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April 2018 Sharp thresholds for contagious sets in random graphs
Omer Angel, Brett Kolesnik
Ann. Appl. Probab. 28(2): 1052-1098 (April 2018). DOI: 10.1214/17-AAP1325


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.


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Omer Angel. Brett Kolesnik. "Sharp thresholds for contagious sets in random graphs." Ann. Appl. Probab. 28 (2) 1052 - 1098, April 2018.


Received: 1 December 2016; Published: April 2018
First available in Project Euclid: 11 April 2018

zbMATH: 06897950
MathSciNet: MR3784495
Digital Object Identifier: 10.1214/17-AAP1325

Primary: 60K35
Secondary: 05C80, 60C05, 82B43

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.28 • No. 2 • April 2018
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