  There has been an explosion of interest in statistical models for analyzing
network data, and considerable interest in the class of exponential random
graph (ERG) models, especially in connection with difficulties in computing
maximum likelihood estimates. The issues associated with these difficulties
relate to the broader structure of discrete exponential families. This paper
re-examines the issues in two parts. First we consider the closure of
$k$-dimensional exponential families of distribution with discrete base measure
and polyhedral convex support $\mathrm{P}$. We show that the normal fan of
$\mathrm{P}$ is a geometric object that plays a fundamental role in deriving
the statistical and geometric properties of the corresponding extended
exponential families. We discuss its relevance to maximum likelihood
estimation, both from a theoretical and computational standpoint. Second, we
apply our results to the analysis of ERG models. In particular, by means of a
detailed example, we provide some characterization of the properties of ERG
models, and, in particular, of certain behaviors of ERG models known as
degeneracy.
