Open Access
October 2017 Contagious sets in random graphs
Uriel Feige, Michael Krivelevich, Daniel Reichman
Ann. Appl. Probab. 27(5): 2675-2697 (October 2017). DOI: 10.1214/16-AAP1254

Abstract

We consider the following activation process in undirected graphs: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least r active neighbors. A contagious set is a set whose activation results with the entire graph being active. Given a graph G, let m(G,r) be the minimal size of a contagious set.

We study this process on the binomial random graph G:=G(n,p) with p:=dn and 1d(nloglognlog2n)r1r. Assuming r>1 to be a constant that does not depend on n, we prove that

m(G,r)=Θ(ndrr1logd), with high probability. We also show that the threshold probability for m(G,r)=r to hold is p=Θ(1(nlogr1n)1/r).

Citation

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Uriel Feige. Michael Krivelevich. Daniel Reichman. "Contagious sets in random graphs." Ann. Appl. Probab. 27 (5) 2675 - 2697, October 2017. https://doi.org/10.1214/16-AAP1254

Information

Received: 1 February 2016; Revised: 1 July 2016; Published: October 2017
First available in Project Euclid: 3 November 2017

zbMATH: 06822203
MathSciNet: MR3719944
Digital Object Identifier: 10.1214/16-AAP1254

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

Keywords: Bootstrap percolation , minimal contagious set , Random graphs

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.27 • No. 5 • October 2017
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