Advertisement

Tight Bounds on the Minimum Size of a Dynamic Monopoly

  • Ahad N. ZehmakanEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11417)

Abstract

Assume that you are given a graph \(G=(V,E)\) with an initial coloring, where each node is black or white. Then, in discrete-time rounds all nodes simultaneously update their color following a predefined deterministic rule. This process is called two-way r-bootstrap percolation, for some integer r, if a node with at least r black neighbors gets black and white otherwise. Similarly, in two-way \(\alpha \)-bootstrap percolation, for some \(0<\alpha <1\), a node gets black if at least \(\alpha \) fraction of its neighbors are black, and white otherwise. The two aforementioned processes are called respectively r-bootstrap and \(\alpha \)-bootstrap percolation if we require that a black node stays black forever.

For each of these processes, we say a node set D is a dynamic monopoly whenever the following holds: If all nodes in D are black then the graph gets fully black eventually. We provide tight upper and lower bounds on the minimum size of a dynamic monopoly.

Keywords

Dynamic monopoly Bootstrap percolation Threshold model Percolating set Target set selection 

References

  1. 1.
    Balister, P., Bollobás, B., Johnson, J.R., Walters, M.: Random majority percolation. Random Struct. Algorithms 36(3), 315–340 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Balogh, J., Bollobás, B., Morris, R.: Bootstrap percolation in high dimensions. Comb. Probab. Comput. 19(5–6), 643–692 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Balogh, J., Pete, G.: Random disease on the square grid. Random Struct. Algorithms 13(3–4), 409–422 (1998)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Balogh, J., Pittel, B.G.: Bootstrap percolation on the random regular graph. Random Struct. Algorithms 30(1–2), 257–286 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chang, C.L., Lyuu, Y.D.: Triggering cascades on strongly connected directed graphs. In: 2012 Fifth International Symposium on Parallel Architectures, Algorithms and Programming (PAAP), pp. 95–99. IEEE (2012)Google Scholar
  6. 6.
    Coja-Oghlan, A., Feige, U., Krivelevich, M., Reichman, D.: Contagious sets in expanders. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1953–1987. Society for Industrial and Applied Mathematics (2015)Google Scholar
  7. 7.
    Feige, U., Krivelevich, M., Reichman, D., et al.: Contagious sets in random graphs. Ann. Appl. Probab. 27(5), 2675–2697 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Flocchini, P., Lodi, E., Luccio, F., Pagli, L., Santoro, N.: Dynamic monopolies in tori. Discret. Appl. Math. 137(2), 197–212 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Freund, D., Poloczek, M., Reichman, D.: Contagious sets in dense graphs. Eur. J. Comb. 68, 66–78 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gärtner, B., N. Zehmakan, A.: 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_29CrossRefzbMATHGoogle Scholar
  11. 11.
    Gärtner, B., Zehmakan, A.N.: Majority model on random regular graphs. In: Bender, M.A., Farach-Colton, M., Mosteiro, M.A. (eds.) LATIN 2018. LNCS, vol. 10807, pp. 572–583. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-77404-6_42CrossRefGoogle Scholar
  12. 12.
    Gunderson, K.: Minimum degree conditions for small percolating sets in bootstrap percolation. arXiv preprint arXiv:1703.10741 (2017)
  13. 13.
    Jeger, C., Zehmakan, A.N.: Dynamic monopolies in reversible bootstrap percolation. arXiv preprint arXiv:1805.07392 (2018)
  14. 14.
    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
  15. 15.
    Peleg, D.: Size bounds for dynamic monopolies. Discret. Appl. Math. 86(2–3), 263–273 (1998)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Reichman, D.: New bounds for contagious sets. Discret. Math. 312(10), 1812–1814 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zehmakan, A.N.: Opinion forming in binomial random graph and expanders. arXiv preprint arXiv:1805.12172 (2018)

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Computer ScienceETH ZürichZürichSwitzerland

Personalised recommendations