Numerical Assessment of the Percolation Threshold Using Complement Networks

  • Giacomo RapisardiEmail author
  • Guido Caldarelli
  • Giulio Cimini
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

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Giacomo Rapisardi
    • 1
    Email author
  • Guido Caldarelli
    • 1
    • 2
  • Giulio Cimini
    • 1
    • 2
  1. 1.IMT School for Advanced StudiesLuccaItaly
  2. 2.Istituto dei Sistemi Complessi (ISC)-CNRRomeItaly

Personalised recommendations