Vulnerability of Interdependent Networks and Networks of Networks

  • Michael M. DanzigerEmail author
  • Louis M. Shekhtman
  • Amir Bashan
  • Yehiel Berezin
  • Shlomo Havlin
Part of the Understanding Complex Systems book series (UCS)


Networks interact with one another in a variety of ways. Even though increased connectivity between networks would tend to make the system more robust, if dependencies exist between networks, these systems are highly vulnerable to random failure or attack. Damage in one network causes damage in another. This leads to cascading failures which amplify the original damage and can rapidly lead to complete system collapse.

Understanding the system characteristics that lead to cascading failures and support their continued propagation is an important step in developing more robust systems and mitigation strategies. Recently, a number of important results have been obtained regarding the robustness of systems composed of random, clustered and spatially embedded networks.

Here we review the recent advances on the role that connectivity and dependency links play in the robustness of networks of networks. We further discuss the dynamics of cascading failures on interdependent networks, including cascade lifetime predictions and explanations of the topological properties which drive the cascade.


Random Network Single Network Dependent Network Random Failure Connectivity Link 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We acknowledge the LINC (No. 289447) and MULTIPLEX (No. 317532) EU projects, the Deutsche Forschungsgemeinschaft (DFG), the Israel Science Foundation, ONR and DTRA for financial support.


  1. 1.
    Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)CrossRefMathSciNetADSzbMATHGoogle Scholar
  2. 2.
    Caldarelli, G.: Scale-Free Networks: Complex Webs in Nature and Technology. Oxford University Press, Oxford (2007)CrossRefzbMATHGoogle Scholar
  3. 3.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440–442 (1998)CrossRefADSGoogle Scholar
  4. 4.
    Amaral, L.A.N., Scala, A., Barthélemy, M., Stanley, H.E.: Classes of small-world networks. Proc. Natl. Acad. Sci. 97(21), 11149–11152 (2000)CrossRefADSGoogle Scholar
  5. 5.
    Newman, M.: Networks: An Introduction. Oxford University Press, Oxford (2010)CrossRefzbMATHGoogle Scholar
  6. 6.
    Cohen, R., Havlin, S.: Complex Networks: Structure, Robustness and Function. Cambridge University Press, New York (2010)CrossRefzbMATHGoogle Scholar
  7. 7.
    Erdős, P., Rényi, A.: On random graphs i. Publ. Math. Debrecen 6, 290 (1959)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Erdős, P., Rényi, A.: On the strength of connectedness of a random graph. Acta Mathematica Academiae Scientiarum Hungaricae 12(1–2), 261–267 (1964)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Bollobás, B.: Modern Graph Theory. Graduate Texts in Mathematics. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  10. 10.
    Pastor-Satorras, R., Vespignani, A.: Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200–3203 (2001)CrossRefADSGoogle Scholar
  11. 11.
    Goldenberg, J., Libai, B., Muller, E.: Talk of the network: a complex systems look at the underlying process of word-of-mouth. Mark. Lett. 12(3), 211–223 (2001)CrossRefGoogle Scholar
  12. 12.
    Cohen, R., Erez, K., ben Avraham, D., Havlin, S.: Resilience of the internet to random breakdowns. Phys. Rev. Lett. 85, 4626–4628 (2000)Google Scholar
  13. 13.
    Li, D., Fu, B., Wang, Y., Lu, G., Berezin, Y., Stanley, H.E., Havlin, S.: Percolation transition in dynamical traffic network with evolving critical bottlenecks. Proc. Natl. Acad. Sci. 1123(3), 669–672 (2015)CrossRefADSGoogle Scholar
  14. 14.
    Yamasaki, K., Gozolchiani, A., Havlin, S.: Climate networks around the globe are significantly affected by El Ni\(\tilde{\mathrm{n}}\) o. Phys. Rev. Lett. 100, 228501 (2008)CrossRefADSGoogle Scholar
  15. 15.
    Ludescher, J., Gozolchiani, A., Bogachev, M.I., Bunde, A., Havlin, S., Schellnhuber, H.J.: Very early warning of next El Niño. Proc. Natl. Acad. Sci. 111(6), 2064–2066 (2014)ADSGoogle Scholar
  16. 16.
