Annals of Applied Probability

Sharp thresholds for contagious sets in random graphs

Omer Angel and Brett Kolesnik

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

Bootstrap percolation cellular automaton phase transition random graph sharp threshold


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.

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