Review Article | Published:

Anomalous critical and supercritical phenomena in explosive percolation

Nature Physics volume 11, pages 531538 (2015) | Download Citation

Abstract

The emergence of large-scale connectivity on an underlying network or lattice, the so-called percolation transition, has a profound impact on the system’s macroscopic behaviours. There is thus great interest in controlling the location of the percolation transition to either enhance or delay its onset and, more generally, in understanding the consequences of such control interventions. Here we review explosive percolation, the sudden emergence of large-scale connectivity that results from repeated, small interventions designed to delay the percolation transition. These transitions exhibit drastic, unanticipated and exciting consequences that make explosive percolation an emerging paradigm for modelling real-world systems ranging from social networks to nanotubes.

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References

  1. 1.

    & Introduction to Percolation Theory (Taylor & Francis, 1994).

  2. 2.

    Applications of Percolation Theory (Taylor & Francis, 1994).

  3. 3.

    & On random graphs I. Math. Debrecen 6, 290–297 (1959).

  4. 4.

    & On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5, 17–61 (1960).

  5. 5.

    Random graphs. Ann. Math. Stat. 30, 1141–1144 (1959).

  6. 6.

    & Kinetic theory of random graphs: From paths to cycles. Phys. Rev. E 71, 026129 (2005).

  7. 7.

    Random Graphs 2nd edn (Cambridge Univ. Press, 2001).

  8. 8.

    , , & Proc. 26th ACM Symp. Theory Comput. 593–602 (1994).

  9. 9.

    , , & Balanced allocations. SIAM J. Comput. 29, 180–200 (1999).

  10. 10.

    , , & Parallel randomized load balancing. Random Struct. Algorithms 13, 159–188 (1998).

  11. 11.

    The power of two choices in randomized load balancing. Parallel Distrib. Syst. 12, 1094–1104 (2001).

  12. 12.

    & Avoiding a giant component. Random Struct. Algorithms 19, 75–85 (2001).

  13. 13.

    Drei Vorträge uber Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen. Z. Phys. 17, 557–585 (1916).

  14. 14.

    Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli 5, 3–48 (1999).

  15. 15.

    & Birth control for giants. Combinatorica 27, 587–628 (2007).

  16. 16.

    , & Explosive percolation in random networks. Science 323, 1453–1455 (2009).

  17. 17.

    & A fast Monte Carlo algorithm for site or bond percolation. Phys. Rev. E 64, 016706 (2001).

  18. 18.

    Explosive growth in biased dynamic percolation on two-dimensional regular lattice networks. Phys. Rev. Lett. 103, 045701 (2009).

  19. 19.

    , , , & Percolation transitions in scale-free networks under the Achlioptas process. Phys. Rev. Lett. 103, 135702 (2009).

  20. 20.

    & Explosive percolation in scale-free networks. Phys. Rev. Lett. 103, 168701 (2009).

  21. 21.

    Scaling behavior of explosive percolation on the square lattice. Phys. Rev. E 82, 051105 (2010).

  22. 22.

    & Explosive percolation: A numerical analysis. Phys. Rev. E 81, 036110 (2010).

  23. 23.

    & Construction and analysis of random networks with explosive percolation. Phys. Rev. Lett. 103, 255701 (2009).

  24. 24.

    & Local cluster aggregation models of explosive percolation. Phys. Rev. Lett. 104, 195702 (2010).

  25. 25.

    & Suppression effect on explosive percolation. Phys. Rev. Lett. 107, 275703 (2011).

  26. 26.

    & Achlioptas processes are not always self-averaging. Phys. Rev. E 86, 011129 (2012).

  27. 27.

    , , & Explosive percolation: Unusual transitions of a simple model. Physica A 407, 54–65 (2014).

  28. 28.

    , & The onset of jamming as the sudden emergence of an infinite k-core cluster. Europhys. Lett. 73, 560–566 (2006).

  29. 29.

    , & Jamming percolation and glass transitions in lattice models. Phys. Rev. Lett. 96, 035702 (2006).

