We define a general class of network formation models, Statistical Exponential Random Graph Models (SERGMs), that nest standard exponential random graph models (ERGMs) as a special case. We provide the first general results on when these models' (including ERGMs) parameters estimated from the observation of a single network are consistent (i.e., become accurate as the number of nodes grows). Next, addressing the problem that standard techniques of estimating ERGMs have been shown to have exponentially slow mixing times for many specifications, we show that by reformulating network formation as a distribution over the space of sufficient statistics instead of the space of networks, the size of the space of estimation can be greatly reduced, making estimation practical and easy. We also develop a related, but distinct, class of models that we call subgraph generation models (SUGMs) that are useful for modeling sparse networks and whose parameter estimates are also directly and easily estimable, consistent, and asymptotically normally distributed. Finally, we show how choice-based (strategic) network formation models can be written as SERGMs and SUGMs, and apply our models and techniques to network data from rural Indian villages.
