A cluster expansion approach to exponential random graph models

TL;DR

This paper introduces a novel approach to exponential random graph models by applying cluster expansion techniques from statistical mechanics, enabling analysis of network behavior in certain parameter regimes.

Contribution

It reformulates exponential random graph models as lattice gas models and derives a convergent power series for the free energy, providing new analytical tools.

Findings

Derived a convergent power series for the free energy in small parameter regimes

Characterized the structure and behavior of the limiting network

Connected exponential random graph models with statistical mechanics methods

Abstract

The exponential family of random graphs is among the most widely-studied network models. We show that any exponential random graph model may alternatively be viewed as a lattice gas model with a finite Banach space norm. The system may then be treated by cluster expansion methods from statistical mechanics. In particular, we derive a convergent power series expansion for the limiting free energy in the case of small parameters. Since the free energy is the generating function for the expectations of other random variables, this characterizes the structure and behavior of the limiting network in this parameter region.