Annals of Applied Probability

Sharp thresholds for contagious sets in random graphs

Omer Angel and Brett Kolesnik

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Abstract

For fixed $r\geq2$, we consider bootstrap percolation with threshold $r$ on the Erdős–Rényi graph $\mathcal{G}_{n,p}$. We identify a threshold for $p$ above which there is with high probability a set of size $r$ that can infect the entire graph. This improves a result of Feige, Krivelevich and Reichman, which gives bounds for this threshold, up to multiplicative constants.

As an application of our results, we obtain an upper bound for the threshold for $K_{4}$-percolation on $\mathcal{G}_{n,p}$, as studied by Balogh, Bollobás and Morris. This bound is shown to be asymptotically sharp in subsequent work.

These thresholds are closely related to the survival probabilities of certain time-varying branching processes, and we derive asymptotic formulae for these survival probabilities, which are of interest in their own right.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 2 (2018), 1052-1098.

Dates
Received: December 2016
First available in Project Euclid: 11 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1523433631

Digital Object Identifier
doi:10.1214/17-AAP1325

Mathematical Reviews number (MathSciNet)
MR3784495

Zentralblatt MATH identifier
06897950

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

Keywords
Bootstrap percolation cellular automaton phase transition random graph sharp threshold

Citation

Angel, Omer; Kolesnik, Brett. Sharp thresholds for contagious sets in random graphs. Ann. Appl. Probab. 28 (2018), no. 2, 1052--1098. doi:10.1214/17-AAP1325. https://projecteuclid.org/euclid.aoap/1523433631


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