Limit theorems for exponential random graphs

TL;DR

This paper analyzes the asymptotic behavior and fluctuations of edge density in the edge-triangle exponential random graph model, providing limit theorems and phase diagram insights using statistical mechanics tools.

Contribution

It offers a comprehensive characterization of the phase diagram and limit theorems for edge density in the edge-triangle model, extending to more general exponential random graphs.

Findings

Limit theorems for edge density in different phases

Asymptotic distribution of edge density as graph size grows

Conjectures on behavior at criticality

Abstract

We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime, the limiting free energy exists together with a complete characterization of the phase diagram of the model. We borrow tools from statistical mechanics to obtain limit theorems for the edge density. First, we investigate the asymptotic distribution of this quantity, as the graph size tends to infinity, in the various phases, and we complement this analysis with a study of the speed of convergence of the average edge density toward its limiting value. Then, we study the fluctuations of the edge density around its average value off the critical curve and formulate conjectures about the behavior at criticality based on the analysis of a mean-field approximation of the model. Some of our results can be extended with no substantial changes to more general classes of exponential random graphs.