Skip to main content
SpringerLink
Log in
Book cover

Interconnected Networks pp 79–99Cite as

Vulnerability of Interdependent Networks and Networks of Networks

  • Chapter
  • First Online:

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

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.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Caldarelli, G.: Scale-Free Networks: Complex Webs in Nature and Technology. Oxford University Press, Oxford (2007)

    Book  MATH  Google Scholar 

  3. Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440–442 (1998)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  5. Newman, M.: Networks: An Introduction. Oxford University Press, Oxford (2010)

    Book  MATH  Google Scholar 

  6. Cohen, R., Havlin, S.: Complex Networks: Structure, Robustness and Function. Cambridge University Press, New York (2010)

    Book  MATH  Google Scholar 

  7. Erdős, P., Rényi, A.: On random graphs i. Publ. Math. Debrecen 6, 290 (1959)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bollobás, B.: Modern Graph Theory. Graduate Texts in Mathematics. Springer, New York (1998)

    Book  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    ADS  Google Scholar 

  16. Bunde, A., Havlin, S.: Fractals and Disordered Systems. Springer, New York (1991)

    Book  MATH  Google Scholar 

  17. Stauffer, D., Aharony, A.: Introduction To Percolation Theory. Taylor & Francis, London/Bristol (1994)

    MATH  Google Scholar 

  18. Goldenfeld, N.: Lectures on Phase Transitions and the Renormalization Group. Frontiers in Physics. Addison-Wesley, Advanced Book Program, Reading (1992)

    MATH  Google Scholar 

  19. Albert, R., Jeong, H., Barabási, A.L.: Error and attack tolerance of complex networks. Nature 406(6794), 378–382 (2000)

    Article  ADS  Google Scholar 

  20. Barthélemy, M.: Spatial networks. Phys. Rep. 499(1–3), 1–101 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  21. Bianconi, G.: Statistical mechanics of multiplex networks: entropy and overlap. Phys. Rev. E 87, 062806 (2013)

    Article  ADS  Google Scholar 

  22. Nicosia, V., Bianconi, G., Latora, V., Barthelemy, M.: Growing multiplex networks. Phys. Rev. Lett. 111, 058701 (2013)

    Article  ADS  Google Scholar 

  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. Kivelä, M., Arenas, A., Barthelemy, M., Gleeson, J.P., Moreno, Y., Porter, M.A.: Multilayer networks. J. Complex Netw. 2(3), 203–271 (2014)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  26. Rinaldi, S., Peerenboom, J., Kelly, T.: Identifying, understanding, and analyzing critical infrastructure interdependencies. Control Syst. IEEE 21(6), 11–25 (2001)

    Article  Google Scholar 

  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)

    Book  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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. 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)

    Article  ADS  Google Scholar 

  31. Gao, J., Buldyrev, S.V., Havlin, S., Stanley, H.E.: Robustness of a network of networks. Phys. Rev. Lett. 107, 195701 (2011)

    Article  ADS  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Bashan, A., Berezin, Y., Buldyrev, S.V., Havlin, S.: The extreme vulnerability of interdependent spatially embedded networks. Nat. Phys. 9, 667–672 (2013)

    Article  Google Scholar 

  35. D’Agostino, G., Scala, A.: Networks of Networks: The Last Frontier of Complexity. Understanding Complex Systems. Springer, Cham (2014)

    Book  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  37. Motter, A.E.: Cascade control and defense in complex networks. Phys. Rev. Lett. 93, 098701 (2004)

    Article  ADS  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  43. Zhou, D., Gao, J., Stanley, H.E., Havlin, S.: Percolation of partially interdependent scale-free networks. Phys. Rev. E 87, 052812 (2013)

    Article  ADS  Google Scholar 

  44. Leicht, E.A., D’Souza, R.M.: Percolation on interacting networks. ArXiv e-prints (2009). http://arxiv.org/abs/0907.0894

    Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  47. Bashan, A., Parshani, R., Havlin, S.: Percolation in networks composed of connectivity and dependency links. Phys. Rev. E 83, 051127 (2011)

    Article  ADS  Google Scholar 

  48. Zhao, J.H., Zhou, H.J., Liu, Y.Y.: Inducing effect on the percolation transition in complex networks. Nat. Commun. 4, 2412 (2013)

