Numerical Assessment of the Percolation Threshold Using Complement Networks

Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 812)


Models of percolation processes on networks currently assume locally tree-like structures at low densities, and are derived exactly only in the thermodynamic limit. Finite size effects and the presence of short loops in real systems however cause a deviation between the empirical percolation threshold \(p_c\) and its model-predicted value \(\pi _c\). Here we show the existence of an empirical linear relation between \(p_c\) and \(\pi _c\) across a large number of real and model networks. Such a putatively universal relation can then be used to correct the estimated value of \(\pi _c\). We further show how to obtain a more precise relation using the concept of the complement graph, by investigating on the connection between the percolation threshold of a network, \(p_c\), and that of its complement, \(\bar{p}_c\).


Percolation theory Complement graphs 



This work was supported by the EU projects CoeGSS (grant no. 676547) and SoBigData (grant no. 654024).


  1. 1.
    Stauffer, D., Aharony, A.: Introduction to Percolation Theory. Taylor and Francis (1994)Google Scholar
  2. 2.
    Callaway, D.S., Newman, M.E.J., Strogatz, S.H., Watts, D.J.: Network robustness and fragility: Percolation on random graphs. Phys. Rev. Lett. 85, 5468 (2000)Google Scholar
  3. 3.
    Newman, M.E.J.: Spread of epidemic disease on networks. Phys. Rev. E 66, 016128 (2002)Google Scholar
  4. 4.
    Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: Critical phenomena in complex networks. Rev. Modern Phys. 80, 1275 (2008)Google Scholar
  5. 5.
    Cohen, R., ben Avraham, D., Havlin, S.: Percolation critical exponents in scale-free networks. Phys. Rev. E 66, 036113 (2002)Google Scholar
  6. 6.
    Serrano, M.A., Boguná, M.: Stability diagram of a few-electron triple dot. Phys. Rev. Lett. 97, 088701 (2006)Google Scholar
  7. 7.
    Newman, M.E.J.: Random graphs with clustering. Phys. Rev. Lett. 103, 058701 (2009)Google Scholar
  8. 8.
    Karrer, B., Newman, M.E.J., Zdeborová, L.: Percolation on sparse networks. Phys. Rev. Lett. 113, 208702 (2014)Google Scholar
  9. 9.
    Hamilton, K.E., Pryadko, L.P.: Tight lower bound for percolation threshold on an infinite graph. Phys. Rev. Lett. 113, 208701 (2014)Google Scholar
  10. 10.
    Hashimoto, K., Namikawa, Y.: Automorphic Forms and Geometry of Arithmetic Varieties. Advanced Studies in Pure Mathematics. Elsevier Science (2014)Google Scholar
  11. 11.
    Radicchi, F., Castellano, C.: Beyond the locally treelike approximation for percolation on real networks. Phys. Rev. E 93, 030302 (2016)Google Scholar
  12. 12.
    Radicchi, F.: Predicting percolation thresholds in networks. Phys. Rev. E 91, 010801 (2015)Google Scholar
  13. 13.
    Timar, G., da Costa, R.A., Dorogovtsev, S.N., Mendes, J.F.F.: Nonbacktracking expansion of finite graphs. Phys. Rev. E 95, 042322 (2017)Google Scholar
  14. 14.
    Newman, M.E.J., Ziff, R.M.: Efficient Monte Carlo algorithm and high-precision results for percolation. Phys. Rev. Lett. 85, 4104 (2000)Google Scholar
  15. 15.
    Radicchi, F., Castellano, C.: Breaking of the site-bond percolation universality in networks. Nature Commun. 6, 10196 (2015)Google Scholar
  16. 16.
    Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45, 167 (2003)Google Scholar
  17. 17.
    Clark, L., Entringer, R.: Smallest maximally nonhamiltonian graphs. Periodica Math. Hung. 14, 57 (1983)Google Scholar
  18. 18.
    Gross, J., Yellen, J.: Graph Theory and Its Applications. Discrete Mathematics and Its Applications. Taylor and Francis (1998)Google Scholar
  19. 19.
    Nordhaus, E.A., Gaddum, J.W.: A Kaleidoscopic view of graph colorings. Am. Math. Mon. 63, 175 (1956)Google Scholar
  20. 20.
    Kao, M.Y., Occhiogrosso, N., Teng, S.-H.: Simple and efficient graph compression schemes for dense and complement graphs. J. Comb. Optim. 2, 351 (1998)Google Scholar
  21. 21.
    Ito, H., Yokoyama, M.: Linear time algorithms for graph search and connectivity determination on complement graphs. Inf. Proc. Lett. 66, 209 (1998)Google Scholar
  22. 22.
    Duan, Z., Liu, C., Chen, G.: Network synchronizability analysis: the theory of subgraphs and complementary graphs. Phys. D Nonlinear Phenom. 237, 1006 (2008)Google Scholar
  23. 23.
    Bermudo, S., Rodrguez, J.M., Sigarreta, J.M., Tours, E.: Hyperbolicity and complement of graphs. Appl. Math. Lett. 24, 1882 (2011)Google Scholar
  24. 24.
    Haas, R., Wexler, T.B.: Bounds on the signed domination number of a graph. Discret. Math. 283, 87 (2004)Google Scholar
  25. 25.
    Akiyama, J., Harary, F.: A graph and its complement with specified properties. Int. J. Math. Math. Sci. 2, 223 (1979)Google Scholar
  26. 26.
    Xu, S.: Some parameter of graph and its component. Discret. Math. 65, 197 (1987)Google Scholar
  27. 27.
    Petrovic, M., Radosavljevic, Z., Simic, S.: A graph and its complement with specified spectral properties. Linear Multilinear Algebra 51, 405 (2003)Google Scholar
  28. 28.
    Bollobas, B., Borgs, C., Chayes, J., Riordan, O.: Percolation on dense graph sequences. Ann. Probab. 38, 150 (2010)Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.IMT School for Advanced StudiesLuccaItaly
  2. 2.Istituto dei Sistemi Complessi (ISC)-CNRRomeItaly

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