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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2305.03607 |
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| _version_ | 1866916443416690688 |
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| author | Hladký, Jan Viswanathan, Gopal |
| author_facet | Hladký, Jan Viswanathan, Gopal |
| contents | Each graphon $W:Ω^2\rightarrow[0,1]$ yields an inhomogeneous random graph model $G(n,W)$. We show that $G(n,W)$ is asymptotically almost surely connected if and only if (i) $W$ is a connected graphon and (ii) the measure of elements of $Ω$ of $W$-degree less than $α$ is $o(α)$ as $α\rightarrow 0$. These two conditions encapsulate the absence of several linear-sized components, and of isolated vertices, respectively.
We study in bigger detail the limit probability of the property that $G(n,W)$ contains an isolated vertex, and, more generally, the limit distribution of the minimum degree of $G(n,W)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_03607 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Connectivity of inhomogeneous random graphs II Hladký, Jan Viswanathan, Gopal Combinatorics Each graphon $W:Ω^2\rightarrow[0,1]$ yields an inhomogeneous random graph model $G(n,W)$. We show that $G(n,W)$ is asymptotically almost surely connected if and only if (i) $W$ is a connected graphon and (ii) the measure of elements of $Ω$ of $W$-degree less than $α$ is $o(α)$ as $α\rightarrow 0$. These two conditions encapsulate the absence of several linear-sized components, and of isolated vertices, respectively. We study in bigger detail the limit probability of the property that $G(n,W)$ contains an isolated vertex, and, more generally, the limit distribution of the minimum degree of $G(n,W)$. |
| title | Connectivity of inhomogeneous random graphs II |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2305.03607 |