On the Geometry and Extremal Properties of the Edge-Degeneracy Model
  The edge-degeneracy model is an exponential random graph model that uses the graph degeneracy, a measure of the graph's connection density, and number of edges in a graph as its sufficient statistics. We show this model is relatively well-behaved by studying the statistical degeneracy of this model through the geometry of the associated polytope. 
Nicolas Kim, Dane Wilburne, Sonja Petrovi\'c and Alessandro Rinaldo
math.ST cs.DM stat.TH