Color War: Cellular Automata with Majority-Rule

  • Bernd Gärtner
  • Ahad N. ZehmakanEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10168)


Consider a graph \(G=(V,E)\) and a random initial vertex-coloring such that each vertex is blue independently with probability \(p_{b}\le 1/2\), and red otherwise. In each step, all vertices change their current color synchronously to the most frequent color in their neighborhood (in the case of a tie, a vertex conserves its current color). We are interested in the behavior of this very natural process, especially in 2-dimensional grids and tori (cellular automata with majority rule). In the present paper, as a main result we prove that a grid \(G_{n,n}\) or a torus \(T_{n,n}\) with 4-neighborhood (8-neighborhood) exhibits a threshold behavior: with high probability, it reaches a red monochromatic configuration in a constant number of steps if \(p_b\ll n^{-\frac{1}{2}}\) (\(p_b\ll n^{-\frac{1}{6}}\)), but \(p_b\gg n^{-\frac{1}{2}}\) (\(p_b\gg n^{-\frac{1}{6}}\)) results in a bichromatic period of configurations of length one or two, after at most \(2n^2\) (\(4n^2\)) steps with high probability.


Cellular automata Majority rule Color war 


  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.: Majority bootstrap percolation on the hypercube. Comb. Probab. Comput. 18(1–2), 17–51 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cardelli, L., Csikász-Nagy, A.: The cell cycle switch computes approximate majority. Sci. Rep. 2 (2012)Google Scholar
  4. 4.
    Einarsson, H., Lengler, J., Panagiotou, K., Mousset, F., Steger, A.: Bootstrap percolation with inhibition. arXiv preprint arXiv:1410.3291 (2014)
  5. 5.
    Fazli, M., Ghodsi, M., Habibi, J., Jalaly, P., Mirrokni, V., Sadeghian, S.: On non-progressive spread of influence through social networks. Theoret. Comput. Sci. 550, 36–50 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Feller, W.: An Introduction to Probability Theory and its Applications: Volume I, vol. 3. Wiley, London (1968)zbMATHGoogle Scholar
  7. 7.
    Flocchini, P., Královič, R., Ružička, P., Roncato, A., Santoro, N.: On time versus size for monotone dynamic monopolies in regular topologies. J. Discret. Algorithms 1(2), 129–150 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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). doi: 10.1007/978-3-642-41527-2_30 CrossRefGoogle Scholar
  9. 9.
    Goles, E., Olivos, J.: Comportement périodique des fonctions à seuil binaires et applications. Discret. Appl. Math. 3(2), 93–105 (1981)CrossRefzbMATHGoogle Scholar
  10. 10.
    Gray, L.: The behavior of processes with statistical mechanical properties. In: Kesten, H. (ed.) Percolation Theory and Ergodic Theory of Infinite Particle Systems, pp. 131–167. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  11. 11.
    Koch, C., Lengler, J.: Bootstrap percolation on geometric inhomogeneous random graphs. arXiv preprint arXiv:1603.02057 (2016)
  12. 12.
    Kozma, R., Puljic, M., Balister, P., Bollobás, B., Freeman, W.J.: Phase transitions in the neuropercolation model of neural populations with mixed local and non-local interactions. Biol. Cybern. 92(6), 367–379 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mitsche, D., Pérez-Giménez, X., Prałat, P.: Strong-majority bootstrap percolation on regular graphs with low dissemination threshold. arXiv preprint arXiv:1503.08310 (2015)
  14. 14.
    Molofsky, J., Durrett, R., Dushoff, J., Griffeath, D., Levin, S.: Local frequency dependence and global coexistence. Theoret. Popul. Biol. 55(3), 270–282 (1999)CrossRefzbMATHGoogle Scholar
  15. 15.
    Moore, C.: Majority-vote cellular automata, ising dynamics, and p-completeness. J. Stat. Phys. 88(3–4), 795–805 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Oliveira, G.M., Martins, L.G., Carvalho, L.B., Fynn, E.: Some investigations about synchronization and density classification tasks in one-dimensional and two-dimensional cellular automata rule spaces. Electron. Notes Theor. Comput. Sci. 252, 121–142 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    de Oliveira, M.J.: Isotropic majority-vote model on a square lattice. J. Stat. Phys. 66(1–2), 273–281 (1992)CrossRefzbMATHGoogle Scholar
  18. 18.
    Peleg, D.: Local majorities, coalitions and monopolies in graphs: a review. Theor. Comput. Sci. 282(2), 231–257 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Perron, E., Vasudevan, D., Vojnovic, M.: Using three states for binary consensus on complete graphs. In: INFOCOM 2009, IEEE, pp. 2527–2535. IEEE (2009)Google Scholar
  20. 20.
    Poljak, S., Sura, M.: On periodical behaviour in societies with symmetric influences. Combinatorica 3(1), 119–121 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Poljak, S., Turzík, D.: On pre-periods of discrete influence systems. Discret. Appl. Math 13(1), 33–39 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    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
  23. 23.
    Shao, J., Havlin, S., Stanley, H.E.: Dynamic opinion model and invasion percolation. Phys. Rev. Lett. 103(1), 018701 (2009)CrossRefGoogle Scholar
  24. 24.
    Spitzer, F.: Interaction of markov processes. Adv. Math. 5, 246–290 (1970)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceETH ZurichZurichSwitzerland

Personalised recommendations