    Bunde, A., Havlin, S.: Fractals and Disordered Systems. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  17. 17.
    Stauffer, D., Aharony, A.: Introduction To Percolation Theory. Taylor & Francis, London/Bristol (1994)zbMATHGoogle Scholar
  18. 18.
    Goldenfeld, N.: Lectures on Phase Transitions and the Renormalization Group. Frontiers in Physics. Addison-Wesley, Advanced Book Program, Reading (1992)zbMATHGoogle Scholar
  19. 19.
    Albert, R., Jeong, H., Barabási, A.L.: Error and attack tolerance of complex networks. Nature 406(6794), 378–382 (2000)CrossRefADSGoogle Scholar
  20. 20.
    Barthélemy, M.: Spatial networks. Phys. Rep. 499(1–3), 1–101 (2011)CrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Bianconi, G.: Statistical mechanics of multiplex networks: entropy and overlap. Phys. Rev. E 87, 062806 (2013)CrossRefADSGoogle Scholar
  22. 22.
    Nicosia, V., Bianconi, G., Latora, V., Barthelemy, M.: Growing multiplex networks. Phys. Rev. Lett. 111, 058701 (2013)CrossRefADSGoogle Scholar
  23. 23.
    De Domenico, M., Solé-Ribalta, A., Cozzo, E., Kivelä, M., Moreno, Y., Porter, M.A., Gómez, S., Arenas, A.: Mathematical formulation of multilayer networks. Phys. Rev. X 3, 041022 (2013)Google Scholar
  24. 24.
    Kivelä, M., Arenas, A., Barthelemy, M., Gleeson, J.P., Moreno, Y., Porter, M.A.: Multilayer networks. J. Complex Netw. 2(3), 203–271 (2014)CrossRefGoogle Scholar
  25. 25.
    Goldenberg, J., Shavitt, Y., Shir, E., Solomon, S.: Distributive immunization of networks against viruses using the ‘honey-pot’ architecture dimension of spatially embedded networks. Nat. Phys. 1(3), 184–188 (2005)CrossRefGoogle Scholar
  26. 26.
    Rinaldi, S., Peerenboom, J., Kelly, T.: Identifying, understanding, and analyzing critical infrastructure interdependencies. Control Syst. IEEE 21(6), 11–25 (2001)CrossRefGoogle Scholar
  27. 27.
    Hokstad, P., Utne, I., Vatn, J.: Risk and Interdependencies in Critical Infrastructures: A Guideline for Analysis. Springer Series in Reliability Engineering. Springer, London (2012)CrossRefGoogle Scholar
  28. 28.
    Buldyrev, S.V., Parshani, R., Paul, G., Stanley, H.E., Havlin, S.: Catastrophic cascade of failures in interdependent networks. Nature 464(7291), 1025–1028 (2010)CrossRefADSGoogle Scholar
  29. 29.
    Danziger, M.M., Bashan, A., Berezin, Y., Shekhtman, L.M., Havlin, S.: An introduction to interdependent networks. In: Mladenov, V., Ivanov, P. (eds.) Nonlinear Dynamics of Electronic Systems. Volume 438 of Communications in Computer and Information Science, pp. 189–202. Springer, Cham (2014)Google Scholar
  30. 30.
    Parshani, R., Buldyrev, S.V., Havlin, S.: Interdependent networks: reducing the coupling strength leads to a change from a first to second order percolation transition. Phys. Rev. Lett. 105, 048701 (2010)CrossRefADSGoogle Scholar
  31. 31.
    Gao, J., Buldyrev, S.V., Havlin, S., Stanley, H.E.: Robustness of a network of networks. Phys. Rev. Lett. 107, 195701 (2011)CrossRefADSGoogle Scholar
  32. 32.
    Kenett, D.Y., Gao, J., Huang, X., Shao, S., Vodenska, I., Buldyrev, S.V., Paul, G., Stanley, H.E., Havlin, S.: Network of interdependent networks: overview of theory and applications. In: D’Agostino, G., Scala, A. (eds.) Networks of Networks: The Last Frontier of Complexity. Understanding Complex Systems, pp. 3–36. Springer, Cham (2014)CrossRefGoogle Scholar
  33. 33.
