Embedding theorems for random graphs with specified degrees
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866913604811358208 |
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| author | Gao, Pu Ohapkin, Yuval |
| author_facet | Gao, Pu Ohapkin, Yuval |
| contents | 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 $Δ({\bf d})$ denote the maximum degree in ${\bf d}$. Our coupling result is optimal when $Δ({\bf d})^2\ll \|{\bf d}\|_1$, 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 $Δ({\bf d})^2\ll \|{\bf d}\|_1$. 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 $i,j\in [n]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_09729 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Embedding theorems for random graphs with specified degrees Gao, Pu Ohapkin, Yuval Combinatorics 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 $Δ({\bf d})$ denote the maximum degree in ${\bf d}$. Our coupling result is optimal when $Δ({\bf d})^2\ll \|{\bf d}\|_1$, 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 $Δ({\bf d})^2\ll \|{\bf d}\|_1$. 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 $i,j\in [n]$. |
| title | Embedding theorems for random graphs with specified degrees |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2302.09729 |