Contagious sets in expanders

Amin Coja-Oghlan, Uriel Feige, Michael Krivelevich, Daniel Reichman

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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, where r > 1 is the activation threshold. 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. It is known that for every d-regular or nearly d-regular graph on n vertices, m(G,r) ≤ O(nr/d). We consider such graphs that additionally have expansion properties, parameterized by the spectral gap and/or the girth of the graphs. The general flavor of our results is that sufficiently strong expansion properties imply that m(G,2) ≤ O(n/d2) (and more generally, m(G,r) ≤ O(n/dr(r-1))). In addition, we demonstrate that rather weak assumptions on the girth and/or the spectral gap suffice in order to imply that m(G,2) ≤ O(n log d/d2). For example, we show this for graphs of girth at least 7, and for graphs with λ(G) ≤ (1-ε)d, provided the graph has no 4-cycles. Our results are algorithmic entailing simple and efficient algorithms for selecting contagious sets.

Original languageEnglish
Title of host publicationProceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015
PublisherAssociation for Computing Machinery
Pages1953-1987
Number of pages35
EditionJanuary
ISBN (Electronic)9781611973747
DOIs
StatePublished - 2015
Event26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 - San Diego, United States
Duration: 4 Jan 20156 Jan 2015

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
NumberJanuary
Volume2015-January

Conference

Conference26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015
Country/TerritoryUnited States
CitySan Diego
Period4/01/156/01/15

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