Contact Processes on Random Regular Graphs

TL;DR

This paper demonstrates that the contact process on large random regular graphs exhibits a sharp cutoff in the supercritical phase, with infection spreading rapidly around a critical logarithmic time scale.

Contribution

It proves the cutoff phenomenon for the contact process on random regular graphs, linking infection rate to rapid phase transition timing.

Findings

The process exhibits a sharp cutoff at logarithmic times.

Infection prevalence jumps from near zero to near equilibrium rapidly.

The cutoff time depends on the infection rate and graph degree.

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 $\epsilon> 0$, when the number $n$ of vertices is large then (a) at times $t< (C - \epsilon) \log n$ the fraction of infected vertices is vanishingly small, but (b) at time $(C + \epsilon) \log n$ the fraction of infected vertices is within $\epsilon$ of $p$, with probability $p$.