Cutoff for random walk on random graphs with a community structure
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866908432519397376 |
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| author | Hermon, Jonathan Šarković, Anđela Sousi, Perla |
| author_facet | Hermon, Jonathan Šarković, Anđela Sousi, Perla |
| contents | We consider a variant of the configuration model with an embedded community structure and study the mixing properties of a simple random walk on it. Every vertex has an internal $\mathrm{deg}^{\text{int}}\geq 3$ and an outgoing $\mathrm{deg}^{\text{out}}$ number of half-edges. Given a stochastic matrix $Q$, we pick a random perfect matching of the half-edges subject to the constraint that each vertex $v$ has $\mathrm{deg}^{\text{int}}(v)$ neighbours inside its community and the proportion of outgoing half-edges from community $i$ matched to a half-edge from community $j$ is $Q(i,j)$. Assuming the number of communities is constant and they all have comparable sizes, we prove the following dichotomy: simple random walk on the resulting graph exhibits cutoff if and only if the product of the Cheeger constant of $Q$ times $\log n$ (where $n$ is the number of vertices) diverges.
In [4], Ben-Hamou established a dichotomy for cutoff for a non-backtracking random walk on a similar random graph model with 2 communities. We prove the same characterisation of cutoff holds for simple random walk. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_04469 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Cutoff for random walk on random graphs with a community structure Hermon, Jonathan Šarković, Anđela Sousi, Perla Probability 60F05, 60G50 We consider a variant of the configuration model with an embedded community structure and study the mixing properties of a simple random walk on it. Every vertex has an internal $\mathrm{deg}^{\text{int}}\geq 3$ and an outgoing $\mathrm{deg}^{\text{out}}$ number of half-edges. Given a stochastic matrix $Q$, we pick a random perfect matching of the half-edges subject to the constraint that each vertex $v$ has $\mathrm{deg}^{\text{int}}(v)$ neighbours inside its community and the proportion of outgoing half-edges from community $i$ matched to a half-edge from community $j$ is $Q(i,j)$. Assuming the number of communities is constant and they all have comparable sizes, we prove the following dichotomy: simple random walk on the resulting graph exhibits cutoff if and only if the product of the Cheeger constant of $Q$ times $\log n$ (where $n$ is the number of vertices) diverges. In [4], Ben-Hamou established a dichotomy for cutoff for a non-backtracking random walk on a similar random graph model with 2 communities. We prove the same characterisation of cutoff holds for simple random walk. |
| title | Cutoff for random walk on random graphs with a community structure |
| topic | Probability 60F05, 60G50 |
| url | https://arxiv.org/abs/2212.04469 |