    ADS  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  50. Gao, J., Buldyrev, S.V., Stanley, H.E., Havlin, S.: Networks formed from interdependent networks. Nat. Phys. 8(1), 40–48 (2012)

    Article  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  53. Zhou, D., Stanley, H.E., D’Agostino, G., Scala, A.: Assortativity decreases the robustness of interdependent networks. Phys. Rev. E 86, 066103 (2012)

    Article  ADS  Google Scholar 

  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. Buldyrev, S.V., Shere, N.W., Cwilich, G.A.: Interdependent networks with identical degrees of mutually dependent nodes. Phys. Rev. E 83, 016112 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  61. Newman, M.E.J.: Random graphs with clustering. Phys. Rev. Lett. 103, 058701 (2009)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  65. Gao, J., Li, D., Havlin, S.: From a single network to a network of networks. Natl. Sci. Rev. 1(3), 346–356 (2014)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. Li, D., Kosmidis, K., Bunde, A., Havlin, S.: Dimension of spatially embedded networks. Nat. Phys. 7(6), 481–484 (2011)

    Article  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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. Nienhuis, B.: Analytical calculation of two leading exponents of the dilute potts model. J. Phys. A: Math. Gen. 15(1), 199 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  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. Danziger, M.M., Shekhtman, L.M., Berezin, Y., Havlin, S.: Two distinct transitions in spatially embedded multiplex networks. ArXiv e-prints (2015). http://arxiv.org/abs/1505.01688

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  78. Schneider, C.M., Yazdani, N., Araújo, N.A., Havlin, S., Herrmann, H.J.: Towards designing robust coupled networks. Sci. Rep. 3, 1969 (2013)

    ADS  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  80. Holland, P.W., Laskey, K.B., Leinhardt, S.: Stochastic blockmodels: first steps. Soc. Netw. 5(2), 109–137 (1983)

    Article  MathSciNet  Google Scholar 

  81. Karrer, B., Newman, M.E.J.: Stochastic blockmodels and community structure in networks. Phys. Rev. E 83, 016107 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  82. Peixoto, T.P., Bornholdt, S.: Evolution of robust network topologies: emergence of central backbones. Phys. Rev. Lett. 109, 118703 (2012)

    Article  ADS  Google Scholar 

  83. Stippinger, M., Kertész, J.: Enhancing resilience of interdependent networks by healing. Phys. A: Stat. Mech. Appl. 416(0), 481–487 (2014)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  86. Shao, S., Huang, X., Stanley, H.E., Havlin, S.: Percolation of localized attack on complex networks. New J. Phys. 17(2), 023049 (2015)

    Article  MathSciNet  ADS  Google Scholar 

  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. Saumell-Mendiola, A., Serrano, M.Á., Boguñá, M.: Epidemic spreading on interconnected networks. Phys. Rev. E 86, 026106 (2012)

    Article  ADS  Google Scholar 

  89. Dickison, M., Havlin, S., Stanley, H.E.: Epidemics on interconnected networks. Phys. Rev. E 85, 066109 (2012)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    ADS  Google Scholar 

  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). http://arxiv.org/abs/1408.0443

    Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  Google Scholar 

  98. Morris, R.G., Barthelemy, M.: Transport on coupled spatial networks. Phys. Rev. Lett. 109, 128703 (2012)

    Article  ADS  Google Scholar 

  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). http://arxiv.org/abs/1502.00244

    Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Department of Physics, Bar Ilan University, Ramat Gan, Israel

    Michael M. Danziger, Louis M. Shekhtman, Yehiel Berezin & Shlomo Havlin

  2. Channing Division of Network Medicine, Brigham Women’s Hospital and Harvard Medical School, Boston, MA, USA

    Amir Bashan

Authors
  1. Michael M. Danziger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Louis M. Shekhtman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Amir Bashan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yehiel Berezin
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Shlomo Havlin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael M. Danziger .

Editor information

Editors and Affiliations

  1. Systems Design, ETH Zürich, Zürich, Switzerland

    Antonios Garas

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Danziger, M.M., Shekhtman, L.M., Bashan, A., Berezin, Y., Havlin, S. (2016). Vulnerability of Interdependent Networks and Networks of Networks. In: Garas, A. (eds) Interconnected Networks. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-23947-7_5

Download citation

Publish with us

Policies and ethics

Search

Navigation