On the phase transition curve in a directed exponential random graph model

TL;DR

This paper analyzes the phase transition behavior in directed exponential random graph models, deriving precise asymptotics for the normalization constant and free energy, revealing a first order phase transition and unusual model behavior along this curve.

Contribution

It provides the first detailed asymptotic analysis of the normalization constant and free energy in directed exponential random graph models, identifying the phase transition curve.

Findings

Asymptotic formulas for the normalization constant and free energy

Identification of a first order phase transition curve

Unusual model behavior along the phase transition curve

Abstract

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.