TY - JOUR
T1 - Contagious sets in random graphs
AU - Feige, Uriel
AU - Krivelevich, Michael
AU - Reichman, Daniel
N1 - Publisher Copyright:
© 2017 Institute of Mathematical Statistics.
PY - 2017/10
Y1 - 2017/10
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. 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 Assuming r > 1 to be a constant that does not depend on n, we prove that with high probability. We also show that the threshold probability for m(G,r) = r to hold.
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. 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 Assuming r > 1 to be a constant that does not depend on n, we prove that with high probability. We also show that the threshold probability for m(G,r) = r to hold.
KW - Bootstrap percolation
KW - Minimal contagious set
KW - Random graphs
UR - https://www.scopus.com/pages/publications/85033675598
U2 - 10.1214/16-AAP1254
DO - 10.1214/16-AAP1254
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AN - SCOPUS:85033675598
SN - 1050-5164
VL - 27
SP - 2675
EP - 2697
JO - Annals of Applied Probability
JF - Annals of Applied Probability
IS - 5
ER -