Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Percolation in real interdependent networks

Abstract

The function of a real network depends not only on the reliability of its own components, but is affected also by the simultaneous operation of other real networks coupled with it. Whereas theoretical methods of direct applicability to real isolated networks exist, the frameworks developed so far in percolation theory for interdependent network layers are of little help in practical contexts, as they are suited only for special models in the limit of infinite size. Here, we introduce a set of heuristic equations that takes as inputs the adjacency matrices of the layers to draw the entire phase diagram for the interconnected network. We demonstrate that percolation transitions in interdependent networks can be understood by decomposing these systems into uncoupled graphs: the intersection among the layers, and the remainders of the layers. When the intersection dominates the remainders, an interconnected network undergoes a smooth percolation transition. Conversely, if the intersection is dominated by the contribution of the remainders, the transition becomes abrupt even in small networks. We provide examples of real systems that have developed interdependent networks sharing cores of ‘high quality’ edges to prevent catastrophic failures.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Figure 1: Decomposition of interconnected networks into uncoupled graphs.
Figure 2: Percolation transition in artificial interconnected networks.
Figure 3: Percolation transition in interdependent biological networks.
Figure 4: Percolation transition in interconnected transportation networks.

References

  1. 1

    Stauffer, D. & Aharony, A. Introduction to Percolation Theory (Taylor and Francis, 1991).

    MATH  Google Scholar 

  2. 2

    Kirkpatrick, S. Percolation and conduction. Rev. Mod. Phys. 45, 574–588 (1973).

    ADS  Article  Google Scholar 

  3. 3

    Berkowitz, B. Analysis of fracture network connectivity using percolation theory. Math. Geol. 27, 467–483 (1995).

    Article  Google Scholar 

  4. 4

    Pastor-Satorras, R. & Vespignani, A. Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200–3203 (2001).

    ADS  Article  Google Scholar 

  5. 5

    Newman, M. E. Spread of epidemic disease on networks. Phys. Rev. E 66, 016128 (2002).

    ADS  MathSciNet  Article  Google Scholar 

  6. 6

    Albert, R., Jeong, H. & Barabási, A-L. Error and attack tolerance of complex networks. Nature 406, 378–382 (2000).

    ADS  Article  Google Scholar 

  7. 7

    Cohen, R., Erez, K., Ben-Avraham, D. & Havlin, S. Resilience of the Internet to random breakdowns. Phys. Rev. Lett. 85, 4626–4628 (2000).

    ADS  Article  Google Scholar 

  8. 8

    Callaway, D. S., Newman, M. E., Strogatz, S. H. & Watts, D. J. Network robustness and fragility: Percolation on random graphs. Phys. Rev. Lett. 85, 5468–5471 (2000).

    ADS  Article  Google Scholar 

  9. 9

    Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. Critical phenomena in complex networks. Rev. Mod. Phys. 80, 1275–1335 (2008).

    ADS  Article  Google Scholar 

  10. 10

    Bollobás, B. et al. Percolation on dense graph sequences. Ann. Prob. 38, 150–183 (2010).

    MathSciNet  Article  Google Scholar 

  11. 11

    Karrer, B., Newman, M. E. J. & Zdeborová, L. Percolation on sparse networks. Phys. Rev. Lett. 113, 208702 (2014).

    ADS  Article  Google Scholar 

  12. 12

    Hamilton, K. E. & Pryadko, L. P. Tight lower bound for percolation threshold on an infinite graph. Phys. Rev. Lett. 113, 208701 (2014).

    ADS  Article  Google Scholar 

  13. 13

    Cohen, R., Havlin, S. & Ben-Avraham, D. Efficient immunization strategies for computer networks and populations. Phys. Rev. Lett. 91, 247901 (2003).

    ADS  Article  Google Scholar 

  14. 14

    Moreira, A. A., Andrade, J. S. Jr, Herrmann, H. J. & Indekeu, J. O. How to make a fragile network robust and vice versa. Phys. Rev. Lett. 102, 018701 (2009).

    ADS  Article  Google Scholar 

  15. 15

    Schneider, C. M., Moreira, A. A., Andrade, J. S., Havlin, S. & Herrmann, H. J. Mitigation of malicious attacks on networks. Proc. Natl Acad. Sci. USA 108, 3838–3841 (2011).

    ADS  Article  Google Scholar 

  16. 16

    Buldyrev, S. V., Parshani, R., Paul, G., Stanley, H. E. & Havlin, S. Catastrophic cascade of failures in interdependent networks. Nature 464, 1025–1028 (2010).

    ADS  Article  Google Scholar 

  17. 17

    Radicchi, F. & Arenas, A. Abrupt transition in the structural formation of interconnected networks. Nature Phys. 9, 717–720 (2013).

    ADS  Article  Google Scholar 

  18. 18

    Szell, M., Lambiotte, R. & Thurner, S. Multirelational organization of large-scale social networks in an online world. Proc. Natl Acad. Sci. USA 107, 13636–13641 (2010).

    ADS  Article  Google Scholar 

  19. 19

    Barthélemy, M. Spatial networks. Phys. Rep. 499, 1–101 (2011).

