Universal gap scaling in percolation


Universality is a principle that fundamentally underlies many critical phenomena, ranging from epidemic spreading to the emergence or breakdown of global connectivity in networks. Percolation, the transition to global connectedness on gradual addition of links, may exhibit substantial gaps in the size of the largest connected network component. We uncover that the largest gap statistics is governed by extreme-value theory. This allows us to unify continuous and discontinuous percolation by virtue of universal critical scaling functions, obtained from normal and extreme-value statistics. Specifically, we show that the universal scaling function of the size of the largest gap is given by the extreme-value Gumbel distribution. This links extreme-value statistics to universality and criticality in percolation.

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Fig. 1: Critical scaling for 2D percolation.
Fig. 2: Universal gap scaling functions for 2D percolation.
Fig. 3: Critical scaling for DCA.
Fig. 4: Universal gap scaling functions for DCA.
Fig. 5: Gap scaling functions for human protein–protein interactions network.

Data availability

The data represented in Figs. 15 are available as Source Data. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

The C++ and Python codes used for the analysis is available on GitHub (https://github.com/fanjingfang/Universal-gap-scaling-in-percolation). All figures are plotted by Origin 2018.

Change history

  • 24 February 2020

    An amendment to this paper has been published and can be accessed via a link at the top of the paper.


  1. 1.

    Sornette, D. Critical Phenomena in Natural Sciences 2nd edn (Springer-Verlag, 2006).

  2. 2.

    Privman, V. & Fisher, M. E. Universal critical amplitudes in finite-size scaling. Phys. Rev. B 30, 322–327 (1984).

  3. 3.

    Privman, V. Finite Size Scaling and Numerical Simulation of Statistical Systems (World Scientific Singapore, 1990).

  4. 4.

    Rhee, I., Gasparini, F. M. & Bishop, D. J. Finite-size scaling of the superfluid density of 4He confined between silicon wafers. Phys. Rev. Lett. 63, 410–413 (1989).

  5. 5.

    Stauffer, D. & Aharony, A. Introduction to Percolation Theory (Taylor and Francis, 2003).

  6. 6.

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

  7. 7.

    Meng, J., Fan, J., Ashkenazy, Y. & Havlin, S. Percolation framework to describe El Nino conditions. Chaos 27, 035807 (2017).

  8. 8.

    Schröder, M., Nagler, J., Timme, M. & Witthaut, D. Hysteretic percolation from locally optimal individual decisions. Phys. Rev. Lett. 120, 248302 (2018).

  9. 9.

    Cohen, R. & Havlin, S. Complex Networks: Structure, Robustness and Function (Cambridge Univ. Press, 2010).

  10. 10.

    Achlioptas, D., D’souza, R. M. & Spencer, J. Explosive percolation in random networks. Science 323, 1453–1455 (2009).

  11. 11.

    da Costa, R. A., Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. Explosive percolation transition is actually continuous. Phys. Rev. Lett. 105, 255701 (2010).

  12. 12.

    Grassberger, P., Christensen, C., Bizhani, G., Son, S.-W. & Paczuski, M. Explosive percolation is continuous, but with unusual finite size behavior. Phys. Rev. Lett. 106, 225701 (2011).

  13. 13.

    Newman, M. E. J. & Ziff, R. M. Fast Monte Carlo algorithm for site or bond percolation. Phys. Rev. E 64, 016706 (2001).

  14. 14.

    Nagler, J., Levina, A. & Timme, M. Impact of single links in competitive percolation. Nat. Phys. 7, 265–270 (2011).

  15. 15.

    D’Souza, R. M. & Nagler, J. Anomalous critical and supercritical phenomena in explosive percolation. Nat. Phys. 11, 531–538 (2015).

  16. 16.

    D’Souza, R. M., Gómez-Gardeñes, J., Nagler, J. & Arenas, A. Explosive phenomena in complex networks. Adv. Phys. 68, 123–223 (2019).

  17. 17.

    Fan, J. & Chen, X. General clique percolation in random networks. Europhys. Lett. 107, 28005 (2014).

  18. 18.

    Nguyen, B. G. Gap exponents for percolation processes with triangle condition. J. Stat. Phys. 49, 235–243 (1987).

  19. 19.

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

  20. 20.

    Chen, W., Schröder, M., D’Souza, R. M., Sornette, D. & Nagler, J. Microtransition cascades to percolation. Phys. Rev. Lett. 112, 155701 (2014).

  21. 21.

    Schröder, M., Chen, W. & Nagler, J. Discrete scale invariance in supercritical percolation. N. J. Phys. 18, 013042 (2016).

  22. 22.

    Newman, M. E. J., Strogatz, S. H. & Watts, D. J. Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026118 (2001).

  23. 23.

    Gumbel, E. J. Statistics of Extremes (Courier Corporation, 2012).

  24. 24.

    Schröder, M., Rahbari, S. H. E. & Nagler, J. Crackling noise in fractional percolation. Nat. Commun. 4, 2222 (2013).

  25. 25.

    Fisher, M. E. Renormalization group theory: its basis and formulation in statistical physics. Rev. Mod. Phys. 70, 653–681 (1998).

