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On Reversible Cascades in Scale-Free and Erdős-Rényi Random Graphs

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Abstract

Consider the following cascading process on a simple undirected graph G(V,E) with diameter Δ. In round zero, a set S⊆V of vertices, called the seeds, are active. In round i+1, i∈ℕ, a non-isolated vertex is activated if at least a ρ∈(0,1] fraction of its neighbors are active in round i; it is deactivated otherwise. For k∈ℕ, let min-seed(k)(G,ρ) be the minimum number of seeds needed to activate all vertices in or before round k. This paper derives upper bounds on min-seed(k)(G,ρ). In particular, if G is connected and there exist constants C>0 and γ>2 such that the fraction of degree-k vertices in G is at most C/k γ for all k∈ℤ+, then min-seed(Δ)(G,ρ)=O(⌈ρ γ−1|V|⌉). Furthermore, for n∈ℤ+, p=Ω((ln(e/ρ))/(ρn)) and with probability 1−exp(−n Ω(1)) over the Erdős-Rényi random graphs G(n,p), min-seed(1)(G(n,p),ρ)=O(ρn).

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References

  1. Ackerman, E., Ben-Zwi, O., Wolfovitz, G.: Combinatorial model and bounds for target set selection. Theor. Comput. Sci. 411(44–46), 4017–4022 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adams, S.S., Brass, Z., Stokes, C., Troxell, D.S.: Irreversible k-threshold and majority conversion processes on complete multipartite graphs and graph products. Technical Report arXiv:1102.5361, Cornell University (2011)

  3. Adams, S.S., Troxell, D.S., Zinnen, S.L.: Dynamic monopolies and feedback vertex sets in hexagonal grids. Comput. Math. Appl. 62(11), 4049–4057 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Balogh, J., Pete, G.: Random disease on the square grid. Random Struct. Algorithms 13(3–4), 409–422 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berger, E.: Dynamic monopolies of constant size. J. Comb. Theory, Ser. B 83(2), 191–200 (2001)

    Article  MATH  Google Scholar 

  6. Bermond, J.-C., Bond, J., Peleg, D., Perennes, S.: The power of small coalitions in graphs. Discrete Appl. Math. 127(3), 399–414 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Blume, L.E.: The statistical mechanics of strategic interaction. Games Econ. Behav. 5(3), 387–424 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bollobás, B.: Random Graphs, 2nd edn. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  9. Bollobás, B., Riordan, O.: The diameter of a scale-free random graph. Combinatorica 24(1), 5–34 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bonato, A.: A survey of models of the Web graph. In: López-Ortiz, A., Hamel, A. (eds.) Proceedings of the 1st Workshop on Combinatorial and Algorithmic Aspects of Networking, pp. 159–172. Springer, Berlin, Heidelberg (2004)

    Google Scholar 

  11. Chang, C.-L., Lyuu, Y.-D.: Spreading messages. Theor. Comput. Sci. 410(27–29), 2714–2724 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chang, C.-L., Lyuu, Y.-D.: Bounding the number of tolerable faults in majority-based systems. In: Calamoneri, T., Díaz, J. (eds.) Proceedings of the 7th International Conference on Algorithms and Complexity, Rome, Italy, pp. 109–119. Springer, Berlin, Heidelberg (2010)

    Google Scholar 

  13. Chang, C.-L., Lyuu, Y.-D.: Spreading of messages in random graphs. Theory Comput. Syst. 48(2), 389–401 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chang, C.-L., Lyuu, Y.-D.: Stable sets of threshold-based cascades on the Erdős-Rényi random graphs. In: Proceedings of the 22nd International Workshop on Combinatorial Algorithms, pp. 96–105 (2011)

    Google Scholar 

  15. Chung, F., Lu, L.: The average distance in a random graph with given expected degrees. Internet Math. 1(1), 91–114 (2004)

