The Annals of Applied Probability

Achlioptas process phase transitions are continuous

Oliver Riordan and Lutz Warnke

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It is widely believed that certain simple modifications of the random graph process lead to discontinuous phase transitions. In particular, starting with the empty graph on $n$ vertices, suppose that at each step two pairs of vertices are chosen uniformly at random, but only one pair is joined, namely, one minimizing the product of the sizes of the components to be joined. Making explicit an earlier belief of Achlioptas and others, in 2009, Achlioptas, D’Souza and Spencer [Science 323 (2009) 1453–1455] conjectured that there exists a $\delta>0$ (in fact, $\delta\ge1/2$) such that with high probability the order of the largest component “jumps” from $o(n)$ to at least $\delta n$ in $o(n)$ steps of the process, a phenomenon known as “explosive percolation.”

We give a simple proof that this is not the case. Our result applies to all “Achlioptas processes,” and more generally to any process where a fixed number of independent random vertices are chosen at each step, and (at least) one edge between these vertices is added to the current graph, according to any (online) rule.

We also prove the existence and continuity of the limit of the rescaled size of the giant component in a class of such processes, settling a number of conjectures. Intriguing questions remain, however, especially for the product rule described above.

Article information

Ann. Appl. Probab., Volume 22, Number 4 (2012), 1450-1464.

First available in Project Euclid: 10 August 2012

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Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 05C80: Random graphs [See also 60B20]

Achlioptas processes explosive percolation random graphs


Riordan, Oliver; Warnke, Lutz. Achlioptas process phase transitions are continuous. Ann. Appl. Probab. 22 (2012), no. 4, 1450--1464. doi:10.1214/11-AAP798.

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  • [1] Achlioptas, D., D’Souza, R. M. and Spencer, J. (2009). Explosive percolation in random networks. Science 323 1453–1455.
  • [2] Bohman, T. (2009). Emergence of connectivity in networks. Science 323 1438–1439.
  • [3] Bohman, T. and Frieze, A. (2001). Avoiding a giant component. Random Structures Algorithms 19 75–85.
  • [4] Bohman, T. and Kravitz, D. (2006). Creating a giant component. Combin. Probab. Comput. 15 489–511.
  • [5] Bollobás, B. (1984). The evolution of random graphs. Trans. Amer. Math. Soc. 286 257–274.
  • [6] da Costa, R. A., Dorogovtsev, S. N., Goltsev, A. V. and Mendes, J. F. F. (2010). Explosive percolation transition is actually continuous. Phys. Rev. Lett. 105 255701.
  • [7] D’Souza, R. M. and Mitzenmacher, M. (2010). Local cluster aggregation models of explosive percolation. Phys. Rev. Lett. 104 195702.
  • [8] Erdős, P. and Rényi, A. (1960). On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 17–61.
  • [9] Friedman, E. J. and Landsberg, A. S. (2009). Construction and analysis of random networks with explosive percolation. Phys. Rev. Lett. 103 255701.
  • [10] Janson, S. and Spencer, J. (2012). Phase transitions for modified Erdős–Rényi processes. Ark. Math. To appear. Available at arXiv:1005.4494.
  • [11] McDiarmid, C. (1989). On the method of bounded differences. In Surveys in Combinatorics, 1989 (Norwich, 1989). London Mathematical Society Lecture Note Series 141 148–188. Cambridge Univ. Press, Cambridge.
  • [12] Nagler, J., Levina, A. and Timme, M. (2011). Impact of single links in competitive percolation. Nature Physics 7 265–270.
  • [13] Riordan, O. and Warnke, L. (2011). Explosive percolation is continuous. Science 333 322–324.
  • [14] Spencer, J. (2010). Potpourri. J. Comb. 1 237–264.
  • [15] Spencer, J. and Wormald, N. (2007). Birth control for giants. Combinatorica 27 587–628.
  • [16] Wormald, N. C. (1999). The differential equation method for random graph processes and greedy algorithms. In Lectures on Approximation and Randomized Algorithms 73–155. PWN, Warsaw.