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Universal gap scaling in percolation

A Publisher Correction to this article was published on 24 February 2020

This article has been updated


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 ( All figures are plotted by Origin 2018.

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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.

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Authors and Affiliations



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.

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Correspondence to Jingfang Fan or Jan Nagler.

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The authors declare no competing interests.

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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

Statistical Source Data

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. 16, 455–461 (2020).

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