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Embedding theorems for random graphs with specified degrees
Published online by Cambridge University Press: 16 October 2024
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 
$jk\in \binom{[n]}{2}$ with probability 
$W_{jk}=W_{kj}$. Given a degree sequence 
$\textbf{d}=(d_1,\ldots, d_n)$, let 
${\mathcal G}(n,\textbf{d})$ denote a uniformly random graph with degree sequence 
$\textbf{d}$. We couple 
${\mathcal G}(n,W)$ and 
${\mathcal G}(n,\textbf{d})$ together so that asymptotically almost surely 
${\mathcal G}(n,W)$ is a subgraph of 
${\mathcal G}(n,\textbf{d})$, where 
$W$ is some function of 
$\textbf{d}$. Let 
$\Delta (\textbf{d})$ denote the maximum degree in 
$\textbf{d}$. Our coupling result is optimal when 
$\Delta (\textbf{d})^2\ll \|\textbf{d}\|_1$, that is, 
$W_{ij}$ is asymptotic to 
${\mathbb P}(ij\in{\mathcal G}(n,\textbf{d}))$ for every 
$i,j\in [n]$. We also have coupling results for 
$\textbf{d}$ that are not constrained by the condition 
$\Delta (\textbf{d})^2\ll \|\textbf{d}\|_1$. For such 
$\textbf{d}$ our coupling result is still close to optimal, in the sense that 
$W_{ij}$ is asymptotic to 
${\mathbb P}(ij\in{\mathcal G}(n,\textbf{d}))$ for most pairs 
$ij\in \binom{[n]}{2}$.
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- © The Author(s), 2024. Published by Cambridge University Press
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