    Foster Jr, J.S., Gjelde, E., Graham, W.R., Hermann, R.J., Kluepfel, H.M., Lawson, R.L., Soper, G.K., Wood, L.L., Woodard, J.B.: Report of the commission to assess the threat to the united states from electromagnetic pulse (emp) attack: critical national infrastructures. Technical report, DTIC Document (2008)Google Scholar
  34. 34.
    Bashan, A., Berezin, Y., Buldyrev, S.V., Havlin, S.: The extreme vulnerability of interdependent spatially embedded networks. Nat. Phys. 9, 667–672 (2013)CrossRefGoogle Scholar
  35. 35.
    D’Agostino, G., Scala, A.: Networks of Networks: The Last Frontier of Complexity. Understanding Complex Systems. Springer, Cham (2014)CrossRefGoogle Scholar
  36. 36.
    Boccaletti, S., Bianconi, G., Criado, R., Del Genio, C., Gómez-Gardeñes, J., Romance, M., Sendina-Nadal, I., Wang, Z., Zanin, M.: The structure and dynamics of multilayer networks. Phys. Rep. 544, 1–122 (2014)CrossRefMathSciNetADSGoogle Scholar
  37. 37.
    Motter, A.E.: Cascade control and defense in complex networks. Phys. Rev. Lett. 93, 098701 (2004)CrossRefADSGoogle Scholar
  38. 38.
    Dobson, I., Carreras, B.A., Lynch, V.E., Newman, D.E.: Complex systems analysis of series of blackouts: cascading failure, critical points, and self-organization. Chaos: Interdiscip. J. Nonlinear Sci. 17(2), 026103 (2007)CrossRefzbMATHGoogle Scholar
  39. 39.
    Baxter, G.J., Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: Avalanche collapse of interdependent networks. Phys. Rev. Lett. 109, 248701 (2012)CrossRefADSGoogle Scholar
  40. 40.
    Zhou, D., Bashan, A., Cohen, R., Berezin, Y., Shnerb, N., Havlin, S.: Simultaneous first- and second-order percolation transitions in interdependent networks. Phys. Rev. E 90, 012803 (2014)CrossRefADSGoogle Scholar
  41. 41.
    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)CrossRefADSGoogle Scholar
  42. 42.
    Watanabe, S., Kabashima, Y.: Cavity-based robustness analysis of interdependent networks: influences of intranetwork and internetwork degree-degree correlations. Phys. Rev. E 89, 012808 (2014)CrossRefADSGoogle Scholar
  43. 43.
    Zhou, D., Gao, J., Stanley, H.E., Havlin, S.: Percolation of partially interdependent scale-free networks. Phys. Rev. E 87, 052812 (2013)CrossRefADSGoogle Scholar
  44. 44.
    Leicht, E.A., D’Souza, R.M.: Percolation on interacting networks. ArXiv e-prints (2009). Google Scholar
  45. 45.
    Hu, Y., Ksherim, B., Cohen, R., Havlin, S.: Percolation in interdependent and interconnected networks: abrupt change from second- to first-order transitions. Phys. Rev. E 84, 066116 (2011)CrossRefADSGoogle Scholar
  46. 46.
    Parshani, R., Buldyrev, S.V., Havlin, S.: Critical effect of dependency groups on the function of networks. Proc. Natl. Acad. Sci. 108(3), 1007–1010 (2011)CrossRefADSGoogle Scholar
  47. 47.
    Bashan, A., Parshani, R., Havlin, S.: Percolation in networks composed of connectivity and dependency links. Phys. Rev. E 83, 051127 (2011)CrossRefADSGoogle Scholar
  48. 48.
    Zhao, J.H., Zhou, H.J., Liu, Y.Y.: Inducing effect on the percolation transition in complex networks. Nat. Commun. 4, 2412 (2013)ADSGoogle Scholar
  49. 49.
    Shao, J., Buldyrev, S.V., Havlin, S., Stanley, H.E.: Cascade of failures in coupled network systems with multiple support-dependence relations. Phys. Rev. E 83, 036116 (2011)CrossRefMathSciNetADSGoogle Scholar
  50. 50.
    Gao, J., Buldyrev, S.V., Stanley, H.E., Havlin, S.: Networks formed from interdependent networks. Nat. Phys. 8(1), 40–48 (2012)CrossRefGoogle Scholar
  51. 51.
