The Annals of Applied Probability

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

Nikolaos Fountoulakis, Mihyun Kang, Christoph Koch, and Tamás Makai

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A bootstrap percolation process on a graph with infection threshold $r\ge1$ 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 et al. [Ann. Appl. Probab. 22 (2012) 1989–2047]. 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.

Article information

Ann. Appl. Probab., Volume 28, Number 2 (2018), 990-1051.

Received: September 2016
Revised: February 2017
First available in Project Euclid: 11 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 60K37: Processes in random environments
Secondary: 82B26: Phase transitions (general) 82B27: Critical phenomena

Bootstrap percolation inhomogeneous random graphs critical phenomena


Fountoulakis, Nikolaos; Kang, Mihyun; Koch, Christoph; Makai, Tamás. A phase transition regarding the evolution of bootstrap processes in inhomogeneous random graphs. Ann. Appl. Probab. 28 (2018), no. 2, 990--1051. doi:10.1214/17-AAP1324.

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