Abstract
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.
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References
D. Stauffer, Introduction to percolation theory (Taylor & Francis, 1985)
M.E.J. Newman, D.J. Watts, Phys. Rev. E 60, 7332 (1999)
R. Cohen, K. Erez, D. ben Avraham, S. Havlin, Phys. Rev. Lett. 85, 4626 (2000)
D.S. Callaway, M.E.J. Newman, S.H. Strogatz, D.J. Watts, Phys. Rev. Lett. 85, 5468 (2000)
C. Moore, M.E.J. Newman, Phys. Rev. E 61, 5678 (2000)
M.E.J. Newman, Networks: An Introduction (Oxford University Press, New York, 2010)
O. Riordan, L. Warnke, Ann. Appl. Probab. 22, 1450 (2012)
Y. Azar, A.Z. Broder, A.R. Karlin, E. Upfal, SIAM J. Comput. 29, 180 (1999)
M. Mitzenmacher, A.W. Richa, R. Sitaraman, in Handbook of Randomized Computing (Kluwer, 2000), pp. 255–312
M. Mitzenmacher, IEEE Trans. Parallel Distrib. Syst. 12, 1094 (2001)
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. 710–723
T. Bohman, A. Frieze, Random Struct. Algorithms 19, 75 (2001)
D. Achlioptas, R.M. D’Souza, J. Spencer, Science 323, 1453 (2009)
R.A. da Costa, S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Phys. Rev. Lett. 105, 255701 (2010)
J. Nagler, A. Levina, M. Timme, Nat. Phys. 7, 265 (2011)
P. Grassberger, C. Christensen, G. Bizhani, S.W. Son, M. Paczuski, Phys. Rev. Lett. 106, 225701 (2011)
S.S. Manna, A. Chatterjee, Physica A 390, 177 (2011)
H.K. Lee, B.J. Kim, H. Park, Phys. Rev. E 84, 020101 (2011)
L. Tian, D.N. Shi, Phys. Lett. A 376, 286 (2012)
O. Riordan, L. Warnke, Science 333, 322 (2011)
W. Chen, R.M. D’Souza, Phys. Rev. Lett. 106, 115701 (2011)
K. Panagiotou, R. Spöhel, A. Steger, H. Thomas, Electron. Notes Discrete Math. 38, 699 (2011)
J. Nagler, T. Tiessen, H.W. Gutch, Phys. Rev. X 2, 031009 (2012)
H. Hwang, J. Herrmann, Y. Cho, B. Kahng, Science 339, 1185 (2013)
J. Spencer, N. Wormald, Combinatorica 27, 587 (2007)
S. Janson, J. Spencer, arXiv:1005.4494v1 [math.CO] (2012)
P. Erdős, A. Rényi, Magyar Tud. Akad. Mat. Kutató Int. Közl 5, 17 (1960)
R. Durrett, Probability: Theory and Examples, Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, 2010)
E.J. Friedman, A.S. Landsberg, Phys. Rev. Lett. 103, 255701 (2009)
D. Aldous, Bernoulli 5, 3 (1999)
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Waagen, A., D’Souza, R.M. Given enough choice, simple local rules percolate discontinuously. Eur. Phys. J. B 87, 304 (2014). https://doi.org/10.1140/epjb/e2014-50278-x
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DOI: https://doi.org/10.1140/epjb/e2014-50278-x