    Gao, J., Buldyrev, S.V., Stanley, H.E., Xu, X., Havlin, S.: Percolation of a general network of networks. Phys. Rev. E 88, 062816 (2013)CrossRefADSGoogle Scholar
  52. 52.
    Gao, J., Buldyrev, S.V., Havlin, S., Stanley, H.E.: Robustness of a network formed by n interdependent networks with a one-to-one correspondence of dependent nodes. Phys. Rev. E 85, 066134 (2012)CrossRefADSGoogle Scholar
  53. 53.
    Zhou, D., Stanley, H.E., D’Agostino, G., Scala, A.: Assortativity decreases the robustness of interdependent networks. Phys. Rev. E 86, 066103 (2012)CrossRefADSGoogle Scholar
  54. 54.
    Parshani, R., Rozenblat, C., Ietri, D., Ducruet, C., Havlin, S.: Inter-similarity between coupled networks. EPL (Europhys. Lett.) 92(6), 68002 (2010)Google Scholar
  55. 55.
    Buldyrev, S.V., Shere, N.W., Cwilich, G.A.: Interdependent networks with identical degrees of mutually dependent nodes. Phys. Rev. E 83, 016112 (2011)CrossRefMathSciNetADSGoogle Scholar
  56. 56.
    Valdez, L.D., Macri, P.A., Stanley, H.E., Braunstein, L.A.: Triple point in correlated interdependent networks. Phys. Rev. E 88, 050803 (2013)CrossRefADSGoogle Scholar
  57. 57.
    Lee, K.M., Kim, J.Y., Cho, W.k., Goh, K.I., Kim, I.M.: Correlated multiplexity and connectivity of multiplex random networks. New J. Phys. 14(3), 033027 (2012)Google Scholar
  58. 58.
    Cellai, D., López, E., Zhou, J., Gleeson, J.P., Bianconi, G.: Percolation in multiplex networks with overlap. Phys. Rev. E 88, 052811 (2013)CrossRefADSGoogle Scholar
  59. 59.
    Li, M., Liu, R.R., Jia, C.X., Wang, B.H.: Critical effects of overlapping of connectivity and dependence links on percolation of networks. New J. Phys. 15(9), 093013 (2013)CrossRefADSGoogle Scholar
  60. 60.
    Hu, Y., Zhou, D., Zhang, R., Han, Z., Rozenblat, C., Havlin, S.: Percolation of interdependent networks with intersimilarity. Phys. Rev. E 88, 052805 (2013)CrossRefADSGoogle Scholar
  61. 61.
    Newman, M.E.J.: Random graphs with clustering. Phys. Rev. Lett. 103, 058701 (2009)CrossRefADSGoogle Scholar
  62. 62.
    Huang, X., Shao, S., Wang, H., Buldyrev, S.V., Eugene Stanley, H., Havlin, S.: The robustness of interdependent clustered networks. EPL 101(1), 18002 (2013)CrossRefADSGoogle Scholar
  63. 63.
    Shao, S., Huang, X., Stanley, H.E., Havlin, S.: Robustness of a partially interdependent network formed of clustered networks. Phys. Rev. E 89, 032812 (2014)CrossRefADSGoogle Scholar
  64. 64.
    Shekhtman, L.M., Berezin, Y., Danziger, M.M., Havlin, S.: Robustness of a network formed of spatially embedded networks. Phys. Rev. E 90, 012809 (2014)CrossRefADSGoogle Scholar
  65. 65.
    Gao, J., Li, D., Havlin, S.: From a single network to a network of networks. Natl. Sci. Rev. 1(3), 346–356 (2014)CrossRefGoogle Scholar
  66. 66.
    Danziger, M.M., Bashan, A., Berezin, Y., Havlin, S.: Percolation and cascade dynamics of spatial networks with partial dependency. J. Complex Netw. 2, 460–474 (2014)CrossRefGoogle Scholar
  67. 67.
    Rosato, V., Issacharoff, L., Tiriticco, F., Meloni, S., Porcellinis, S.D., Setola, R.: Modelling interdependent infrastructures using interacting dynamical models. Int. J. Crit. Infrastruct. 4(1/2), 63 (2008)CrossRefGoogle Scholar
  68. 68.
