Competitive Diffusion on Weighted Graphs

  • Takehiro Ito
  • Yota Otachi
  • Toshiki Saitoh
  • Hisayuki Satoh
  • Akira Suzuki
  • Kei UchizawaEmail author
  • Ryuhei Uehara
  • Katsuhisa Yamanaka
  • Xiao Zhou
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)


Consider an undirected and vertex-weighted graph modeling a 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.


Nash Equilibrium Graph Class Pure Nash Equilibrium Weighted Path Information Processing Letter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Alon, N., Feldman, M., Procaccia, A.D., Tennenholtz, M.: A note on competitive diffusion through social networks. Information Processing Letters 110(6), 221–225 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Apt, K.R., Markakis, E.: Social networks with competing products. Fundamenta Informaticae 129(3), 225–250 (2014)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bharathi, S., Kempe, D., Salek, M.: Competitive influence maximization in social networks. In: Deng, X., Graham, F.C. (eds.) WINE 2007. LNCS, vol. 4858, pp. 306–311. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  4. 4.
    Borodin, A., Braverman, M., Lucier, B., Oren, J.: Strategyproof mechanisms for competitive influence in networks. In: Proc. of the 22nd International Conference on World Wide Web, pp. 141–151 (2013)Google Scholar
  5. 5.
    Borodin, A., Filmus, Y., Oren, J.: Threshold models for competitive influence in social networks. In: Saberi, A. (ed.) WINE 2010. LNCS, vol. 6484, pp. 539–550. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  6. 6.
    Bulteau, L., Froese, V., Talmon, N.: Multi-Player diffusion games on graph classes. In: Proc. of the 12th Annual Conference on Theory and Applications of Models of Computation (to appear)Google Scholar
  7. 7.
    Clark, A., Poovendran, R.: Maximizing influence in competitive environments: a game-theoretic approach. In: Proc. of the Second International Conference on Decision and Game Theory for Security, pp. 151–162Google Scholar
  8. 8.
    Cooper, C., Uehara, R.: Scale Free Properties of Random \(k\)-Trees. Mathematics in Computer Science 3(4), 489–496 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Domingos, P., Richardson, M.: Mining the network value of customers. In: Proc. of the 7th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 57–66 (2001)Google Scholar
  10. 10.
    Draief, M., Heidari, H., Kearns, M.: New models for competitive contagion. In: Proc. of the 28th AAAI Conference on Artificial Intelligence, pp. 637–644 (2014)Google Scholar
  11. 11.
    Dürr, C., Thang, N.K.: Nash equilibria in Voronoi games on graphs. In: Proc. of the 15th Annual European Symposium on Algorithms, pp. 17–28 (2007)Google Scholar
  12. 12.
    Etesami, S.R., Basar, T.: Complexity of equilibrium in diffusion games on social networks. 1403.3881
  13. 13.
    Feldmann, R., Mavronicolas, M., Monien, B.: Nash equilibria for voronoi games on transitive graphs. In: Leonardi, S. (ed.) WINE 2009. LNCS, vol. 5929, pp. 280–291. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  14. 14.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series (2006)Google Scholar
  15. 15.
    Goyal, S., Heidari, H., Kearns, M.: Competitive contagion in networks. Games and Economic Behavior (to appear)Google Scholar
  16. 16.
    He, X., Kempe, D.: Price of anarchy for the N-player competitive cascade game with submodular activation functions. In: Proc. of the 9th International Workshop on Internet and Network Economics, pp. 232–248 (2013)Google Scholar
  17. 17.
    Heggernes, P., Kratsch, D.: Linear-time certifying recognition algorithms and forbidden induced subgraphs. Nordic Journal of Computing 14, 87–108 (2008)MathSciNetGoogle Scholar
  18. 18.
    Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer-Verlag (2004)Google Scholar
  19. 19.
    Kempe, D., Kleinberg, J., Tardos, E.: Maximizing the spread of influence through a social network. In: Proc. of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 137–146 (2003)Google Scholar
  20. 20.
    Kempe, D., Kleinberg, J.M., Tardos, É.: Influential nodes in a diffusion model for social networks. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1127–1138. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  21. 21.
    Mavronicolas, M., Monien, B., Papadopoulou, V.G., Schoppmann, F.: Voronoi games on cycle graphs. In: Proc. of the 33rd International Symposium on Mathematical Foundations of Computer Science, pp. 503–514 (2008)Google Scholar
  22. 22.
    Richardson, M., Domingos, P.: Mining knowledge-sharing sites for viral marketing. In: Proc. of the 8th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 61–70 (2002)Google Scholar
  23. 23.
    Small, L., Mason, O.: Nash Equilibria for competitive information diffusion on trees. Information Processing Letters 113(7), 217–219 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Takehara, R., Hachimori, M., Shigeno, M.: A comment on pure-strategy Nash equilibria in competitive diffusion games. Information Processing Letters 112(3), 59–60 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Teramoto, S., Demaine, E.D., Uehara, R.: The Voronoi game on graphs and its complexity. Journal of Graph Algorithms and Applications 15(4), 485–501 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Takehiro Ito
    • 1
    • 2
  • Yota Otachi
    • 3
  • Toshiki Saitoh
    • 2
    • 4
  • Hisayuki Satoh
    • 1
  • Akira Suzuki
    • 1
    • 2
  • Kei Uchizawa
    • 5
    Email author
  • Ryuhei Uehara
    • 3
  • Katsuhisa Yamanaka
    • 6
  • Xiao Zhou
    • 1
  1. 1.Tohoku UniversitySendaiJapan
  2. 2.CREST, JSTSaitamaJapan
  3. 3.Japan Advanced Institute of Science and TechnologyNomiJapan
  4. 4.Kobe UniversityKobeJapan
  5. 5.Yamagata UniversityYonezawaJapan
  6. 6.Iwate UniversityMoriokaJapan

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