Contact Processes on Random Regular Graphs

2015/02/26 by Steven P. Lalley, Wei Su, Lalley, Steven +1 · 4 citations
Mathematics · #FOS: Mathematics #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1502.07421

openalex publication_date 2015/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the contact process on a random d-regular graph initiated by a single infected vertex obeys the "cutoff phenomenon" in its supercritical phase. In particular, we prove that when the infection rate is larger than the critical value of the contact process on the infinite d-regular tree there are positive constants C, p depending on the infection rate such that for sufficiently small ε> 0, when the number n of vertices is large then (a) at times t< (C - ε) log n the fraction of infected vertices is vanishingly small, but (b) at time (C + ε) log n the fraction of infected vertices is within ε of p, with probability p.

Cited by

• The extinction of the contact process in a one-dimensional random environment with long-range interactions
• Exponential extinction time of the contact process on rank-one inhomogeneous random graphs
• Phase transitions for contact processes on sparse random graphs via metastability and local limits
• The contact process on Scale-Free Percolation

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• The contact process over a dynamical d-regular graph
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• The contact process on dynamic regular graphs: monotonicity and subcritical phase
• A Metastability Result for the Contact Process on a Random Regular Graph