Journal Article Competitive Diffusion on Weighted Graphs

Ito, Takehiro  ,  Otachi, Yota  ,  Saitoh, Toshiki  ,  Satoh, Hisayuki  ,  Suzuki, Akira  ,  Uchizawa, Kei  ,  Uehara, Ryuhei  ,  Yamanaka, Katsuhisa  ,  Zhou, Xiao

9214pp.422 - 433 , 2015-08-05 , Springer
Consider an undirected and vertex-weighted graph modelinga social network, where the vertices represent individuals, the edges do connections among them, and weights do levels of importance of individuals. In the competitive diffusion game, each of a number of players chooses a vertex as a seed to propagate his/her idea which spreads along the edges in the graph. The objective of every player is to maximize the sum of weights of vertices infected by his/her idea. In this paper, we study a computational problem of asking whether a pure Nash equilibrium exists in a given graph, and present several negative and positive results with regard to graph classes. We first prove that the problem is W[1]-hard when parameterized by the number of players even for unweighted graphs. We also show that the problem is NP-hard even for series-parallel graphs with positive integer weights, and is NP-hard even for forests with arbitrary integer weights. Furthermore, we show that the problem for forests of paths with arbitrary weights is solvable in pseudo-polynomial time; and it is solvable in quadratic time if a given graph is unweighted. We also prove that the problem is solvable in polynomial time for chain graphs, cochain graphs, and threshold graphs with arbitrary integer weights.
Algorithms and Data Structures, 14th International Symposium, WADS 2015, Victoria, BC, Canada, August 5-7, 2015. Proceedings

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