  26. 26.

    Cho, Y. S., Kahng, B. & Kim, D. Cluster aggregation model for discontinuous percolation transitions. Phys. Rev. E 81, 030103 (2010).

  27. 27.

    Cho, Y. S., Mazza, M. G., Kahng, B. & Nagler, J. Genuine non-self-averaging and ultraslow convergence in gelation. Phys. Rev. E 94, 022602 (2016).

  28. 28.

    Cho, Y., Lee, J., Herrmann, H. & Kahng, B. Hybrid percolation transition in cluster merging processes: continuously varying exponents. Phys. Rev. Lett. 116, 025701 (2016).

  29. 29.

    Du, C., Satik, C. & Yortsos, Y. C. Percolation in a fractional Brownian motion lattice. AIChE J. 42, 2392–2395 (1996).

  30. 30.

    Isichenko, M. B. Percolation, statistical topography, and transport in random media. Rev. Mod. Phys. 64, 961–1043 (1992).

  31. 31.

    Nijs, M. P. Md A relation between the temperature exponents of the eight-vertex and q-state Potts model. J. Phys. A 12, 1857–1868 (1979).

  32. 32.

    Essam, I. W., Gaunt, D. S. & Guttmann, A. J. Percolation theory at the critical dimension. J. Phys. A 11, 1983–1990 (1978).

  33. 33.

    Bollobás, B. Random Graphs (Cambridge Univ. Press, 2001).

  34. 34.

    Riordan, O. & Warnke, L. Explosive percolation is continuous. Science 333, 322–324 (2011).

  35. 35.

    Cho, Y. S., Hwang, S., Herrmann, H. J. & Kahng, B. Avoiding a spanning cluster in percolation models. Science 339, 1185–1187 (2013).

  36. 36.

    Smoluchowski, M. Drei vorträge über diffusion, brownsche bewegung und koagulation von kolloidteilchen. Z. Phys. 17, 557–585 (1916).

  37. 37.

    Family, F. & Landau, D. P. Kinetics of Aggregation and Gelation (Elsevier, 2012).

  38. 38.

    Lee, D., Choi, W., Kertész, J. & Kahng, B. Universal mechanism for hybrid percolation transitions. Sci. Rep. 7, 5723 (2017).

  39. 39.

    Fan, J., Meng, J. & Saberi, A. A. Percolation framework of the Earth’s topography. Phys. Rev. E 99, 022304 (2019).

  40. 40.

    Pikovsky, A., Rosenblum, M. & Kurths, J. Synchronization: A Universal Concept in Nonlinear Sciences 1st edn (Cambridge Univ. Press, 2003).

  41. 41.

    Romualdo, P.-S., Castellano, C., Van Mieghem, P. & Vespignani, A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015).

  42. 42.

    Strenski, P. N., Bradley, R. M. & Debierre, J.-M. Scaling behavior of percolation surfaces in three dimensions. Phys. Rev. Lett. 66, 1330–1333 (1991).

  43. 43.

    Erdős, P. & Rényi, A. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci 5, 17–60 (1960).

  44. 44.

    Nachmias, A. & Peres, Y. Critical percolation on random regular graphs. Random Struct. Algorithms 36, 111–148 (2010).

  45. 45.

    Essam, J. W. Percolation theory. Rep. Prog. Phys. 43, 833–912 (1980).

  46. 46.

    Levenberg, K. A method for the solution of certain non-linear problems in least squares. Q. Appl. Math. 2, 164–168 (1944).

  47. 47.

    Marquardt, D. W. An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math. 11, 431–441 (1963).

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We acknowledge the ‘East Africa Peru India Climate Capacities — EPICC’ project, which is part of the International Climate Initiative (IKI). The Federal Ministry for the Environment, Nature Conservation and Nuclear Safety (BMU) supports this initiative on the basis of a decision adopted by the German Bundestag. The Potsdam Institute for Climate Impact Research (PIK) is leading the execution of the project together with its project partners The Energy and Resources Institute (TERI) and the Deutscher Wetterdienst (DWD). A.A.S. acknowledges support from the Alexander von Humboldt Foundation and partial financial support from the research council of the University of Tehran.

Author information

J.F., J.M., A.A.S., J.K. and J.N. designed the research, conceived the study, carried out the analysis and prepared the manuscript. J.F., J.M. and J.N. analysed data. J.F., J.M., Y.L., A.A.S. and J.N. discussed results and contributed to writing the manuscript.

Correspondence to Jingfang Fan or Jan Nagler.

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Peer review information Nature Physics thanks Filippo Radicchiand the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Figs. 1–20, discussion and Table 1.

Source data

Source Data Fig. 1

Statistical Source Data

Source Data Fig. 2

Statistical Source Data

Source Data Fig. 3

Statistical Source Data

Source Data Fig. 4

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Source Data Fig. 5

Statistical Source Data

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Fan, J., Meng, J., Liu, Y. et al. Universal gap scaling in percolation. Nat. Phys. (2020). https://doi.org/10.1038/s41567-019-0783-2

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