New bounds for contagious sets

We consider the following activation process: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least k active neighbors (k is identical for all vertices of the graph). Our goal is to find a set of minimum size whose activation will result in the entire graph being activated. Call such a set contagious. We give new upper bounds for the size of contagious sets in terms of the degree sequence of the graph. In particular, we prove that if G=(V,E) is an undirected graph then the size of a contagious set is bounded by @?"v"@?"Vmin{1,kd(v)+1} (where d(v) is the degree of v). Our proof techniques lead to a new proof for a known theorem regarding induced k-degenerate subgraphs in arbitrary graphs.

[1]  J. Kleinberg.  Algorithmic Game Theory: Cascading Behavior in Networks: Algorithmic and Economic Issues , 2007 .

[2]  J. Balogh,et al.  Random disease on the square grid , 1998 .

[3]  H. Duminil-Copin,et al.  The sharp threshold for bootstrap percolation in all dimensions , 2010, 1010.3326.

[4]  B. Bollob'as,et al.  Bootstrap percolation in three dimensions , 2008, 0806.4485.

[5]  Jacob Goldenberg,et al.  Using Complex Systems Analysis to Advance Marketing Theory Development , 2001 .

[6]  Béla Bollobás,et al.  Majority Bootstrap Percolation on the Hypercube , 2007, Combinatorics, Probability and Computing.

[7]  Béla Bollobás,et al.  Bootstrap percolation on the hypercube , 2006 .

[8]  Ning Chen,et al.  On the approximability of influence in social networks , 2008, SODA '08.

[9]  P. Leath,et al.  Bootstrap percolation on a Bethe lattice , 1979 .

[10]  Yuval Peres,et al.  Bootstrap Percolation on Infinite Trees and Non-Amenable Groups , 2003, Combinatorics, Probability and Computing.

[11]  Svante Janson,et al.  Majority bootstrap percolation on the random graph G(n,p) , 2010, 1012.3535.

[12]  Jon Kleinberg,et al.  Maximizing the spread of influence through a social network , 2003, KDD '03.

[13]  József Balogh,et al.  Bootstrap percolation on the random regular graph , 2007, Random Struct. Algorithms.

[14]  Noga Alon,et al.  Large induced degenerate subgraphs , 1987, Graphs Comb..

[15]  Jerrold R. Griggs.  Lower bounds on the independence number in terms of the degrees , 1983, J. Comb. Theory, Ser. B.

[16]  Eyal Ackerman,et al.  Combinatorial model and bounds for target set selection , 2010, Theor. Comput. Sci..

[17]  Svante Janson,et al.  On percolation in random graphs with given vertex degrees , 2008, 0804.1656.

[18]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.