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Second-order Approximation of Exponential Random Graph Models

Wen-Yi Ding, Xiao Fang

TL;DR

It is proved that the second-order approximation of ERGMs using two-stars and triangles achieves second-order accuracy in the triangle-free case and is formally obtained by the Hoeffding decomposition and rigorously justified using Stein's method.

Abstract

Exponential random graph models (ERGMs) are flexible probability models allowing edge dependency. However, it is known that, to a first-order approximation, many ERGMs behave like Erdös-Rényi random graphs, where edges are independent. In this paper, to distinguish ERGMs from Erdös-Rényi random graphs, we consider second-order approximations of ERGMs using two-stars and triangles. We prove that the second-order approximation indeed achieves second-order accuracy in the triangle-free case. The new approximation is formally obtained by Hoeffding decomposition and rigorously justified using Stein's method.

Second-order Approximation of Exponential Random Graph Models

TL;DR

It is proved that the second-order approximation of ERGMs using two-stars and triangles achieves second-order accuracy in the triangle-free case and is formally obtained by the Hoeffding decomposition and rigorously justified using Stein's method.

Abstract

Exponential random graph models (ERGMs) are flexible probability models allowing edge dependency. However, it is known that, to a first-order approximation, many ERGMs behave like Erdös-Rényi random graphs, where edges are independent. In this paper, to distinguish ERGMs from Erdös-Rényi random graphs, we consider second-order approximations of ERGMs using two-stars and triangles. We prove that the second-order approximation indeed achieves second-order accuracy in the triangle-free case. The new approximation is formally obtained by Hoeffding decomposition and rigorously justified using Stein's method.
Paper Structure (7 sections, 2 theorems, 75 equations)

This paper contains 7 sections, 2 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Second-order expansion
  4. Justification and discussion
  5. Organization
  6. Formal expansion
  7. Justification of the formal expansion

Key Result

Proposition 3.1

When $X$ is the rectangle ERGM rect in the subcritical region and $Z$ is the Erdös--Rényi graph $G(n,p)$ with $p$ satisfying rect2, there exist some parameter values $\beta_1, \beta_2$ satisfying $\Phi'(1)<2$, where $\Phi(a)$ was defined in varph, and a sequence of functions $h_n: \{0,1\}^{\mathcal{

Theorems & Definitions (5)

  • Definition 2.1
  • Proposition 3.1
  • Theorem 3.1
  • proof : Proof of \ref{['thm:main']}
  • proof : Proof of \ref{['prop:sharp']}