On the phase transition curve in a directed exponential random graph   model
  We consider a family of directed exponential random graph models parametrized by edges and outward stars. Much of the important statistical content of such models is given by the normalization constant of the models, and in particular, an appropriately scaled limit of the normalization, which is called the free energy. We derive precise asymptotics for the normalization constant for finite graphs. We use this to derive a formula for the free energy. The limit is analytic everywhere except along a curve corresponding to a first order phase transition. We examine unusual behavior of the model along the phase transition curve. 
David Aristoff and Lingjiong Zhu
math.PR cond-mat.stat-mech math.CO