, we show that multilevel structure facilitates local computing of $M$-estimators and in doing so reduces computing time. Taken together, these results suggest that exponential-family random graph models with multilevel structure constitute a promising direction of statistical network analysis.