Advertisement

Majority Model on Random Regular Graphs

  • Bernd Gärtner
  • Ahad N. Zehmakan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

Abstract

Consider a graph \(G=(V,E)\) and an initial random coloring where each vertex \(v \in V\) is blue with probability \(P_b\) and red otherwise, independently from all other vertices. In each round, all vertices simultaneously switch their color to the most frequent color in their neighborhood and in case of a tie, a vertex keeps its current color. The main goal of the present paper is to analyze the behavior of this basic and natural process on the random d-regular graph \(\mathbb {G}_{n,d}\). It is shown that for \(\epsilon >0\), \(P_b \le 1/2-\epsilon \) results in final complete occupancy by red in \(\mathcal {O}(\log _d\log n)\) rounds with high probability, provided that \(d\ge c/\epsilon ^2\) for a sufficiently large constant c. We argue that the bound \(\mathcal {O}(\log _d\log n)\) is asymptomatically tight. Furthermore, we show that with high probability, \(\mathbb {G}_{n,d}\) is immune; i.e., the smallest dynamic monopoly is of linear size. A dynamic monopoly is a subset of vertices that can “take over” in the sense that a commonly chosen initial color eventually spreads throughout the whole graph, irrespective of the colors of other vertices. This answers an open question of Peleg [22].

Keywords

Majority model Random regular graph Bootstrap percolation Density classification Threshold behavior Dynamic monopoly 

Notes

Acknowledgments

The authors would like to thank Mohsen Ghaffari for several stimulating conversations and Jozsef Balogh and Nick Wormald for referring to some relevant prior results.

References

  1. 1.
    Aizenman, M., Lebowitz, J.L.: Metastability effects in bootstrap percolation. J. Phys. A: Math. Gen. 21(19), 3801 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amini, H., Draief, M., Lelarge, M.: Flooding in weighted sparse random graphs. SIAM J. Discrete Math. 27(1), 1–26 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Balogh, J., Bollobás, B., Morris, R.: Majority bootstrap percolation on the hypercube. Comb. Probab. Comput. 18(1–2), 17–51 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Balogh, J., Pittel, B.G.: Bootstrap percolation on the random regular graph. Random Struct. Algorithms 30(1–2), 257–286 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bender, E.A., Canfield, E.R.: The asymptotic number of labeled graphs with given degree sequences. J. Comb. Theory Ser. A 24(3), 296–307 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berger, E.: Dynamic monopolies of constant size. J. Comb. Theory Ser. B 83(2), 191–200 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bollobás, B., Fernandez de la Vega, W.: The diameter of random regular graphs. Combinatorica 2(2), 125–134 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    de Oliveira, M.J.: Isotropic majority-vote model on a square lattice. J. Stat. Phys. 66(1), 273–281 (1992)CrossRefzbMATHGoogle Scholar
  9. 9.
    Feller, W.: An Introduction to Probability Theory and Its Applications: Volume I, vol. 3. Wiley, New York (1968)zbMATHGoogle Scholar
  10. 10.
    Flocchini, P., Lodi, E., Luccio, F., Pagli, L., Santoro, N.: Dynamic monopolies in tori. Discrete Appl. Math. 137(2), 197–212 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fountoulakis, N., Panagiotou, K.: Rumor spreading on random regular graphs and expanders. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX/RANDOM-2010. LNCS, vol. 6302, pp. 560–573. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-15369-3_42 CrossRefGoogle Scholar
  12. 12.
    Frischknecht, S., Keller, B., Wattenhofer, R.: Convergence in (social) influence networks. In: Afek, Y. (ed.) DISC 2013. LNCS, vol. 8205, pp. 433–446. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-41527-2_30 CrossRefGoogle Scholar
  13. 13.
    Gärtner, B., Zehmakan, A.N.: (Biased) majority rule cellular automata. arXiv preprint arXiv:1711.10920 (2017)
  14. 14.
    Gärtner, B., Zehmakan, A.N.: Color war: cellular automata with majority-rule. In: Drewes, F., Martín-Vide, C., Truthe, B. (eds.) LATA 2017. LNCS, vol. 10168, pp. 393–404. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-53733-7_29 CrossRefGoogle Scholar
  15. 15.
    Goles, E., Olivos, J.: Comportement périodique des fonctions à seuil binaires et applications. Discrete Appl. Math. 3(2), 93–105 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Janson, S., Luczak, T., Rucinski, A.: Random Graphs, vol. 45. Wiley, Hoboken (2011)zbMATHGoogle Scholar
  17. 17.
    Kaaser, D., Mallmann-Trenn, F., Natale, E.: On the voting time of the deterministic majority process. arXiv preprint arXiv:1508.03519 (2015)
  18. 18.
    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
  19. 19.
    Land, M., Belew, R.K.: No perfect two-state cellular automata for density classification exists. Phys. Rev. Lett. 74(25), 5148 (1995)CrossRefGoogle Scholar
  20. 20.
    Mourrat, J.-C., Valesin, D., et al.: Phase transition of the contact process on random regular graphs. Electron. J. Probab. 21 (2016)Google Scholar
  21. 21.
    Peleg, D.: Local majority voting, small coalitions and controlling monopolies in graphs: a review. In: Proceedings of 3rd Colloquium on Structural Information and Communication Complexity, pp. 152–169 (1997)Google Scholar
  22. 22.
    Peleg, D.: Immunity against local influence. In: Dershowitz, N., Nissan, E. (eds.) Language, Culture, Computation. Computing - Theory and Technology. LNCS, vol. 8001, pp. 168–179. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-642-45321-2_8 CrossRefGoogle Scholar
  23. 23.
    Poljak, S., Turzík, D.: On pre-periods of discrete influence systems. Discrete Appl. Math. 13(1), 33–39 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Schonmann, R.H.: Finite size scaling behavior of a biased majority rule cellular automaton. Phys. A: Stat. Mech. Appl. 167(3), 619–627 (1990)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Spitzer, F.: Interaction of Markov processes. Adv. Math. 5(2), 246–290 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Stefánsson, S.Ö., Vallier, T.: Majority bootstrap percolation on the random graph G(n, p). arXiv preprint arXiv:1503.07029 (2015)

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceETH ZurichZürichSwitzerland

Personalised recommendations