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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Balister, P., Bollobás, B., Johnson, J.R., Walters, M.: Random majority percolation. Random Struct. Algorithms 36(3), 315–340 (2010)
Balogh, J., Bollobás, B., Morris, R.: Bootstrap percolation in high dimensions. Comb. Probab. Comput. 19(5–6), 643–692 (2010)
Balogh, J., Pete, G.: Random disease on the square grid. Random Struct. Algorithms 13(3–4), 409–422 (1998)
Balogh, J., Pittel, B.G.: Bootstrap percolation on the random regular graph. Random Struct. Algorithms 30(1–2), 257–286 (2007)
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)
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)
Feige, U., Krivelevich, M., Reichman, D., et al.: Contagious sets in random graphs. Ann. Appl. Probab. 27(5), 2675–2697 (2017)
Flocchini, P., Lodi, E., Luccio, F., Pagli, L., Santoro, N.: Dynamic monopolies in tori. Discret. Appl. Math. 137(2), 197–212 (2004)
Freund, D., Poloczek, M., Reichman, D.: Contagious sets in dense graphs. Eur. J. Comb. 68, 66–78 (2018)
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_29
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_42
Gunderson, K.: Minimum degree conditions for small percolating sets in bootstrap percolation. arXiv preprint arXiv:1703.10741 (2017)
Jeger, C., Zehmakan, A.N.: Dynamic monopolies in reversible bootstrap percolation. arXiv preprint arXiv:1805.07392 (2018)
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)
Peleg, D.: Size bounds for dynamic monopolies. Discret. Appl. Math. 86(2–3), 263–273 (1998)
Reichman, D.: New bounds for contagious sets. Discret. Math. 312(10), 1812–1814 (2012)
Zehmakan, A.N.: Opinion forming in binomial random graph and expanders. arXiv preprint arXiv:1805.12172 (2018)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Zehmakan, A.N. (2019). Tight Bounds on the Minimum Size of a Dynamic Monopoly. In: Martín-Vide, C., Okhotin, A., Shapira, D. (eds) Language and Automata Theory and Applications. LATA 2019. Lecture Notes in Computer Science(), vol 11417. Springer, Cham. https://doi.org/10.1007/978-3-030-13435-8_28
Download citation
DOI: https://doi.org/10.1007/978-3-030-13435-8_28
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-13434-1
Online ISBN: 978-3-030-13435-8
eBook Packages: Computer ScienceComputer Science (R0)