Space Geometry Effect over the Internet as a Physical-Logical Interdependent Network

  • Ivana BachmannEmail author
  • Francisco Sanhueza
  • Javier Bustos-Jiménez
Conference paper
Part of the Springer Proceedings in Complexity book series (SPCOM)


In this article we study the Internet’s robustness under physical node failures, given that the physical layer is built over spaces with geometry/shape restrictions. This is of special interest for countries prone to natural catastrophes, and long and narrow geographies such as Chile and Japan. We model the Internet as an interdependent network composed of the Internet’s physical layer (Internet backbone) and he Internet’s logical layer (Autonomous System level network) coupled. Here, the robustness is tested by measuring the amount of functional nodes on the logical network after randomly removing physical nodes. In this work, we tested six different spatially constrained network models to generate the Internet’s physical layer (Yao graphs, geometric preferential attachment, Erdős-Rényi, n-nearest neighbours, Gabriel graphs, and Modified relative neighbourhood model), and three different geometries with width to lengths ratios going from a square geometry to a Chile-like space geometry. Additionally, we study the relation between the amount of physical edges and the Internet’s robustness. Our findings suggest that both: the edge addition strategy (i.e. the physical network model used) and the amount of physical edges play an important role on the Internet’s robustness. Our results also suggest that Internet based interdependent systems whose robustness is affected by the space geometry (geometry-sensitive) can become more robust by randomly adding few edges. Furthermore, these interdependent systems can become geometry-insensitive after the edge addition, meaning that the robustness of the interdependent system is no longer affected by the space geometry.


Interdependent networks Robustness Spatial networks Internet 



This work was partially funded by CONICYT Doctorado Nacional 21170165.