  30. 30.

    & Force-balance percolation. Phys. Rev. E 81, 011134 (2010).

  31. 31.

    & Correlated percolation and tricriticality. Phys. Rev. E 86, 061131 (2012).

  32. 32.

    , & Impact of single links in competitive percolation. Nature Phys. 7, 265–270 (2011).

  33. 33.

    & A new route to explosive percolation. Physica A 390, 177–182 (2011).

  34. 34.

    , , & Explosive percolation transition is actually continuous. Phys. Rev. Lett. 105, 255701 (2010).

  35. 35.

    , , , & Explosive percolation is continuous, but with unusual finite size behavior. Phys. Rev. Lett. 106, 225701 (2011).

  36. 36.

    , & Continuity of the explosive percolation transition. Phys. Rev. E 84, 020101 (2011).

  37. 37.

    & The nature of explosive percolation phase transition. Phys. Lett. A 376, 286–289 (2012).

  38. 38.

    & Explosive percolation is continuous. Science 333, 322–324 (2011).

  39. 39.

    & Criterion for explosive percolation transitions on complex networks. Phys. Rev. E 83, 032101 (2011).

  40. 40.

    & Achlioptas process phase transitions are continuous. Ann. Appl. Probab. 22, 1450–1464 (2012).

  41. 41.

    , & Continuous percolation with discontinuities. Phys. Rev. X 2, 031009 (2012).

  42. 42.

    , & Crackling noise in fractional percolation. Nature Commun. 4, 2222 (2013).

  43. 43.

    , , & Avoiding a spanning cluster in percolation models. Science 339, 1185–1187 (2013).

  44. 44.

    , & Explosive percolation in the human protein homology network. Eur. Phys. J. B 75, 305–310 (2010).

  45. 45.

    & Explosive percolation with multiple giant components. Phys. Rev. Lett. 106, 115701 (2011).

  46. 46.

    & Explosive percolation via control of the largest cluster. Phys. Rev. Lett. 105, 035701 (2010).

  47. 47.

    & Types of discontinuous percolation transitions in cluster merging processes. Sci. Rep. (in the press).

  48. 48.

    , & Avoidance of a giant component in half the edge set of a random graph. Random Struct. Algorithms 25, 432–449 (2004).

  49. 49.

    et al. Phase transitions in supercritical Explosive Percolation. Phys. Rev. E 87, 052130 (2013).

  50. 50.

    et al. Unstable supercritical discontinuous percolation transitions. Phys. Rev. E 88, 042152 (2013).

  51. 51.

    , & Deriving an underlying mechanism for discontinuous percolation. Europhys. Lett. 100, 66006 (2012).

  52. 52.

    , , & Explosive percolation in Erdős-Rényi-like random graph processes. Electron. Notes Discrete Math. 38, 699–704 (2011).

  53. 53.

    , & Ordinary percolation with discontinuous transitions. Nature Commun. 3, 787 (2012).

  54. 54.

    , , , & Hamiltonian approach for explosive percolation. Phys. Rev. E 81, 040101(R) (2010).

  55. 55.

    , , , & Catastrophic cascade of failures in interdependent networks. Nature 464, 1025–1028 (2010).

  56. 56.

    , , , & Explosive ising. J. Stat. Mech. 2012, L06002 (2012).

  57. 57.

    , , & Explosive synchronization transitions in scale-free networks. Phys. Rev. Lett. 106, 128701 (2011).

  58. 58.

    , , & Explosive synchronization in adaptive and multilayer networks. Phys. Rev. Lett. 114, 038701 (2015).

  59. 59.

    , & Discontinuous percolation transitions in epidemic processes, surface depinning in random media, and Hamiltonian random graphs. Phys. Rev. E 86, 011128 (2012).

  60. 60.

    , & A Kinetic View of Statistical Physics (Cambridge Univ. Press, 2010).

  61. 61.

    , & Cluster aggregation model for discontinuous percolation transition. Phys. Rev. E 81, 030103(R) (2010).

  62. 62.

