Advertisement

Stable Sets of Threshold-Based Cascades on the Erdős-Rényi Random Graphs

  • Ching-Lueh Chang
  • Yuh-Dauh Lyuu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7056)

Abstract

Consider the following reversible cascade on the Erdős-Rényi random graph G(n,p). In round zero, a set of vertices, called the seeds, are active. For a given ρ ∈ ( 0,1 ], a non-isolated vertex is activated (resp., deactivated) in round t ∈ ℤ +  if the fraction f of its neighboring vertices that were active in round t − 1 satisfies f ≥ ρ (resp., f < ρ). An irreversible cascade is defined similarly except that active vertices cannot be deactivated. A set of vertices, S, is said to be stable if no vertex will ever change its state, from active to inactive or vice versa, once the set of active vertices equals S. For both the reversible and the irreversible cascades, we show that for any constant ε > 0, all p ∈ [ (1 + ε) (ln (e/ρ))/n,1 ] and with probability 1 − n − Ω(1), every stable set of G(n,p) has size O(⌈ρn⌉) or n − O(⌈ρn⌉).

Keywords

Random Graph Neighboring Vertex Discrete Apply Mathematic Active Vertex Simple Undirected Graph 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ackerman, E., Ben-Zwi, O., Wolfovitz, G.: Combinatorial model and bounds for target set selection. Theoretical Computer Science (forthcoming 2010), doi:10.1016/j.tcs.2010.08.021Google Scholar
  2. 2.
    Agur, Z.: Resilience and variability in pathogens and hosts. IMA Journal on Mathematical Medicine and Biology 4(4), 295–307 (1987)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Agur, Z.: Fixed points of majority rule cellular automata with application to plasticity and precision of the immune system. Complex Systems 5(3), 351–357 (1991)zbMATHGoogle Scholar
  4. 4.
    Agur, Z., Fraenkel, A.S., Klein, S.T.: The number of fixed points of the majority rule. Discrete Mathematics 70(3), 295–302 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Balogh, J., Bollobás, B., Morris, R.: Bootstrap percolation in high dimensions. Combinatorics, Probability and Computing 19(5-6), 643–692 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berger, E.: Dynamic monopolies of constant size. Journal of Combinatorial Theory Series B 83(2), 191–200 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Blume, L.E.: The statistical mechanics of strategic interaction. Games and Economic Behavior 5(3), 387–424 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bollobás, B.: Random Graphs, 2nd edn. Cambridge University Press (2001)Google Scholar
  9. 9.
    Chang, C.-L., Lyuu, Y.-D.: Spreading messages. Theoretical Computer Science 410(27-29), 2714–2724 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chang, C.-L., Lyuu, Y.-D.: Bounding the Number of Tolerable Faults in Majority-Based Systems. In: Calamoneri, T., Diaz, J. (eds.) CIAC 2010. LNCS, vol. 6078, pp. 109–119. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Chang, C.-L., Lyuu, Y.-D.: Spreading of messages in random graphs. Theory of Computing Systems 48(2), 389–401 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen, N.: On the approximability of influence in social networks. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1029–1037 (2008)Google Scholar
  13. 13.
    Dreyer, P.A., Roberts, F.S.: Irreversible k-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ellison, G.: Learning, local interaction, and coordination. Econometrica 61(5), 1047–1071 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Flocchini, P.: Contamination and decontamination in majority-based systems. Journal of Cellular Automata 4(3), 183–200 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Flocchini, P., Geurts, F., Santoro, N.: Optimal irreversible dynamos in chordal rings. Discrete Applied Mathematics 113(1), 23–42 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Flocchini, P., Královič, R., Ružička, 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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Flocchini, P., Lodi, E., Luccio, F., Pagli, L., Santoro, N.: Dynamic monopolies in tori. Discrete Applied Mathematics 137(2), 197–212 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ginosar, Y., Holzman, R.: The majority action on infinite graphs: Strings and puppets. Discrete Mathematics 215(1-3), 59–71 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Goles, E., Olivos, J.: Periodic behavior of generalized threshold functions. Discrete Mathematics 30(2), 187–189 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Goles-Chacc, E., Fogelman-Soulie, F., Pellegrin, D.: Decreasing energy functions as a tool for studying threshold networks. Discrete Applied Mathematics 12(3), 261–277 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Granville, A.: On a paper by Agur, Fraenkel and Klein. Discrete Mathematics 94(2), 147–151 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kempe, D., Kleinberg, J., Tardos, É.: Maximizing the spread of influence through a social network. In: Proceedings of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 137–146 (2003)Google Scholar
