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Normal approximation for exponential random graphs

Xiao Fang, Song-Hao Liu, Qi-Man Shao, Yi-Kun Zhao

TL;DR

The paper resolves the open question of whether the central limit theorem holds for the total number of edges in exponential random graph models (ERGMs) within the subcritical region, establishing a quantitative CLT with non-asymptotic error bounds and an explicit asymptotic variance. It develops a general Stein's method framework for normal approximation of functionals of nonlinear exponential families, including a higher-order concentration component, and applies it to ERGMs to obtain a CLT and finite-sample error bounds. The authors also extend the CLT to general subgraph counts and illustrate the approach with the Curie–Weiss model as a canonical example. The results offer a robust toolkit for normal approximation in nonlinear exponential-family models and provide precise rates that enhance understanding of ERGM behavior in the subcritical regime.

Abstract

The question of whether the central limit theorem (CLT) holds for the total number of edges in exponential random graph models (ERGMs) in the subcritical region of parameters has remained an open problem. In this paper, we establish the CLT. As a result of our proof, we also derive a convergence rate for the CLT, an explicit formula for the asymptotic variance, and the CLT for general subgraph counts. To establish our main result, we develop Stein's method for the normal approximation of general functionals of nonlinear exponential families of random variables, which is of independent interest. In addition to ERGMs, our general theorem can also be applied to other models. A key ingredient needed in our proof for the ERGM is a higher-order concentration inequality, which was known in a subset of the subcritical region called Dobrushin's uniqueness region. We use Stein's method to partially generalize such inequalities to the subcritical region.

Normal approximation for exponential random graphs

TL;DR

The paper resolves the open question of whether the central limit theorem holds for the total number of edges in exponential random graph models (ERGMs) within the subcritical region, establishing a quantitative CLT with non-asymptotic error bounds and an explicit asymptotic variance. It develops a general Stein's method framework for normal approximation of functionals of nonlinear exponential families, including a higher-order concentration component, and applies it to ERGMs to obtain a CLT and finite-sample error bounds. The authors also extend the CLT to general subgraph counts and illustrate the approach with the Curie–Weiss model as a canonical example. The results offer a robust toolkit for normal approximation in nonlinear exponential-family models and provide precise rates that enhance understanding of ERGM behavior in the subcritical regime.

Abstract

The question of whether the central limit theorem (CLT) holds for the total number of edges in exponential random graph models (ERGMs) in the subcritical region of parameters has remained an open problem. In this paper, we establish the CLT. As a result of our proof, we also derive a convergence rate for the CLT, an explicit formula for the asymptotic variance, and the CLT for general subgraph counts. To establish our main result, we develop Stein's method for the normal approximation of general functionals of nonlinear exponential families of random variables, which is of independent interest. In addition to ERGMs, our general theorem can also be applied to other models. A key ingredient needed in our proof for the ERGM is a higher-order concentration inequality, which was known in a subset of the subcritical region called Dobrushin's uniqueness region. We use Stein's method to partially generalize such inequalities to the subcritical region.
Paper Structure (6 sections, 9 theorems, 178 equations)

This paper contains 6 sections, 9 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. CLT for nonlinear exponential families
  3. Normal approximation for ERGM
  4. Proof of Theorem \ref{['theorem:1']}
  5. Proof of Theorem \ref{['thm:ERGM']}
  6. Proof of Example \ref{['ex:CW']}

Key Result

Theorem 2.1

For $W=f(Y)$ defined above and for the distributional distances $d_{\text{Wass}}$ and $d_{\text{Kol}}$ defined in eq:d2dist and eq:dk respectively, we have where $C$ is an absolute constant.

Theorems & Definitions (21)

  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Example 2.1: Curie--Weiss model
  • Theorem 3.1
  • Remark 3.1
  • Corollary 3.1: CLT for general subgraph counts
  • proof
  • proof : Proof of \ref{['theorem:1']}
  • Lemma 4.1
  • ...and 11 more