  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.
