Contagious Sets in Random Graphs
نویسندگان
چکیده
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 2 active neighbors. A contagious set is a set whose activation results with the entire graph being active. Given a graph G, let m(G, 2) be the minimal size of a contagious set. We consider the binomial random graph G := G(n, p) with p := d n and w(n) < d < n 1 2−ε, where ε > 0 is an arbitrary constant and w(n) is an arbitrary function tending to infinity with n. We prove that m(G, 2) = Θ ( n d2 log d ) with high probability. ∗Faculty of Computer Science and Applied Mathematics, the Weizmann Institute, Rehovot, 76100, Israel. [email protected]. Work supported in part by the Israel Science Foundation (grant No. 621/12), by the I-CORE Program of the Planning and Budgeting Committee and The Israel Science Foundation (grant No. 4/11), and by the Citi foundation. †School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 6997801, Israel. [email protected] . Research supported in part by: USA-Israel BSF Grant 2010115 and by grant 912/12 from the Israel Science Foundation. ‡Faculty of Computer Science and Applied Mathematics, the Weizmann Institute, Rehovot, 76100, Israel. [email protected]. Supported in part by The Israel Science Foundation (grant No. 621/12).
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