Embedding theorems for random graphs with specified degrees

TL;DR

This paper establishes optimal coupling between random graphs with specified degree sequences and inhomogeneous Erdős–Rényi graphs, providing new insights into their structural similarities under various degree constraints.

Contribution

It introduces coupling theorems that relate graphs with given degrees to inhomogeneous Erdős–Rényi models, extending previous results to broader degree conditions.

Findings

Coupling is optimal when maximum degree squared is much less than total degree sum.

Results hold for less constrained degree sequences, still closely approximating edge probabilities.

Provides a framework for understanding the embedding of degree-constrained graphs within inhomogeneous models.

Abstract

Given an $n\times n$ symmetric matrix $W\in [0,1]^{[n]\times [n]}$, let $\mathcal{G}(n,W)$ be the random graph obtained by independently including each edge $jk$ with probability $W_{jk}$. Given a degree sequence ${\bf d}=(d_1,\ldots, d_n)$, let $\mathcal{G}(n,{\bf d})$ denote a uniformly random graph with degree sequence ${\bf d}$. We couple $\mathcal{G}(n,W)$ and $\mathcal{G}(n,{\bf d})$ together so that a.a.s. $\mathcal{G}(n,W)$ is a subgraph of $\mathcal{G}(n,{\bf d})$, where $W$ is some function of ${\bf d}$. Let $\Delta({\bf d})$ denote the maximum degree in ${\bf d}$. Our coupling result is optimal when $\Delta({\bf d})^2\ll \|{\bf d}\|_1$, i.e.\ $W_{ij}$ is asymptotic to $\mathbb{P}(ij\in \mathcal{G}(n,{\bf d}))$ for every $i,j\in [n]$. We also have coupling results for ${\bf d}$ that are not constrained by the condition $\Delta({\bf d})^2\ll \|{\bf d}\|_1$. For such ${\bf d}$ our coupling result is still close to optimal, in the sense that $W_{ij}$ is asymptotic to $\mathbb{P}(ij\in \mathcal{G}(n,{\bf d}))$ for most pairs $i,j\in [n]$.