Critical phenomena in exponential random graphs

TL;DR

This paper analyzes phase transitions in exponential random graph models, revealing a comprehensive phase diagram with first and second order transitions influenced by local subgraph features.

Contribution

It derives the full phase diagram for a broad class of 3-parameter exponential random graph models, identifying the nature of their phase transitions.

Findings

Existence of a phase transition in exponential random graphs.

Identification of a first order surface transition and a second order critical curve.

Complete phase diagram for a family of models with attraction parameters.

Abstract

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.