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Critical nodes in interdependent networks with deterministic and probabilistic cascading failures

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Abstract

We consider optimization problems of identifying critical nodes in coupled interdependent networks, that is, choosing a subset of nodes whose deletion causes the maximum network fragmentation (quantified by an appropriate metric) in the presence of deterministic or probabilistic cascading failure propagations. We use two commonly considered network fragmentation metrics: total number of disabled nodes and total number of disabled pair-wise connectivities. First, we discuss computational complexity issues and develop linear mixed integer programming (MIP) formulations for the corresponding optimization problems in the deterministic case. We then extend these problems to the case with probabilistic failure propagations using Conditional Value-at-Risk measure. We develop a scenario-based linear MIP model and propose an exact Markov chain-based algorithm to solve these problems. Finally, we perform a series of computational experiments on synthetic and semi-synthetic networks and discuss some interesting insights that illustrate the properties of the proposed models.

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Notes

  1. In this subsection only, the parameter p is used in the context of the classical G(n, p) model, rather than the parameter denoting the probability of failure propagation throughout other sections of the paper.

References

  1. Amin, M.: Toward secure and resilient interdependent infrastructures. J. Infrastruct. Syst. 8, 67 (2002)

    Article  Google Scholar 

  2. Arulselvan, A., Commander, C., Elefteriadou, L., Pardalos, P.: Detecting critical nodes in sparse graphs. Comput. Oper. Res. 36(7), 2193–2200 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Badanidiyuru, A., Kleinberg, R., Lee, H.: Approximating low-dimensional coverage problems. In: Proceedings of the 2012 Symposuim on Computational Geometry, pp. 161–170. ACM (2012)

  4. Bak, P., Chen, K., Tang, C.: A forest-fire model and some thoughts on turbulence. Phys. Lett. A 147(5–6), 297–300 (1990)

    Article  Google Scholar 

  5. Barrett, C., Beckman, R., Channakeshava, K., Huang, F., Kumar, V., Marathe, A., Marathe, M., Pei, G.: Cascading failures in multiple infrastructures: from transportation to communication network. In: 2010 5th International Conference on Critical Infrastructure (CRIS), pp. 1–8 (2010). https://doi.org/10.1109/CRIS.2010.5617569

  6. Bienstock, D.: Optimal control of cascading power grid failures. In: 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), pp. 2166–2173 (2011). https://doi.org/10.1109/CDC.2011.6160415

  7. Brummitt, C.D., DSouza, R.M., Leicht, E.: Suppressing cascades of load in interdependent networks. Proc. Natl. Acad. Sci. 109(12), E680–E689 (2012)

    Article  Google Scholar 

  8. Buldyrev, S., Parshani, R., Paul, G., Stanley, H., Havlin, S.: Catastrophic cascade of failures in interdependent networks. Nature 464(7291), 1025–1028 (2010)

    Article  Google Scholar 

  9. Buldyrev, S.V., Shere, N.W., Cwilich, G.A.: Interdependent networks with identical degrees of mutually dependent nodes. Phys. Rev. E 83(1), 016112 (2011)

    Article  MathSciNet  Google Scholar 

  10. Carreras, B.A., Newman, D.E., Gradney, P., Lynch, V.E., Dobson, I.: interdependent risk in interacting infrastructure systems. In: 40th Annual Hawaii International Conference on System Sciences, 2007. HICSS 2007, pp. 112–112. IEEE (2007)

  11. Di Summa, M., Grosso, A., Locatelli, M.: Branch and cut algorithms for detecting critical nodes in undirected graphs. Comput. Optim. Appl. 53(3), 649–680 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dinh, T.N., Thai, M.T.: Precise structural vulnerability assessment via mathematical programming. In: Military Communications Conference, 2011-MILCOM 2011, pp. 1351–1356. IEEE (2011)

  13. 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  Google Scholar 

  14. Drossel, B., Schwabl, F.: Forest-fire model with immune trees. Phys. A Stat. Mech. Appl. 199(2), 183–197 (1993)

    Article  Google Scholar 

  15. Dueñas-Osorio, L., Craig, J., Goodno, B.: Probabilistic response of interdependent infrastructure networks. In: 2nd Annual Meeting of the Asian-Pacific Network of Centers for Earthquake Engineering Research (ANCER). Honolulu, Hawaii, July pp. 28–30 (2004)

  16. Dueñas-Osorio, L., Craig, J., Goodno, B., Bostrom, A.: Interdependent response of networked systems. J. Infrastruct. Syst. 13, 185 (2007)