    Hines, P., Blumsack, S., Cotilla Sanchez, E., Barrows, C.: The topological and electrical structure of power grids. In: 2010 43rd Hawaii International Conference on System Sciences (HICSS), Koloa, pp. 1–10 (2010)Google Scholar
  69. 69.
    Li, D., Kosmidis, K., Bunde, A., Havlin, S.: Dimension of spatially embedded networks. Nat. Phys. 7(6), 481–484 (2011)CrossRefGoogle Scholar
  70. 70.
    Li, W., Bashan, A., Buldyrev, S.V., Stanley, H.E., Havlin, S.: Cascading failures in interdependent lattice networks: the critical role of the length of dependency links. Phys. Rev. Lett. 108, 228702 (2012)CrossRefADSGoogle Scholar
  71. 71.
    Berezin, Y., Bashan, A., Danziger, M.M., Li, D., Havlin, S.: Localized attacks on spatially embedded networks with dependencies. Sci. Rep. 5, 8934 (2015)CrossRefADSGoogle Scholar
  72. 72.
    Danziger, M.M., Bashan, A., Berezin, Y., Havlin, S.: Interdependent spatially embedded networks: dynamics at percolation threshold. In: 2013 International Conference on Signal-Image Technology Internet-Based Systems (SITIS), Kyoto, pp. 619–625 (2013)Google Scholar
  73. 73.
    Nienhuis, B.: Analytical calculation of two leading exponents of the dilute potts model. J. Phys. A: Math. Gen. 15(1), 199 (1982)CrossRefMathSciNetADSGoogle Scholar
  74. 74.
    Danziger, Michael M., Bashan, Amir, Havlin, Shlomo: Interdependent resistor networks with process-based dependency. New J. Phys. 17(4), 043046 (2015)Google Scholar
  75. 75.
    Danziger, M.M., Shekhtman, L.M., Berezin, Y., Havlin, S.: Two distinct transitions in spatially embedded multiplex networks. ArXiv e-prints (2015).
  76. 76.
    Huang, X., Gao, J., Buldyrev, S.V., Havlin, S., Stanley, H.E.: Robustness of interdependent networks under targeted attack. Phys. Rev. E 83, 065101 (2011)CrossRefADSGoogle Scholar
  77. 77.
    Dong, G., Gao, J., Du, R., Tian, L., Stanley, H.E., Havlin, S.: Robustness of network of networks under targeted attack. Phys. Rev. E 87, 052804 (2013)CrossRefADSGoogle Scholar
  78. 78.
    Schneider, C.M., Yazdani, N., Araújo, N.A., Havlin, S., Herrmann, H.J.: Towards designing robust coupled networks. Sci. Rep. 3, 1969 (2013)ADSGoogle Scholar
  79. 79.
    Valdez, L.D., Macri, P.A., Braunstein, L.A.: A triple point induced by targeted autonomization on interdependent scale-free networks. J. Phys. A: Math. Theor. 47(5), 055002 (2014)CrossRefADSzbMATHGoogle Scholar
  80. 80.
    Holland, P.W., Laskey, K.B., Leinhardt, S.: Stochastic blockmodels: first steps. Soc. Netw. 5(2), 109–137 (1983)CrossRefMathSciNetGoogle Scholar
  81. 81.
    Karrer, B., Newman, M.E.J.: Stochastic blockmodels and community structure in networks. Phys. Rev. E 83, 016107 (2011)CrossRefMathSciNetADSGoogle Scholar
  82. 82.
    Peixoto, T.P., Bornholdt, S.: Evolution of robust network topologies: emergence of central backbones. Phys. Rev. Lett. 109, 118703 (2012)CrossRefADSGoogle Scholar
  83. 83.
    Stippinger, M., Kertész, J.: Enhancing resilience of interdependent networks by healing. Phys. A: Stat. Mech. Appl. 416(0), 481–487 (2014)CrossRefGoogle Scholar
  84. 84.
    Agarwal, P.K., Efrat, A., Ganjugunte, S., Hay, D., Sankararaman, S., Zussman, G.: The resilience of WDM networks to probabilistic geographical failures. In: 2011 Proceedings IEEE INFOCOM, Shanghai, pp. 1521–1529 (2011)Google Scholar
  85. 85.
