Given enough choice, simple local rules percolate discontinuously

  • Alex WaagenEmail author
  • Raissa M. D’Souza
Regular Article


There is still much to discover about the mechanisms and nature of discontinuous percolation transitions. Much of the past work considers graph evolution algorithms known as Achlioptas processes in which a single edge is added to the graph from a set of k randomly chosen candidate edges at each timestep until a giant component emerges. Several Achlioptas processes seem to yield a discontinuous percolation transition, but it was proven by Riordan and Warnke that the transition must be continuous in the thermodynamic limit. However, they also proved that if the number k(n) of candidate edges increases with the number of nodes, then the percolation transition may be discontinuous. Here we attempt to find the simplest such process which yields a discontinuous transition in the thermodynamic limit. We introduce a process which considers only the degree of candidate edges and not component size. We calculate the critical point \(t_c = (1 - \theta (\tfrac{1} {k}))n\) and rigorously show that the critical window is of size \(O\left( {\tfrac{n} {{k(n)}}} \right)\). If k(n) grows very slowly, for example k(n) = log n, the critical window is barely sublinear and hence the phasetransition is discontinuous but appears continuous in finite systems. We also present arguments that Achlioptas processes with bounded size rules will always have continuous percolation transitions even with infinite choice.


Statistical and Nonlinear Physics 


  1. 1.
    D. Stauffer, Introduction to percolation theory (Taylor & Francis, 1985)Google Scholar
  2. 2.
    M.E.J. Newman, D.J. Watts, Phys. Rev. E 60, 7332 (1999)ADSCrossRefGoogle Scholar
  3. 3.
    R. Cohen, K. Erez, D. ben Avraham, S. Havlin, Phys. Rev. Lett. 85, 4626 (2000)ADSCrossRefGoogle Scholar
  4. 4.
    D.S. Callaway, M.E.J. Newman, S.H. Strogatz, D.J. Watts, Phys. Rev. Lett. 85, 5468 (2000)ADSCrossRefGoogle Scholar
  5. 5.
    C. Moore, M.E.J. Newman, Phys. Rev. E 61, 5678 (2000)ADSCrossRefGoogle Scholar
  6. 6.
    M.E.J. Newman, Networks: An Introduction (Oxford University Press, New York, 2010)Google Scholar
  7. 7.
    O. Riordan, L. Warnke, Ann. Appl. Probab. 22, 1450 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Y. Azar, A.Z. Broder, A.R. Karlin, E. Upfal, SIAM J. Comput. 29, 180 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    M. Mitzenmacher, A.W. Richa, R. Sitaraman, in Handbook of Randomized Computing (Kluwer, 2000), pp. 255–312Google Scholar
  10. 10.
    M. Mitzenmacher, IEEE Trans. Parallel Distrib. Syst. 12, 1094 (2001)CrossRefGoogle Scholar
  11. 11.
    A. Sinclair, D. Vilenchik, Delaying satisfiability for random 2SAT, in Proceedings of the 13th International Conference on Approximation, and 14th International Conference on Randomization, and combinatorial optimization: algorithms and techniques (Springer-Verlag, Berlin, Heidelberg, 2010), pp. 710–723Google Scholar
  12. 12.
    T. Bohman, A. Frieze, Random Struct. Algorithms 19, 75 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    D. Achlioptas, R.M. D’Souza, J. Spencer, Science 323, 1453 (2009)ADSCrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    R.A. da Costa, S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Phys. Rev. Lett. 105, 255701 (2010)ADSCrossRefGoogle Scholar
  15. 15.
    J. Nagler, A. Levina, M. Timme, Nat. Phys. 7, 265 (2011)CrossRefGoogle Scholar
  16. 16.
    P. Grassberger, C. Christensen, G. Bizhani, S.W. Son, M. Paczuski, Phys. Rev. Lett. 106, 225701 (2011)ADSCrossRefGoogle Scholar
  17. 17.
    S.S. Manna, A. Chatterjee, Physica A 390, 177 (2011)ADSCrossRefGoogle Scholar
  18. 18.
    H.K. Lee, B.J. Kim, H. Park, Phys. Rev. E 84, 020101 (2011)ADSGoogle Scholar
  19. 19.
    L. Tian, D.N. Shi, Phys. Lett. A 376, 286 (2012)ADSCrossRefzbMATHGoogle Scholar
  20. 20.
    O. Riordan, L. Warnke, Science 333, 322 (2011)ADSCrossRefGoogle Scholar
  21. 21.
    W. Chen, R.M. D’Souza, Phys. Rev. Lett. 106, 115701 (2011)ADSCrossRefGoogle Scholar
  22. 22.
    K. Panagiotou, R. Spöhel, A. Steger, H. Thomas, Electron. Notes Discrete Math. 38, 699 (2011)CrossRefGoogle Scholar
  23. 23.
    J. Nagler, T. Tiessen, H.W. Gutch, Phys. Rev. X 2, 031009 (2012)Google Scholar
  24. 24.
    H. Hwang, J. Herrmann, Y. Cho, B. Kahng, Science 339, 1185 (2013)ADSCrossRefGoogle Scholar
  25. 25.
    J. Spencer, N. Wormald, Combinatorica 27, 587 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    S. Janson, J. Spencer, arXiv:1005.4494v1 [math.CO] (2012)Google Scholar
  27. 27.
    P. Erdős, A. Rényi, Magyar Tud. Akad. Mat. Kutató Int. Közl 5, 17 (1960)Google Scholar
  28. 28.
    R. Durrett, Probability: Theory and Examples, Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, 2010)Google Scholar
  29. 29.
    E.J. Friedman, A.S. Landsberg, Phys. Rev. Lett. 103, 255701 (2009)ADSCrossRefGoogle Scholar
  30. 30.
    D. Aldous, Bernoulli 5, 3 (1999)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.University of CaliforniaDavisUSA
  2. 2.The Santa Fe InstituteSanta FeUSA

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