    Article  Google Scholar 

  17. Dueñas-Osorio, L., Craig, J.I., Goodno, B.J.: Seismic response of critical interdependent networks. Earthq. Eng. Struct. Dyn. 36(2), 285–306 (2007). https://doi.org/10.1002/eqe.626

    Article  Google Scholar 

  18. Erdős, P., Rényi, A.: On random graphs. Publ. Math. Debr. 6, 290–297 (1959)

    MATH  Google Scholar 

  19. \(\text{FICO}^{{\rm TM}}\) Xpress Optimization Suite 7.6. http://www.fico.com/ (2013)

  20. Gao, C., Li, X., Zhang, X., Li, K.: Network immunization for interdependent networks. J. Comput. Inf. Syst. 9(16), 6661–6668 (2013)

    Google Scholar 

  21. Gao, J., Buldyrev, S.V., Havlin, S., Stanley, H.E.: Robustness of a network of networks. Phys. Rev. Lett. 107, 195701 (2011). https://doi.org/10.1103/PhysRevLett.107.195701

    Article  Google Scholar 

  22. Gao, J., Buldyrev, S.V., Stanley, H.E., Havlin, S.: Networks formed from interdependent networks. Nat. Phys. 8(1), 40–48 (2011)

    Article  Google Scholar 

  23. Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman and Co., New York (1979)

    MATH  Google Scholar 

  24. Gong, J., Mitchell, J.E., Krishnamurthy, A., Wallace, W.A.: An Interdependent Layered Network Model for a Resilient Supply Chain. Omega 46, 104–116 (2013)

    Article  Google Scholar 

  25. Huang, X., Gao, J., Buldyrev, S., Havlin, S., Stanley, H.: Robustness of interdependent networks under targeted attack. Phys. Rev. E 83(6), 065101 (2011)

    Article  Google Scholar 

  26. Johansson, J., Hassel, H.: An approach for modelling interdependent infrastructures in the context of vulnerability analysis. Reliab. Eng. Syst. Saf. 95(12), 1335–1344 (2010)

    Article  Google Scholar 

  27. Kempe, D., Kleinberg, J., Tardos, É.: Maximizing the spread of influence through a social network. In: Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 137–146. ACM (2003)

  28. Khuller, S., Moss, A., Naor, J.S.: The budgeted maximum coverage problem. Inf. Process. Lett. 70(1), 39–45 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. 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 

  30. Krokhmal, P., Palmquist, J., Uryasev, S.: Portfolio optimization with conditional value-at-risk objective and constraints. J. Risk 4, 43–68 (2002)

    Article  Google Scholar 

  31. Lalou, M., Tahraoui, M.A., Kheddouci, H.: The critical node detection problem in networks: a survey. Comput. Sci. Rev. 28, 92–117 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Little, R.: Controlling cascading failure: understanding the vulnerabilities of interconnected infrastructures. J. Urban Technol. 9(1), 109–123 (2002)

    Article  Google Scholar 

  33. Luiijf, E., Nieuwenhuijs, A., Klaver, M., van Eeten, M., Cruz, E.: Empirical findings on critical infrastructure dependencies in Europe. In: Setola, R., Geretshuber, S. (eds.) Critical Information Infrastructure Security, Lecture Notes in Computer Science, vol. 5508, pp. 302–310. Springer, Berlin (2009)

    Google Scholar 

  34. Newman, D., Nkei, B., Carreras, B., Dobson, I., Lynch, V., Gradney, P.: Risk assessment in complex interacting infrastructure systems. In: HICSS ’05. Proceedings of the 38th Annual Hawaii International Conference on System Sciences, 2005, p. 63c (2005). https://doi.org/10.1109/HICSS.2005.524

  35. Newman, D.E., Nkei, B., Carreras, B.A., Dobson, I., Lynch, V.E., Gradney, P.: Risk assessment in complex interacting infrastructure systems. In: Proceedings of the 38th Annual Hawaii International Conference on System Sciences, 2005. HICSS’05, pp. 63c–63c. IEEE (2005)

  36. Nguyen, D., Shen, Y., Thai, M.: Detecting critical nodes in interdependent power networks for vulnerability assessment. IEEE Trans. Smart Grid 4(1), 151–159 (2013)

    Article  Google Scholar 

  37. Oosten, M., Rutten, J., Spieksma, F.: Disconnecting graphs by removing vertices: a polyhedral approach. Stat. Neerl. 61(1), 35–60 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ouyang, M.: Review on modeling and simulation of interdependent critical infrastructure systems. Reliab. Eng. Syst. Saf. 121, 43–60 (2014)

