---
abstract: 'We study a mean field model of a complex network, focusing on edge and triangle densities. Our first result is the derivation of a variational characterization of the entropy density, compatible with the infinite node limit. We then determine the optimizing graphs for small triangle density and a range of edge density, though we can only prove they are local, not global, maxima of the entropy density. With this assumption we then prove that the resulting entropy density must lose its analyticity in various regimes. In particular this implies the existence of a phase transition between distinct heterogeneous multipartite phases at low triangle density, and a phase transition between these phases and the disordered phase at high triangle density.'
address:
- |
    Charles Radin\
    Department of Mathematics\
    The University of Texas at Austin\
    Austin, TX 78712
- |
    Lorenzo Sadun\
    Department of Mathematics\
    The University of Texas at Austin\
    Austin, TX 78712
author:
- Charles Radin and Lorenzo Sadun
title: Phase transitions in a complex network
---

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Introduction
============

Exponential random graph models are a well known class of complex networks; see [@Ne] and references therein. Using the language of statistical mechanics they are mean field models, 