research-article

Contagious sets in expanders

Authors:

Amin Coja-Oghlan,

Michael Krivelevich,

Daniel ReichmanAuthors Info & Claims

SODA '15: Proceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms

Pages 1953 - 1987

Published: 04 January 2015 Publication History

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/d^r/(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.

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Cited By

Freund DPoloczek MReichman D(2018)Contagious sets in dense graphsEuropean Journal of Combinatorics10.1016/j.ejc.2017.07.01168:C(66-78)Online publication date: 1-Feb-2018
https://dl.acm.org/doi/10.1016/j.ejc.2017.07.011
Cordasco GGargano LMecchia MRescigno AVaccaro U(2018)Discovering Small Target Sets in Social NetworksAlgorithmica10.1007/s00453-017-0390-580:6(1804-1833)Online publication date: 1-Jun-2018
https://dl.acm.org/doi/10.1007/s00453-017-0390-5
Cordasco GGargano LRescigno A(2015)Influence Propagation over Large Scale Social NetworksProceedings of the 2015 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining 201510.1145/2808797.2808888(1531-1538)Online publication date: 25-Aug-2015
https://dl.acm.org/doi/10.1145/2808797.2808888
Show More Cited By

Contagious sets in expanders
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
2. Theory of computation

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cover image ACM Other conferences

SODA '15: Proceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms

January 2015

2056 pages

Program Chair:
Piotr Indyk
Massachusetts Institute of Technology

Sponsors

SIAM: Society for Industrial and Applied Mathematics

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SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 04 January 2015

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SODA '15

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SODA '15: ACM SIAM Symposium on Discrete Algorithms

January 4 - 6, 2015

California, San Diego

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SODA '15 Paper Acceptance Rate 137 of 495 submissions, 28%;

Overall Acceptance Rate 411 of 1,322 submissions, 31%

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Cited By

Freund DPoloczek MReichman D(2018)Contagious sets in dense graphsEuropean Journal of Combinatorics10.1016/j.ejc.2017.07.01168:C(66-78)Online publication date: 1-Feb-2018
https://dl.acm.org/doi/10.1016/j.ejc.2017.07.011
Cordasco GGargano LMecchia MRescigno AVaccaro U(2018)Discovering Small Target Sets in Social NetworksAlgorithmica10.1007/s00453-017-0390-580:6(1804-1833)Online publication date: 1-Jun-2018
https://dl.acm.org/doi/10.1007/s00453-017-0390-5
Cordasco GGargano LRescigno A(2015)Influence Propagation over Large Scale Social NetworksProceedings of the 2015 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining 201510.1145/2808797.2808888(1531-1538)Online publication date: 25-Aug-2015
https://dl.acm.org/doi/10.1145/2808797.2808888
Cordasco GGargano LMecchia MRescigno AVaccaro U(2015)A Fast and Effective Heuristic for Discovering Small Target Sets in Social NetworksProceedings of the 9th International Conference on Combinatorial Optimization and Applications - Volume 948610.1007/978-3-319-26626-8_15(193-208)Online publication date: 18-Dec-2015
https://dl.acm.org/doi/10.1007/978-3-319-26626-8_15
Cordasco GGargano LRescigno AVaccaro U(2015)Optimizing Spread of Influence in Social Networks via Partial IncentivesPost-Proceedings of the 22nd International Colloquium on Structural Information and Communication Complexity - Volume 943910.1007/978-3-319-25258-2_9(119-134)Online publication date: 14-Jul-2015
https://dl.acm.org/doi/10.1007/978-3-319-25258-2_9

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