TY - GEN

T1 - Contagious sets in expanders

AU - Coja-Oghlan, Amin

AU - Feige, Uriel

AU - Krivelevich, Michael

AU - Reichman, Daniel

N1 - Publisher Copyright:
Copyright © 2015 by the Society for Industrial and Applied Mathmatics.

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84938263073&partnerID=8YFLogxK

U2 - 10.1137/1.9781611973730.131

DO - 10.1137/1.9781611973730.131

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AN - SCOPUS:84938263073

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 1953

EP - 1987

BT - Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015

PB - Association for Computing Machinery

Y2 - 4 January 2015 through 6 January 2015

ER -