On the Phase Transition Curve in a Directed Exponential Random Graph Model

Abstract. We consider a family of directed exponential random graph models parametrized by edges and outward stars. Essentially all of the statistical content of such models is given by the free energy density, which is an appropriately scaled version of the probability normalization. We derive precise asymptotics for the free energy density of finite graphs. We use this to rederive a formula for the limiting free energy density first obtained by Chatterjee and Diaconis [3]. The limit is analytic everywhere except along a phase transition curve first identified by Radin and Yin [18]. Building on their results, we carefully study the model along the phase transition curve. In particular, we give precise scaling laws for the variance and covariance of edge and outward star densities, and we obtain an exact formula for the limiting edge probabilities, both on and off the phase transition curve.