    Article  MathSciNet  Google Scholar 

  16. Chung, F., Lu, L.: The small world phenomenon in hybrid power law graphs. In: Ben-Naim, E., Frauenfelder, H., Toroczkai, Z. (eds.) Complex Networks, pp. 89–104. Springer, Berlin, Heidelberg (2004)

    Chapter  Google Scholar 

  17. Cohen, R., Havlin, S.: Scale-free networks are ultrasmall. Phys. Rev. Lett. 90(5), 058701 (2003)

    Article  Google Scholar 

  18. Donato, D., Laura, L., Leonardi, S., Millozzi, S.: Simulating the Webgraph: A comparative analysis of models. Comput. Sci. Eng. 6(6), 84–89 (2004)

    Article  Google Scholar 

  19. Dreyer, P.A., Roberts, F.S.: Irreversible k-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Appl. Math. 157(7), 1615–1627 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ellison, G.: Learning, local interaction, and coordination. Econometrica 61(5), 1047–1071 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  21. Flocchini, P., Geurts, F., Santoro, N.: Optimal irreversible dynamos in chordal rings. Discrete Appl. Math. 113(1), 23–42 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Flocchini, P., Královič, R., Ružička, P., Roncato, A., Santoro, N.: On time versus size for monotone dynamic monopolies in regular topologies. J. Discrete Algorithms 1(2), 129–150 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Flocchini, P., Lodi, E., Luccio, F., Pagli, L., Santoro, N.: Dynamic monopolies in tori. Discrete Appl. Math. 137(2), 197–212 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Granovetter, M.: Threshold models of collective behavior. Am. J. Sociol. 83(6), 1420–1443 (1978)

    Article  Google Scholar 

  25. Kleinberg, J.: Cascading behavior in networks: Algorithmic and economic issues. In: Nisan, N., Roughgarden, T., Tardos, É., Vazirani, V. (eds.) Algorithmic Game Theory. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  26. 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)

  27. Linial, N., Peleg, D., Rabinovich, Y., Saks, M.: Sphere packing and local majorities in graphs. In: Proceedings of the 2nd Israel Symposium on Theory of Computing Systems, pp. 141–149 (1993)

    Chapter  Google Scholar 

  28. Luccio, F.: Almost exact minimum feedback vertex set in meshes and butterflies. Inf. Process. Lett. 66(2), 59–64 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  29. Luccio, F., Pagli, L., Sanossian, H.: Irreversible dynamos in butterflies. In: Proceedings of the 6th International Colloquium on Structural Information & Communication Complexity, pp. 204–218 (1999)

    Google Scholar 

  30. 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)

    Chapter  Google Scholar 

  31. Morris, S.: Contagion. Rev. Econ. Stud. 67(1), 57–78 (2000)

    Article  MATH  Google Scholar 

  32. Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  33. Peleg, D.: Graph immunity against local influence. Technical Report CS96-11, Weizmann Science Press of Israel (1996)

  34. Peleg, D.: Size bounds for dynamic monopolies. Discrete Appl. Math. 86(2–3), 263–273 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Peleg, D.: Local majorities, coalitions and monopolies in graphs: A review. Theor. Comput. Sci. 282(2), 231–257 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Pike, D.A., Zou, Y.: Decycling Cartesian products of two cycles. SIAM J. Discrete Math. 19(3), 651–663 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Watts, D.J.: A simple model of global cascades on random networks. Proc. Natl. Acad. Sci. 99(9), 5766–5771 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice-Hall, New York (2001)

    Google Scholar 

  39. 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 

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Yuan Ze University, Taoyuan, Taiwan

    Ching-Lueh Chang & Chao-Hong Wang

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  1. Ching-Lueh Chang
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Correspondence to Ching-Lueh Chang.

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The authors are supported in part by the National Science Council of Taiwan under grant 99-2218-E-155-014-MY2.

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Chang, CL., Wang, CH. On Reversible Cascades in Scale-Free and Erdős-Rényi Random Graphs. Theory Comput Syst 52, 303–318 (2013). https://doi.org/10.1007/s00224-012-9387-2

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