On the Non-progressive Spread of Influence through Social Networks

  • MohammadAmin Fazli
  • Mohammad Ghodsi
  • Jafar Habibi
  • Pooya Jalaly Khalilabadi
  • Vahab Mirrokni
  • Sina Sadeghian Sadeghabad
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7256)


The spread of influence in social networks is studied in two main categories: the progressive model and the non-progressive model (see e.g. the seminal work of Kempe, Kleinberg, and Tardos in KDD 2003). While the progressive models are suitable for modeling the spread of influence in monopolistic settings, non-progressive are more appropriate for modeling non-monopolistic settings, e.g., modeling diffusion of two competing technologies over a social network. Despite the extensive work on the progressive model, non-progressive models have not been studied well. In this paper, we study the spread of influence in the nonprogressive model under the strict majority threshold: given a graph G with a set of initially infected nodes, each node gets infected at time τ iff a majority of its neighbors are infected at time τ – 1. Our goal in the MinPTS problem is to find a minimum-cardinality initial set of infected nodes that would eventually converge to a steady state where all nodes of G are infected.

We prove that while the MinPTS is NP-hard for a restricted family of graphs, it admits an improved constant-factor approximation algorithm for power-law graphs. We do so by proving lower and upper bounds in terms of the minimum and maximum degree of nodes in the graph. The upper bound is achieved in turn by applying a natural greedy algorithm. Our experimental evaluation of the greedy algorithm also shows its superior performance compared to other algorithms for a set of realworld graphs as well as the random power-law graphs. Finally, we study the convergence properties of these algorithms and show that the nonprogressive model converges in at most O(|E(G)|) steps.


