  The exponential family of random graphs is one of the most promising class of
network models. Dependence between the random edges is defined through certain
finite subgraphs, analogous to the use of potential energy to provide
dependence between particle states in a grand canonical ensemble of statistical
physics. By adjusting the specific values of these subgraph densities, one can
analyze the influence of various local features on the global structure of the
network. Loosely put, a phase transition occurs when a singularity arises in
the limiting free energy density, as it is the generating function for the
limiting expectations of all thermodynamic observables. We derive the full
phase diagram for a large family of 3-parameter exponential random graph models
with attraction and show that they all consist of a first order surface phase
transition bordered by a second order critical curve.