    , & Protein homology network families reveal step-wise diversification of Type III and Type IV secretion systems. PLoS Comput. Biol. 2, e173 (2006).

  63. 63.

    The strength of weak ties. Am. J. Soc.1360–1380 (1973).

  64. 64.

    , , , & Using explosive percolation in analysis of real-world networks. Phys. Rev. E 83, 046112 (2011).

  65. 65.

    Topological Evolution of Networks: Case Studies in the US Airlines and Language Wikipedias PhD thesis, Massachusetts Institute of Technology (2009)

  66. 66.

    & Diffusion-limited aggregation, a kinetic critical phenomenon. Phys. Rev. Lett. 47, 1400–1403 (1981).

  67. 67.

    & Discontinuous percolation transitions in real physical systems. Phys. Rev. E 84, 050102 (2011).

  68. 68.

    , , & Explosive electric breakdown due to conducting-particle deposition on an insulating substrate. Phys. Rev. Lett. 113, 155701 (2014).

  69. 69.

    , & Explosive percolation in a nanotube-based system. Phys. Rev. E 82, 061105 (2010).

  70. 70.

    , , , & Are randomly grown graphs really random? Phys. Rev. E 64, 041902 (2001).

  71. 71.

    , , & Growth dominates choice in network percolation. Phys. Rev. E 88, 032141 (2013).

  72. 72.

    , , & Percolation properties of growing networks under an Achlioptas process. Europhys. Lett. 103, 26004 (2013).

  73. 73.

    , , , & Microtransition cascades to percolation. Phys. Rev. Lett. 112, 155701 (2014).

  74. 74.

    Discrete scale invariance and complex dimensions. Phys. Rep. 297, 239–270 (1998).

  75. 75.

    , , & Solution of the explosive percolation quest: Scaling functions and critical exponents. Phys. Rev. E 90, 022145 (2014).

  76. 76.

    & Tight lower bound for percolation threshold on an infinite graph. Phys. Rev. Lett. 113, 208701 (2014).

  77. 77.

    , & Percolation on sparse networks. Phys. Rev. Lett. 113, 208702 (2014).

  78. 78.

    Predicting percolation thresholds in networks. Phys. Rev. E 91, 010801 (2015).

  79. 79.

    et al. Weakly explosive percolation in directed networks. Phys. Rev. E 87, 052127 (2013).

  80. 80.

    The giant component: The golden anniversary. Not. AMS 57, 720–724 (2010).

  81. 81.

    & Percolation with multiple giant clusters. J. Phys. A 38, L417 (2005).

  82. 82.

    & The evolution of subcritical Achlioptas processes. Random Struct. Algorithms (2014).

  83. 83.

    , , , & Recent advances and open challenges in percolation. Eur. Phys. J. Spec. Top. 223, 2307–2321 (2014).

  84. 84.

    Recent advances in percolation theory and its applications. Phys. Rep. 578, 1–32 (2015).

  85. 85.

    Globally networked risks and how to respond. Nature 497, 51–59 (2013).

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Acknowledgements

We thank M. Schröder and A. Witt for valuable discussions and assistance in the preparation of the figures, and C. D’Souza and L. Nagler-Deutsch for invaluable input. R.M.D’S. gratefully acknowledges support from the US Army Research Office MURI Award No. W911NF-13-1-0340 and Cooperative Agreement No. W911NF-09-2-0053 and the Defense Threat Reduction Agency Basic Research Award HDTRA1-10-1-0088.

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Affiliations

  1. Department of Computer Science and Department of Mechanical and Aerospace Engineering at University of California, Davis, California 95616, USA

    • Raissa M. D’Souza
  2. Computational Physics for Engineering Materials, IfB, ETH Zürich, Wolfgang-Pauli-Strasse 27 CH 8093 Zürich, Switzerland

    • Jan Nagler

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The authors declare no competing financial interests.

Corresponding authors

Correspondence to Raissa M. D’Souza or Jan Nagler.

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DOI

https://doi.org/10.1038/nphys3378

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