Subcritical epidemics on random graphs
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866910765181566976 |
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| author | Nguyen, Oanh Sly, Allan |
| author_facet | Nguyen, Oanh Sly, Allan |
| contents | We study the contact process on random graphs with low infection rate $λ$. For random $d$-regular graphs, it is known that the survival time is $O(\log n)$ below the critical $λ_c$. By contrast, on the Erdős-Rényi random graphs $\mathcal G(n,d/n)$, rare high-degree vertices result in much longer survival times. We show that the survival time is governed by high-density local configurations. In particular, we show that there is a long string of high-degree vertices on which the infection lasts for time $n^{λ^{2+o(1)}}$. To establish a matching upper bound, we introduce a modified version of the contact process which ignores infections that do not lead to further infections and allows for a shaper recursive analysis on branching process trees, the local-weak limit of the graph. Our methods, moreover, generalize to random graphs with given degree distributions that have exponential moments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_03551 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Subcritical epidemics on random graphs Nguyen, Oanh Sly, Allan Probability We study the contact process on random graphs with low infection rate $λ$. For random $d$-regular graphs, it is known that the survival time is $O(\log n)$ below the critical $λ_c$. By contrast, on the Erdős-Rényi random graphs $\mathcal G(n,d/n)$, rare high-degree vertices result in much longer survival times. We show that the survival time is governed by high-density local configurations. In particular, we show that there is a long string of high-degree vertices on which the infection lasts for time $n^{λ^{2+o(1)}}$. To establish a matching upper bound, we introduce a modified version of the contact process which ignores infections that do not lead to further infections and allows for a shaper recursive analysis on branching process trees, the local-weak limit of the graph. Our methods, moreover, generalize to random graphs with given degree distributions that have exponential moments. |
| title | Subcritical epidemics on random graphs |
| topic | Probability |
| url | https://arxiv.org/abs/2205.03551 |