Abstract
Many real-world complex systems are represented not by single networks but rather by sets of interdependent networks. In these specific networks, vertices in each network mutually depend on vertices in other networks. In the simplest representative case, interdependent networks are equivalent to the so-called multiplex networks containing vertices of one sort but several kinds of edges. Connectivity properties of these networks and their robustness against damage differ sharply from ordinary networks. Connected components in ordinary networks are naturally generalized to viable clusters in multiplex networks whose vertices are connected by paths passing over each individual sort of their edges. We examine the robustness of the giant viable cluster to random damage. We show that random damage to these systems can lead to the avalanche collapse of the viable cluster, and that this collapse is a hybrid phase transition combining a discontinuity and the critical singularity. For this transition we identify latent critical clusters associated with the avalanches triggered by a removal of single vertices. Divergence of their mean size signals the approach to the hybrid phase transition from one side, while there are no critical precursors on the other side. We find that this discontinuous transition occurs in scale-free multiplex networks whenever the mean degree of at least one of the interdependent networks does not diverge.
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References
Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: Critical phenomena in complex networks. Rev. Mod. Phys. 80, 1275–1335 (2008)
Dorogovtsev, S.N., Mendes, J., Samukhin, A., Zyuzin, A.: Organization of modular networks. Phys. Rev. E 78, 056106 (2008)
Leicht, E.A., D’Souza, R.M.: Percolation on interacting networks (2009). arXiv:0908.0894
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)
Pocock, M.J.O., Evans, D.M., Memmott, J.: The robustness and restoration of a network of ecological networks. Science 335, 973–976 (2012)
Kurant, M., Thiran, P.: Layered complex networks. Phys. Rev. Lett. 96, 138701 (2006)
Rinaldi, S.M., Peerenboom, J.P., Kelly, T.K.: Identifying, understanding, and analyzing critical infrastructure interdependencies. IEEE Control Syst. Mag. 21, 11–25 (2001)
Dueñas, L., Cragin, J.I., Goodno, B.J.: Seismic response of critical interdependent networks. Earthq. Eng. Struct. Dyn. 36, 285–306 (2007)
Poljans̆ek, K., Bono, F., Gutiérrez, E.: Seismic risk assessment of interdependent critical infrastructure systems: The case of european gas and electricity networks. Earthq. Eng. Struct. Dyn. 41, 61–79 (2012)
Buldyrev, S.V., Parshani, R., Paul, G., Stanley, H.E., Havlin, S.: Catastrophic casdade of failures in interdependent networks. Nature 464, 08932 (2010)
Gao, J., Buldyrev, S.V., Havlin, S., Stanley, H.E.: Robustness of a network of networks. Phys. Rev. Lett. 107, 195701 (2011)
Son, S.W., Bizhani, G., Christensen, C., Grassberger, P., Paczuski, M.: Percolation theory on interdependent networks based on epidemic spreading. EPL 97, 16006 (2012)
Baxter, G.J., Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: Avalanche collapse of interdependent networks. Phys. Rev. Lett. 109, 248701 (2012)
Baxter, G.J., Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: Bootstrap percolation on complex networks. Phys. Rev. E 82, 011103 (2010)
Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: \(k\)-core organisation of complex networks. Phys. Rev. Lett. 96, 040601 (2006)
Eckmann, J.P., Feinerman, O., Gruendlinger, L., Moses, E., Soriano, J., Tlusty, T.: The physics of living neural networks. Phys. Rep. 449, 54–76 (2007)
Dai, L., Vorselen, D., Korolev, K.S., Gore, J.: Generic indicators for loss of resilience before a tipping point leading to population collapse. Science 336, 1175–7 (2012)
Klimek, P., Thurner, S., Hanel, R.: Pruning the tree of life: \(k\)-core percolation as selection mechanism. J. Theoret. Biol. 256, 142–146 (2009)
Sellitto, M., Biroli, G., Toninell, C.: Facilitated spin models on bethe lattice: Bootstrap percolation, mode-coupling transition and glassy dynamics. Europhys. Lett. 69(4), 496–502 (2005)
Toninelli, C., Biroli, G., Fisher, D.S.: Jamming percolation and glass transitions in lattice models. Phys. Rev. Lett. 96(3), 035702 (2006)
Sabhapandit, S., Dhar, D., Shukla, P.: Hysteresis in the random-field Ising model and bootstrap percolation. Phys. Rev. Lett. 88, 197202 (2002)
Stauffer, D., Aharony, A.: Introduction to percolation theory, 2nd edn. Taylor and Francis, London (1992)
Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: \(k\)-core architecture and \(k\)-core percolation on complex networks. Physica D 224, 7–19 (2006)
Cohen, R., ben Avraham, D., Havlin, S.: Percolation critical exponents in scale-free networks. Phys. Rev. E 66, 036113 (2002)
Albert, R., Jeong, H., Barabási, A.L.: Error and attack tolerance in complex networks. Nature 401, 378–382 (2000)
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, 5468–5471 (2000)
Acknowledgments
This work was partially supported by FET IP Project MULTIPLEX 317532 and by the PTDC projects SAU-NEU/103904/2008, FIS/108476/2008, MAT/114515/2009 and PEst-C/CTM/LA0025/2011, and post-doctoral fellowship SFRH/BPD/74040/2010.
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Baxter, G.J., Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F. (2014). Avalanches in Multiplex and Interdependent Networks. In: D'Agostino, G., Scala, A. (eds) Networks of Networks: The Last Frontier of Complexity. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-03518-5_2
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DOI: https://doi.org/10.1007/978-3-319-03518-5_2
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