Contact processes with random recovery rates and edge weights on complete graphs

TL;DR

This paper analyzes a contact process with random recovery rates and edge weights on complete graphs, identifying a critical threshold and characterizing the process's extinction and survival times in large networks.

Contribution

It introduces a model with random recovery and edge weights on complete graphs and derives the critical value and asymptotic behaviors for extinction and survival times.

Findings

Critical value inversely proportional to mean edge weight and inverse mean recovery rate

Subcritical process dies out in O(log n) time with high probability

Supercritical process survives in exponential time with high probability

Abstract

In this paper we are concerned with the contact process with random recovery rates and edge weights on complete graph with $n$ vertices. We show that the model has a critical value which is inversely proportional to the product of the mean of the edge weight and the mean of the inverse of the recovery rate. In the subcritical case, the process dies out before a moment with order $O(\log n)$ with high probability as $n\rightarrow+\infty$. In the supercritical case, the process survives at a moment with order $\exp\{O(n)\}$ with high probability as $n\rightarrow+\infty$. Our proof for the subcritical case is inspired by the graphical method introduced in \cite{Har1978}. Our proof for the supercritical case is inspired by approach introduced in \cite{Pet2011}, which deal with the case where the contact process is with random vertex weights.