From survival to extinction of the contact process by the removal of a single edge

TL;DR

This paper constructs a specific tree structure demonstrating that removing a single edge can cause the contact process to transition from survival to extinction, highlighting the critical role of certain edges.

Contribution

It introduces a novel tree construction showing how a single edge removal can drastically change the contact process's behavior from survival to extinction.

Findings

Contact process survives on the constructed tree for any positive infection rate.

Removing a specific edge causes the process to die out for infection rates below 1/4.

Highlights the importance of individual edges in the dynamics of contact processes.

Abstract

We give a construction of a tree in which the contact process with any positive infection rate survives but, if a certain privileged edge $e^*$ is removed, one obtains two subtrees in which the contact process with infection rate smaller than $1/4$ dies out.