A phase transition regarding the evolution of bootstrap processes in inhomogeneous random graphs

TL;DR

This paper investigates phase transitions in bootstrap percolation on inhomogeneous Chung-Lu random graphs, identifying critical conditions under which small initial infections lead to widespread outbreaks, especially when the degree distribution follows a power law with exponent 3.

Contribution

It characterizes the conditions for phase transitions in bootstrap percolation on Chung-Lu graphs, focusing on the influence of the weight distribution tail and identifying the critical initial infection density.

Findings

Critical phenomena occur mainly when the weight distribution tail dominates a power law with exponent 3.

A critical initial infection density is determined for the phase transition.

Small initial infections can cause large outbreaks under specific weight distribution conditions.

Abstract

A bootstrap percolation process on a graph with infection threshold $r\ge 1$ is a dissemination process that evolves in time steps. The process begins with a subset of infected vertices and in each subsequent step every uninfected vertex that has at least $r$ infected neighbours becomes infected and remains so forever. Critical phenomena in bootstrap percolation processes were originally observed by Aizenman and Lebowitz in the late 1980s as finite-volume phase transitions in $\mathbb{Z}^d$ that are caused by the accumulation of small local islands of infected vertices. They were also observed in the case of dense (homogeneous) random graphs by Janson, \L uczak, Turova and Valier (2012). In this paper, we consider the class of inhomogeneous random graphs known as the Chung-Lu model: each vertex is equipped with a positive weight and each pair of vertices appears as an edge with probability proportional to the product of the weights. In particular, we focus on the sparse regime, where the number of edges is proportional to the number of vertices. The main results of this paper determine those weight sequences for which a critical phenomenon occurs: there is a critical density of vertices that are infected at the beginning of the process, above which a small (sublinear) set of infected vertices creates an avalanche of infections that in turn leads to an outbreak. We show that this occurs essentially only when the tail of the weight distribution dominates a power law with exponent 3 and we determine the critical density in this case.