    ADS  MathSciNet  Article  Google Scholar 

  20. 20

    De Domenico, M., Solé-Ribalta, A., Gómez, S. & Arenas, A. Navigability of interconnected networks under random failures. Proc. Natl Acad. Sci. USA 111, 8351–8356 (2014).

    ADS  MathSciNet  Article  Google Scholar 

  21. 21

    Gao, J., Buldyrev, S. V., Stanley, H. E. & Havlin, S. Networks formed from interdependent networks. Nature Phys. 8, 40–48 (2012).

    ADS  Article  Google Scholar 

  22. 22

    Son, S-W., Bizhani, G., Christensen, C., Grassberger, P. & Paczuski, M. Percolation theory on interdependent networks based on epidemic spreading. Europhys. Lett. 97, 16006 (2012).

    ADS  Article  Google Scholar 

  23. 23

    Radicchi, F. Driving interconnected networks to supercriticality. Phys. Rev. X 4, 021014 (2014).

    Google Scholar 

  24. 24

    Reis, S. D. et al. Avoiding catastrophic failure in correlated networks of networks. Nature Phys. 10, 762–767 (2014).

    ADS  Article  Google Scholar 

  25. 25

    Hashimoto, K-i. Automorphic Forms and Geometry of Arithmetic Varieties 211–280 (Kinokuniya Company Ltd., 1989).

    Google Scholar 

  26. 26

    Krzakala, F. et al. Spectral redemption in clustering sparse networks. Proc. Natl Acad. Sci. USA 110, 20935–20940 (2013).

    ADS  MathSciNet  Article  Google Scholar 

  27. 27

    Hu, Y. et al. Percolation of interdependent networks with intersimilarity. Phys. Rev. E 88, 052805 (2013).

    ADS  Article  Google Scholar 

  28. 28

    Cellai, D., López, E., Zhou, J., Gleeson, J. P. & Bianconi, G. Percolation in multiplex networks with overlap. Phys. Rev. E 88, 052811 (2013).

    ADS  Article  Google Scholar 

  29. 29

    Min, B., Lee, S., Lee, K-M. & Goh, K-I. Link overlap, viability, and mutual percolation in multiplex networks. Chaos Solitons Fractals 72, 49–58 (2015).

    ADS  MathSciNet  Article  Google Scholar 

  30. 30

    Melnik, S., Hackett, A., Porter, M. A., Mucha, P. J. & Gleeson, J. P. The unreasonable effectiveness of tree-based theory for networks with clustering. Phys. Rev. E 83, 036112 (2011).

    ADS  MathSciNet  Article  Google Scholar 

  31. 31

    Schneider, C. M., Araújo, N. A. M. & Herrmann, H. J. Algorithm to determine the percolation largest component in interconnected networks. Phys. Rev. E 87, 043302 (2013).

    ADS  Article  Google Scholar 

  32. 32

    Hwang, S., Choi, S., Lee, D. & Kahng, B. Efficient algorithm to compute mutually connected components in interdependent networks. Phys. Rev. E 91, 022814 (2015).

    ADS  Article  Google Scholar 

  33. 33

    Pittel, B., Spencer, J. & Wormald, N. Sudden emergence of a giant k-core in a random graph. J. Comb. Theory B 67, 111–151 (1996).

    MathSciNet  Article  Google Scholar 

  34. 34

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

    ADS  Article  Google Scholar 

  35. 35

    Bianconi, G. & Dorogovtsev, S. N. Percolation in networks of networks with random matching of nodes in different layers. Preprint at http://arXiv.org/abs/1411.4160 (2014).

  36. 36

    Stark, C. et al. Biogrid: A general repository for interaction datasets. Nucleic Acids Res. 34, D535–D539 (2006).

    Article  Google Scholar 

  37. 37

    De Domenico, M., Porter, M. A. & Arenas, A. Muxviz: A tool for multilayer analysis and visualization of networks. J. Complex Netw. 3, 159–176 (2015).

    Article  Google Scholar 

  38. 38

    Guimerà, R., Mossa, S., Turtschi, A. & Amaral, L. N. The worldwide air transportation network: Anomalous centrality, community structure, and cities’ global roles. Proc. Natl Acad. Sci. USA 102, 7794–7799 (2005).

    ADS  MathSciNet  Article  Google Scholar 

  39. 39

    Colizza, V., Barrat, A., Barthélemy, M. & Vespignani, A. The role of the airline transportation network in the prediction and predictability of global epidemics. Proc. Natl Acad. Sci. USA 103, 2015–2020 (2006).

    ADS  Article  Google Scholar 

  40. 40

    TranStats (United States Department of Transportation, accessed 18 January 2015); http://www.transtats.bts.gov

Download references

Acknowledgements

The author thanks C. Castellano and A. Flammini for comments and suggestions.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Filippo Radicchi.

Ethics declarations

Competing interests

The author declares no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 592 kb)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Radicchi, F. Percolation in real interdependent networks. Nature Phys 11, 597–602 (2015). https://doi.org/10.1038/nphys3374

Download citation

Further reading

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing