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October 2012 Bootstrap percolation on the random graph $G_{n,p}$
Svante Janson, Tomasz Łuczak, Tatyana Turova, Thomas Vallier
Ann. Appl. Probab. 22(5): 1989-2047 (October 2012). DOI: 10.1214/11-AAP822

Abstract

Bootstrap percolation on the random graph $G_{n,p}$ is a process of spread of “activation” on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least $r\geq2$ active neighbors become active as well.

We study the size $A^{\ast}$ of the final active set. The parameters of the model are, besides $r$ (fixed) and $n$ (tending to $\infty$), the size $a=a(n)$ of the initially active set and the probability $p=p(n)$ of the edges in the graph. We show that the model exhibits a sharp phase transition: depending on the parameters of the model, the final size of activation with a high probability is either $n-o(n)$ or it is $o(n)$. We provide a complete description of the phase diagram on the space of the parameters of the model. In particular, we find the phase transition and compute the asymptotics (in probability) for $A^{\ast}$; we also prove a central limit theorem for $A^{\ast}$ in some ranges. Furthermore, we provide the asymptotics for the number of steps until the process stops.

Citation

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Svante Janson. Tomasz Łuczak. Tatyana Turova. Thomas Vallier. "Bootstrap percolation on the random graph $G_{n,p}$." Ann. Appl. Probab. 22 (5) 1989 - 2047, October 2012. https://doi.org/10.1214/11-AAP822

Information

Published: October 2012
First available in Project Euclid: 12 October 2012

zbMATH: 1254.05182
MathSciNet: MR3025687
Digital Object Identifier: 10.1214/11-AAP822

Subjects:
Primary: 05C80, 60C05, 60K35

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.22 • No. 5 • October 2012
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