The Annals of Applied Probability

Bootstrap percolation on the random graph $G_{n,p}$

Svante Janson, Tomasz Łuczak, Tatyana Turova, and Thomas Vallier

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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.

Article information

Source
Ann. Appl. Probab., Volume 22, Number 5 (2012), 1989-2047.

Dates
First available in Project Euclid: 12 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1350067992

Digital Object Identifier
doi:10.1214/11-AAP822

Mathematical Reviews number (MathSciNet)
MR3025687

Zentralblatt MATH identifier
1254.05182

Subjects
Primary: 05C80: Random graphs [See also 60B20] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60C05: Combinatorial probability

Keywords
Bootstrap percolation random graph sharp threshold

Citation

Janson, Svante; Łuczak, Tomasz; Turova, Tatyana; Vallier, Thomas. Bootstrap percolation on the random graph $G_{n,p}$. Ann. Appl. Probab. 22 (2012), no. 5, 1989--2047. doi:10.1214/11-AAP822. https://projecteuclid.org/euclid.aoap/1350067992


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