Social Network Bipartite Graph Greedy Algorithm Convergence Time Infected Node 
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.
    Ackerman, E., Ben-Zwi, O., Wolfovitz, G.: Combinatorial Model and Bounds for Target Set Selection. Theoretical Computer Science (2010)Google Scholar
  2. 2.
    Allan, R., Laskar, R.: On domination and independent domination numbers of a graph. Discrete Mathematics 23(2), 73–76 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ben-Zwi, O., Hermelin, D., Lokshtanov, D., Newman, I.: An exact almost optimal algorithm for target set selection in social networks. In: Proceedings of the Tenth ACM Conference on Electronic Commerce, pp. 355–362. ACM (2009)Google Scholar
  4. 4.
    Blume, L.: The statistical mechanics of strategic interaction. Games and Economic Behavior 5, 387–424 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brown, J., Reingen, P.: Social ties and word-of-mouth referral behavior. The Journal of Consumer Research 14(3), 350–362 (1987)CrossRefGoogle Scholar
  6. 6.
    Chang, C.: On reversible cascades in scale-free and Erdos Renyi random graphs. Arxiv preprint arXiv:1011.0653 (2010)Google Scholar
  7. 7.
    Chang, C., Lyuu, Y.: On irreversible dynamic monopolies in general graphs. Arxiv preprint arXiv:0904.2306 (2009)Google Scholar
  8. 8.
    Chang, C., Lyuu, Y.: Spreading messages. Theoretical Computer Science 410(27-29), 2714–2724 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Chen, N.: On the approximability of influence in social networks. In: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1029–1037. Society for Industrial and Applied Mathematics (2008)Google Scholar
  10. 10.
    Chen, W., Wang, Y., Yang, S.: Efficient influence maximization in social networks. In: Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 199–208. ACM (2009)Google Scholar
  11. 11.
    Clauset, A., Shalizi, C., Newman, M.: Power-law distributions in empirical data. SIAM Review 51(4), 661–703 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Dezső, Z., Barabási, A.: Halting viruses in scale-free networks. Physical Review E 65(5), 55103 (2002)CrossRefGoogle Scholar
  13. 13.
    Domingos, P., Richardson, M.: Mining the network value of customers. In: KDD 2001: Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, August 26-29, p. 57. Assn. for Computing Machinery, San Francisco (2001)CrossRefGoogle Scholar
  14. 14.
    Ellison, G.: Learning, local interaction, and coordination. Econometrica 61, 1047–1071 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Flocchini, P., Geurts, F., Santoro, N.: Optimal irreversible dynamos in chordal rings. Discrete Applied Mathematics 113(1), 23–42 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Flocchini, P., Královi, R., Ruika, P., Roncato, A., Santoro, N.: On time versus size for monotone dynamic monopolies in regular topologies. Journal of Discrete Algorithms 1(2), 129–150 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Flocchini, P., Lodi, E., Luccio, F., Pagli, L., Santoro, N.: Dynamic monopolies in tori. Discrete Applied Mathematics 137(2), 197–212 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Freeman, L.: The development of social network analysis. Empirical Press, Vancouver (2004)Google Scholar
  19. 19.
    Goyal, A., Bonchi, F., Lakshmanan, L., Balcan, M., Harvey, N., Lapus, R., Simon, F., Tittmann, P., Ben-Shimon, S., Ferber, A., et al.: Approximation Analysis of Influence Spread in Social Networks. Arxiv preprint arXiv:1008.2005 (2010)Google Scholar
  20. 20.
    Immorlica, N., Kleinberg, J., Mahdian, M., Wexler, T.: The role of compatibility in the diffusion of technologies through social networks. In: Proceedings of the 8th ACM Conference on Electronic Commerce, EC 2007, pp. 75–83. ACM, New York (2007)CrossRefGoogle Scholar
  21. 21.
    Ivic, A.: Riemann zeta-function. John Wiley & Sons, Inc., One Wiley Drive, Somerset, NJ 08873 (USA), 340 (1985)Google Scholar
  22. 22.
    Jackson, M., Yariv, L.: Diffusion on social networks. Economie Publique 16, 69–82 (2005)Google Scholar
  23. 23.
    Kempe, D., Kleinberg, J., Tardos, É.: Maximizing the spread of influence through a social network. In: Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 137–146. ACM (2003)Google Scholar
  24. 24.
    Luccio, F., Pagli, L., Sanossian, H.: Irreversible dynamos in butterflies. In: Proc. of 6th Colloquium on Structural Information and Communication Complexity, pp. 204–218. Citeseer (1999)Google Scholar
  25. 25.
    Mossel, E., Roch, S.: On the submodularity of influence in social networks. In: Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, pp. 128–134. ACM (2007)Google Scholar
  26. 26.
    Mossel, E., Schoenebeck, G.: Reaching consensus on social networks. In: Innovations in Computer Science, ICS (2009)Google Scholar
  27. 27.
    Pastor-Satorras, R., Vespignani, A.: Epidemic spreading in scale-free networks. Physical Review Letters 86(14), 3200–3203 (2001)CrossRefGoogle Scholar
  28. 28.
    Peleg, D.: Local majorities, coalitions and monopolies in graphs: a review. Theoretical Computer Science 282(2), 231–257 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Pike, D., Zou, Y.: Decycling Cartesian products of two cycles. SIAM Journal on Discrete Mathematics 19, 651 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Richardson, M., Domingos, P.: Mining knowledge-sharing sites for viral marketing. In: Proceedings of the Eighth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 61–70. ACM (2002)Google Scholar
  31. 31.
    Tang, J., Sun, J., Wang, C., Yang, Z.: Social influence analysis in large-scale networks. In: Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 807–816. ACM (2009)Google Scholar
  32. 32.
    Wilson, D.: Levels of selection: An alternative to individualism in biology and the human sciences. Social Networks 11(3), 257–272 (1989)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • MohammadAmin Fazli
    • 1
  • Mohammad Ghodsi
    • 1
    • 3
  • Jafar Habibi
    • 1
  • Pooya Jalaly Khalilabadi
    • 1
  • Vahab Mirrokni
    • 2
  • Sina Sadeghian Sadeghabad
    • 1
  1. 1.Computer Engineering DepartmentSharif University of TechnologyTehranIran
  2. 2.Google Research NYCNewYorkUSA
  3. 3.Institute for Research in Fundamental Sciences (IPM)TehranIran

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