    Bernstein, A., Bienstock, D., Hay, D., Uzunoglu, M., Zussman, G.: Sensitivity analysis of the power grid vulnerability to large-scale cascading failures. SIGMETRICS Perform. Eval. Rev. 40(3), 33–37 (2012)CrossRefGoogle Scholar
  86. 86.
    Shao, S., Huang, X., Stanley, H.E., Havlin, S.: Percolation of localized attack on complex networks. New J. Phys. 17(2), 023049 (2015)CrossRefMathSciNetADSGoogle Scholar
  87. 87.
    Son, S.W., Bizhani, G., Christensen, C., Grassberger, P., Paczuski, M.: Percolation theory on interdependent networks based on epidemic spreading. EPL (Europhys. Lett.) 97(1), 16006 (2012)Google Scholar
  88. 88.
    Saumell-Mendiola, A., Serrano, M.Á., Boguñá, M.: Epidemic spreading on interconnected networks. Phys. Rev. E 86, 026106 (2012)CrossRefADSGoogle Scholar
  89. 89.
    Dickison, M., Havlin, S., Stanley, H.E.: Epidemics on interconnected networks. Phys. Rev. E 85, 066109 (2012)CrossRefADSGoogle Scholar
  90. 90.
    Wang, H., Li, Q., D’Agostino, G., Havlin, S., Stanley, H.E., Van Mieghem, P.: Effect of the interconnected network structure on the epidemic threshold. Phys. Rev. E 88, 022801 (2013)CrossRefADSGoogle Scholar
  91. 91.
    Erez, T., Hohnisch, M., Solomon, S.: Statistical economics on multi-variable layered networks. In: Salzano, M., Kirman, A. (eds.) Economics: Complex Windows. New Economic Windows, pp. 201–217. Springer, Milan (2005)CrossRefGoogle Scholar
  92. 92.
    Huang, X., Vodenska, I., Havlin, S., Stanley, H.E.: Cascading failures in bi-partite graphs: model for systemic risk propagation. Sci. Rep. 3, 1219 (2013)ADSGoogle Scholar
  93. 93.
    Li, W., Kenett, D.Y., Yamasaki, K., Stanley, H.E., Havlin, S.: Ranking the economic importance of countries and industries. ArXiv e-prints (2014). Google Scholar
  94. 94.
    Bashan, A., Bartsch, R.P., Kantelhardt, J.W., Havlin, S., Ivanov, P.C.: Network physiology reveals relations between network topology and physiological function. Nat. Commun. 3, 702 (2012)CrossRefADSGoogle Scholar
  95. 95.
    Pocock, M.J.O., Evans, D.M., Memmott, J.: The robustness and restoration of a network of ecological networks. Science 335(6071), 973–977 (2012)CrossRefADSGoogle Scholar
  96. 96.
    Donges, J., Schultz, H., Marwan, N., Zou, Y., Kurths, J.: Investigating the topology of interacting networks. Eur. Phys. J. B 84(4), 635–651 (2011)CrossRefADSGoogle Scholar
  97. 97.
    Reis, S.D.S., Hu, Y., Babino, A., Andrade Jr, J.S., Canals, S., Sigman, M., Makse, H.A.: Avoiding catastrophic failure in correlated networks of networks. Nat. Phys. 10(10), 762–767 (2014)CrossRefGoogle Scholar
  98. 98.
    Morris, R.G., Barthelemy, M.: Transport on coupled spatial networks. Phys. Rev. Lett. 109, 128703 (2012)CrossRefADSGoogle Scholar
  99. 99.
    Majdandzic, A., Braunstein, L.A., Curme, C., Vodenska, I., Levy-Carciente, S., Stanley, H.E., Havlin, S.: Multiple tipping points and optimal repairing in interacting networks. ArXiv e-prints (2015). Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Michael M. Danziger
    • 1
    Email author
  • Louis M. Shekhtman
    • 1
  • Amir Bashan
    • 2
  • Yehiel Berezin
    • 1
  • Shlomo Havlin
    • 1
  1. 1.Department of PhysicsBar Ilan UniversityRamat GanIsrael
  2. 2.Channing Division of Network MedicineBrigham Women’s Hospital and Harvard Medical SchoolBostonUSA

Personalised recommendations