Subcritical epidemics on random graphs

Oanh Nguyen; Allan Sly

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arXiv · 2205.03551

Subcritical epidemics on random graphs

Oanh Nguyen·Allan Sly

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Abstract

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.

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BibTeX RIS

Oanh Nguyen, Allan Sly. 2024-12-28. Subcritical epidemics on random graphs. https://arxiv.org/abs/2205.03551

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