  24. 24.
    Kynčl, J., Lidický, B., Vyskočil, T.: Irreversible 2-conversion set is NP-complete. Technical Report KAM-DIMATIA Series 2009-933, Department of Applied Mathematics, Charles University, Prague, Czech Republic (2009)Google Scholar
  25. 25.
    Luccio, F.: Almost exact minimum feedback vertex set in meshes and butterflies. Information Processing Letters 66(2), 59–64 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Luccio, F., Pagli, L., Sanossian, H.: Irreversible dynamos in butterflies. In: Proceedings of the 6th International Colloquium on Structural Information and Communication Complexity, pp. 204–218 (1999)Google Scholar
  27. 27.
    Mitzenmacher, M., Upfal, E.: Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press (2005)Google Scholar
  28. 28.
    Montanari, A., Saberi, A.: Convergence to equilibrium in local interaction games. In: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, pp. 303–312 (2009)Google Scholar
  29. 29.
    Moran, G.: Parametrization for stationary patterns of the r-majority operators on 0-1 sequences. Discrete Mathematics 132(1-3), 175–195 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Moran, G.: The r-majority vote action on 0-1 sequences. Discrete Mathematics 132(1-3), 145–174 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Moran, G.: On the period-two property of the majority operator in infinite graphs. Transactions of the American Mathematical Society 347(5), 1649–1667 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Morris, S.: Contagion. Review of Economic Studies 67(1), 57–78 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press (1995)Google Scholar
  34. 34.
    Mustafa, N.H., Pekec, A.: Majority Consensus and the Local Majority Rule. In: Yu, Y., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 530–542. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  35. 35.
    Peleg, D.: Size bounds for dynamic monopolies. Discrete Applied Mathematics 86(2-3), 263–273 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Peleg, D.: Local majorities, coalitions and monopolies in graphs: A review. Theoretical Computer Science 282(2), 231–257 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Pike, D.A., Zou, Y.: Decycling Cartesian products of two cycles. SIAM Journal on Discrete Mathematics 19(3), 651–663 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Poljak, S., Sura, M.: On periodical behavior in societies with symmetric influences. Combinatorica 3(1), 119–121 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Poljak, S., Turzik, D.: On an application of convexity to discrete systems. Discrete Applied Mathematics 13(1), 27–32 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Reddy, T.V.T., Krishna, D.S., Rangan, C.P.: Variants of spreading messages. In: Proceedings of the 4th Workshop on Algorithms and Computation, pp. 240–251 (2010)Google Scholar
  41. 41.
    Stauffer, D., Aharony, A.: Introduction to Percolation Theory, 2nd edn. Taylor & Francis (1994)Google Scholar
  42. 42.
    West, D.B.: Introduction to Graph Theory, 3rd edn. Prentice-Hall, Upper Saddle River (2007)Google Scholar
  43. 43.
    Young, H.P.: The diffusion of innovations in social networks. In: Blume, L.E., Durlauf, S.N. (eds.) Economy as an Evolving Complex System. Proceedings Volume in the Santa Fe Institute Studies in the Sciences of Complexity, vol. 3, pp. 267–282. Oxford University Press, New York (2006)Google Scholar
  44. 44.
    Zollman, K.J.S.: Social structure and the effects of conformity. Humanities, Social Sciences and Law 172(3), 317–340 (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Ching-Lueh Chang
    • 1
  • Yuh-Dauh Lyuu
    • 2
    • 3
  1. 1.Department of Computer Science and EngineeringYuan Ze UniversityTaoyuanTaiwan
  2. 2.Department of Computer Science and Information EngineeringNational Taiwan UniversityTaipeiTaiwan
  3. 3.Department of FinanceNational Taiwan UniversityTaipeiTaiwan

Personalised recommendations