Conditional central limit theorems for exponential random graphs

TL;DR

This paper establishes a conditional Central Limit Theorem for the number of two stars in exponential random graph models, extending results from Erdős–Rényi graphs and introducing new techniques for analyzing subgraph counts.

Contribution

It develops a novel conditional CLT for ERGMs using exchangeable pairs, with explicit formulas and new concentration inequalities, applicable to general subgraph counts.

Findings

Proves a conditional CLT for two-star counts in ERGMs.

Develops a new exchangeable pairs method for conditional CLTs.

Provides concentration inequalities for subgraph counts.

Abstract

In this paper, we study the Exponential Random Graph Models (ERGMs) conditioning on the number of edges. In subcritical region of model parameters, we prove a conditional Central Limit Theorem (CLT) with explicit mean and variance for the number of two stars. This generalizes the corresponding result in the literature for the Erd\H{o}s--R\'enyi random graph. To prove our main result, we develop a new conditional CLT via exchangeable pairs based on the ideas of Dey and Terlov. Our key technical contributions in the application to ERGMs include establishing a linearity condition for an exchangeable pair involving two star counts, a local CLT for edge counts, as well as new higher-order concentration inequalities. Our approach also works for general subgraph counts, and we give a conjectured form of their conditional CLT.