    Article  Google Scholar 

  39. Ouyang, M., Hong, L., Mao, Z.J., Yu, M.H., Qi, F.: A methodological approach to analyze vulnerability of interdependent infrastructures. Simul. Modell. Pract. Theory 17(5), 817–828 (2009). https://doi.org/10.1016/j.simpat.2009.02.001. http://www.sciencedirect.com/science/article/pii/S1569190X0900015X

  40. 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  Google Scholar 

  41. Parshani, R., Rozenblat, C., Ietri, D., Ducruet, C., Havlin, S.: Inter-similarity between coupled networks. EPL (Europhys. Lett.) 92(6), 68002 (2010). http://stacks.iop.org/0295-5075/92/i=6/a=68002

  42. Pavlikov, K.: Improved formulations for minimum connectivity network interdiction problems. Comput. Oper. Res. 97, 48–57 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  43. Reed, D., Kapur, K., Christie, R.: Methodology for assessing the resilience of networked infrastructure. IEEE Sys. J. 3(2), 174–180 (2009). https://doi.org/10.1109/JSYST.2009.2017396

    Article  Google Scholar 

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

    Article  Google Scholar 

  45. Rinaldi, S.M.: Modeling and simulating critical infrastructures and their interdependencies. In: Proceedings of the 37th Annual Hawaii International Conference on System Sciences, 2004, pp. 8–pp. IEEE (2004)

  46. Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–41 (2000)

    Article  Google Scholar 

  47. Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26(7), 1443–1471 (2002)

    Article  Google Scholar 

  48. Rockafellar, R.T., Uryasev, S.: The fundamental risk quadrangle in risk management, optimization and statistical estimation. Surv. Oper. Res. Manag. Sci. 18(1), 33–53 (2013)

    MathSciNet  Google Scholar 

  49. Santos, D., de Sousa, A., Monteiro, P.: Compact models for critical node detection in telecommunication networks. Electron. Notes Discrete Math. 64, 325–334 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

  51. 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(3), 036116 (2011)

    Article  MathSciNet  Google Scholar 

  52. Shen, S., Smith, J.C., Goli, R.: Exact interdiction models and algorithms for disconnecting networks via node deletions. Discrete Optim. 9(3), 172–188 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  53. Simonsen, I., Buzna, L., Peters, K., Bornholdt, S., Helbing, D.: Transient dynamics increasing network vulnerability to cascading failures. Phys. Rev. Lett. 100(21), 218701 (2008)

    Article  Google Scholar 

  54. Veremyev, A., Boginski, V., Pasiliao, E.L.: Exact identification of critical nodes in sparse networks via new compact formulations. Optim. Lett. 8(4), 1245–1259 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  55. Veremyev, A., Prokopyev, O.A., Pasiliao, E.L.: An integer programming framework for critical elements detection in graphs. J. Comb. Optim. 28(1), 233–273 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  56. Veremyev, A., Sorokin, A., Boginski, V., Pasiliao, E.L.: Minimum vertex cover problem for coupled interdependent networks with cascading failures. Eur. J. Oper. Res. 232(3), 499–511 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  57. Wang, S., Hong, L., Chen, X.: Vulnerability analysis of interdependent infrastructure systems: a methodological framework. Phys. A Stat. Mech. Appl. 391(11), 3323–3335 (2012)

    Article  Google Scholar 

  58. Yagan, O., Qian, D., Zhang, J., Cochran, D.: Optimal allocation of interconnecting links in cyber-physical systems: interdependence, cascading failures, and robustness. IEEE Trans. Parallel Distrib. Syst. 23(9), 1708–1720 (2012)

    Article  Google Scholar 

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Acknowledgements

This material is based upon work supported by the AFRL Mathematical Modeling and Optimization Institute. M.T. Thai’s and V. Boginski’s research is supported in part by NSF award EFRI-1441231.

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Authors and Affiliations

  1. Department of Industrial Engineering and Management Systems, University of Central Florida, Orlando, FL, 32816, USA

    Alexander Veremyev & Vladimir Boginski

  2. Department of Business and Economics, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark

    Konstantin Pavlikov

  3. Munitions Directorate, Air Force Research Laboratory, Building 13, Eglin AFB, FL, 32542, USA

    Eduardo L. Pasiliao

  4. Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL, 32611, USA

    My T. Thai

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  1. Alexander Veremyev
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  2. Konstantin Pavlikov
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  4. My T. Thai
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  5. Vladimir Boginski
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Correspondence to Vladimir Boginski.

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Veremyev, A., Pavlikov, K., Pasiliao, E.L. et al. Critical nodes in interdependent networks with deterministic and probabilistic cascading failures. J Glob Optim 74, 803–838 (2019). https://doi.org/10.1007/s10898-018-0703-5

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