  1. 1.
    Adler, C.O., Dagli, C.H.: Study of the use of a genetic algorithm to improve networked system-of-systems resilience. Proc. Comput. Sci. 36, 49–56 (2014)CrossRefGoogle Scholar
  2. 2.
    Autonomous system (1930). Accessed 29 Sep 2019
  3. 3.
    Bachmann, I., Bustos-Jiménez, J.: Improving the Chilean Internet robustness: increase the interdependencies or change the shape of the country? In: International Conference on Complex Networks and Their Applications, pp. 646–657. Springer, Berlin (2017)Google Scholar
  4. 4.
    Berezin, Y., Bashan, A., Danziger, M.M., Li, D., Havlin, S.: Localized attacks on spatially embedded networks with dependencies. Sci. Rep. 5, 8934 (2015)ADSCrossRefGoogle Scholar
  5. 5.
    Border gateway protocol (2006). Accessed 29 Sep 2019
  6. 6.
    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)ADSCrossRefGoogle Scholar
  7. 7.
    Cai, Y., Li, Y., Cao, Y., Li, W., Zeng, X.: Modeling and impact analysis of interdependent characteristics on cascading failures in smart grids. Int. J. Electr. Power Energy Syst. 89, 106–114 (2017)CrossRefGoogle Scholar
  8. 8.
    Carlson, J.M., Doyle, J.: Complexity and robustness. Proc. Natl. Acad. Sci. 99(suppl. 1), 2538–2545 (2002)ADSCrossRefGoogle Scholar
  9. 9.
    Chattopadhyay, S., Dai, H.: Towards optimal link patterns for robustness of interdependent networks against cascading failures. In: 2015 IEEE Global Communications Conference (GLOBECOM), pp. 1–6. IEEE, Piscataway (2015)Google Scholar
  10. 10.
    Chen, Z., Wu, J., Xia, Y., Zhang, X.: Robustness of interdependent power grids and communication networks: a complex network perspective. IEEE Trans. Circ. Syst. Express Briefs 65(1), 115–119 (2017)ADSCrossRefGoogle Scholar
  11. 11.
    Cowie, J.H., Ogielski, A.T., Premore, B., Smith, E.A., Underwood, T.: Impact of the 2003 blackouts on internet communications. Preliminary Report, Renesys Corporation (updated March 1, 2004) (2003)Google Scholar
  12. 12.
    Danziger, M.M., Bashan, A., Berezin, Y., Havlin, S. (2013). Interdependent spatially embedded networks: dynamics at percolation threshold. In: 2013 International Conference on Signal-Image Technology & Internet-Based Systems (SITIS), pp. 619–625. IEEE, Piscataway (2013)Google Scholar
  13. 13.
    Dong, G., Tian, L., Du, R., Fu, M., Stanley, H.E.: Analysis of percolation behaviors of clustered networks with partial support-dependence relations. Phys. A: Stat. Mech. Appl. 394, 370–378 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Eppstein, D., Paterson, M.S., Yao, F.F.: On nearest-neighbor graphs. Discrete Comput. Geom.17(3), 263–282 (1997)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Erdős, P.,Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci 5(1), 17–60 (1960)Google Scholar
  16. 16.
    Fabrikant, A., Koutsoupias, E., Papadimitriou, C.H.: Heuristically optimized trade-offs: a new paradigm for power laws in the internet. In: International Colloquium on Automata, Languages, and Programming, pp. 110–122. Springer, Berlin (2002)CrossRefGoogle Scholar
  17. 17.
    Faloutsos, M., Faloutsos, P., Faloutsos, C.: On power-law relationships of the internet topology. In: ACM SIGCOMM Computer Communication Review, vol. 29, pp. 251–262. ACM, New York (1999)CrossRefGoogle Scholar
  18. 18.
    Flaxman, A.D., Frieze, A.M., Vera, J.: A geometric preferential attachment model of networks. Internet Math. 3(2), 187–205 (2006)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gabriel, K.R., Sokal, R.R.: A new statistical approach to geographic variation analysis. Syst. Zool. 18(3), 259–278 (1969)CrossRefGoogle Scholar
  20. 20.
    Han, Y., Li, Z., Guo, C., Tang, Y.: Improved percolation theory incorporating power flow analysis to model cascading failures in cyber-physical power system. In: Power and Energy Society General Meeting (PESGM), 2016, pp. 1–5. IEEE, Piscataway (2016)Google Scholar
  21. 21.
    Huang, X., Gao, J., Buldyrev, S.V., Havlin, S., Stanley, H.E.: Robustness of interdependent networks under targeted attack. Phys. Rev. E 83(6), 065101 (2011)ADSCrossRefGoogle Scholar
  22. 22.
    Huang, Z., Wang, C., Nayak, A., Stojmenovic, I.: Small cluster in cyber physical systems: Network topology, interdependence and cascading failures. IEEE Trans. Parallel Distrib. Syst. 26(8), 2340–2351 (2014)CrossRefGoogle Scholar
  23. 23.
    Kazawa, Y., Tsugawa, S.: On the effectiveness of link addition for improving robustness of multiplex networks against layer node-based attack. In: 2017 IEEE 41st Annual Computer Software and Applications Conference (COMPSAC), vol. 1, pp. 697–700. IEEE, Piscataway (2017)Google Scholar
  24. 24.
    Li, L., Alderson, D., Doyle, J.C., Willinger, W.: Towards a theory of scale-free graphs: definition, properties, and implications. Internet Math. 2(4), 431–523 (2005)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Matsui, Y., Kojima, H., Tsuchiya, T.: Modeling the interaction of power line and scada networks. In: 2014 IEEE 15th International Symposium on High-Assurance Systems Engineering, pp. 261–262. IEEE, Piscataway (2014)Google Scholar
  26. 26.
    Nguyen, D.T., Shen, Y., Thai, M.T.: Detecting critical nodes in interdependent power networks for vulnerability assessment. IEEE Trans. Smart Grid 4(1), 151–159 (2013)CrossRefGoogle Scholar
  27. 27.
    Parandehgheibi, M., Modiano, E.: Robustness of interdependent networks: the case of communication networks and the power grid. In: 2013 IEEE Global Communications Conference (GLOBECOM), pp. 2164–2169. IEEE, Piscataway (2013)Google Scholar
  28. 28.
    Qiu, Y.: The effect of clustering-based and degree-based weighting on robustness in symmetrically coupled heterogeneous interdependent networks. In: 2013 IEEE International Conference on Systems, Man, and Cybernetics, pp. 3984–3988. IEEE, Piscataway (2013)Google Scholar
  29. 29.
    Radicchi, F.: Percolation in real interdependent networks. Nat. Phys. 11(7), 597–602 (2015)CrossRefGoogle Scholar
  30. 30.
    Ramiro, V., Piquer, J., Barros, T., Sepúlveda, P.: The Chilean Internet: did it survive the earthquake? WIT Trans. State-of-the-art in Sci. Eng. 58, 133–151 (2012)CrossRefGoogle Scholar
  31. 31.
    Reis, S.D., 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
  32. 32.
    Rosato, V., Issacharoff, L., Tiriticco, F., Meloni, S., Porcellinis, S., Setola, R.: Modelling interdependent infrastructures using interacting dynamical models. Int. J. Crit. Infrastruct. 4(1–2), 63–79 (2008)CrossRefGoogle Scholar
  33. 33.
    Schneider, C.M., Yazdani, N., Araújo, N.A., Havlin, S., Herrmann, H.J.: Towards designing robust coupled networks. Sci. Rep. 3, 1969 (2013)ADSCrossRefGoogle Scholar
  34. 34.
    Wang, X., Kooij, R.E., Van Mieghem, P.: Modeling region-based interconnection for interdependent networks. Phys. Rev. E 94(4), 042315 (2016)ADSCrossRefGoogle Scholar
  35. 35.
    Watanabe, S., Kabashima,Y.: Cavity-based robustness analysis of interdependent networks: influences of intranetwork and internetwork degree-degree correlations. Phys. Rev. E 89(1), 012808 (2014)ADSCrossRefGoogle Scholar
  36. 36.
    Willinger, W., Roughan, M.: Internet topology research redux. ACM SIGCOMM eBook: Recent Advances in Networking. CiteSeer (2013)Google Scholar
  37. 37.
    Willinger, W., Alderson, D., Doyle, J.C.: Mathematics and the internet: a source of enormous confusion and great potential. Notices Am. Math. Soc. 56(5), 586–599 (2009)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Yao, A.C.C.: On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM J. Comp. 11(4), 721–736 (1982)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Zhou, D., Stanley, H.E., D’Agostino, G., Scala, A.: Assortativity decreases the robustness of interdependent networks. Phys. Rev. E 86(6), 066103 (2012)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Ivana Bachmann
    • 1
    Email author
  • Francisco Sanhueza
    • 1
  • Javier Bustos-Jiménez
    • 1
  1. 1.NIC LabsUniversity of ChileSantiagoChile

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