Modeling Interdependent Networks as Random Graphs: Connectivity and Systemic Risk

  • R. M. D’SouzaEmail author
  • C. D. Brummitt
  • E. A. Leicht
Part of the Understanding Complex Systems book series (UCS)


Idealized models of interconnected networks can provide a laboratory for studying the consequences of interdependence in real-world networks, in particular those networks constituting society’s critical infrastructure. Here we show how random graph models of connectivity between networks can provide insights into shifts in percolation properties and into systemic risk. Tradeoffs abound in many of our results. For instance, edges between networks confer global connectivity using relatively few edges, and that connectivity can be beneficial in situations like communication or supplying resources, but it can prove dangerous if epidemics were to spread on the network. For a specific model of cascades of load in the system (namely, the sandpile model), we find that each network minimizes its risk of undergoing a large cascade if it has an intermediate amount of connectivity to other networks. Thus, connections among networks confer benefits and costs that balance at optimal amounts. However, what is optimal for minimizing cascade risk in one network is suboptimal for minimizing risk in the collection of networks. This work provides tools for modeling interconnected networks (or single networks with mesoscopic structure), and it provides hypotheses on tradeoffs in interdependence and their implications for systemic risk.


Random Graph Degree Distribution Power Grid Interconnected Network Critical Infrastructure 
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.


  1. 1.
    Review of Selected 1996 Electric System Disturbances in North America.North American Electric Reliability Council, 2002.Google Scholar
  2. 2.
    F. Allen and D. Gale.Financial contagion.Journal of Political Economy, 108(1):1–33, 2000.Google Scholar
  3. 3.
    M. Amin.National infrastructure as complex interactive networks. In T. Samad and J. Weyrauch, editors, Automation, control and complexity: an integrated approach, pages 263–286. John Wiley & Sons,Inc., 2000.Google Scholar
  4. 4.
    M. Anghel, Z. Toroczkai, K. Bassler, and G. Korniss. Competition-Driven Network Dynamics: Emergence of a Scale-Free Leadership Structure and Collective Efficiency. Physical Review Letters, 92(5):058701, Feb. 2004.Google Scholar
  5. 5.
    P. Bak, C. Tang, and K. Wiesenfeld. Self-organized criticality: An explanation of \(1/f\) noise. Physical Review Letters, 59(4):381–384, 1987.Google Scholar
  6. 6.
    P. Bak, C. Tang, and K. Wiesenfeld. Self-organized criticality. Physical Review A, 38(1):364–374, 1988.Google Scholar
  7. 7.
    R. Baldick, B. Chowdhury, I. Dobson, Z. Dong, B. Gou, D. Hawkins, H. Huang, M. Joung, D. Kirschen, F. Li, J. Li, Z. Li, C.-C. Liu, L. Mili, S. Miller, R. Podmore, K. Schneider, K. Sun, D. Wang, Z. Wu, P. Zhang, W. Zhang, and X. Zhang. Initial review of methods for cascading failure analysis in electric power transmission systems ieee pes cams task force on understanding, prediction, mitigation and restoration of cascading failures. In Power and Energy Society General Meeting - Conversion and Delivery of Electrical Energy in the 21st Century, 2008 IEEE, pages 1–8, July 2008.Google Scholar
  8. 8.
    S. Battiston, D. D. Gatti, M. Gallegati, B. Greenwald, and J. E. Stiglitz. Default cascades: When does risk diversification increase stability? Journal of Financial Stability, 8(3):138–149, Sept. 2012.Google Scholar
  9. 9.
    S. Battiston, D. D. Gatti, M. Gallegati, B. Greenwald, and J. E. Stiglitz. Liaisons dangereuses: Increasing connectivity, risk sharing, and systemic risk. Journal of Economic Dynamics and Control, 36(8):1121–1141, Aug. 2012.Google Scholar
  10. 10.
    B. Bollobás. A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European Journal of Combinatorics, 1:311, 1980.Google Scholar
  11. 11.
    E. Bonabeau. Sandpile dynamics on random graphs. Journal of the Physical Society of Japan, 64(1):327–328, 1995.Google Scholar
  12. 12.
    D. Braess, A. Nagurney, and T. Wakolbinger. On a Paradox of Traffic Planning. Transportation Science, 39(4):446–450, Nov. 2005.Google Scholar
  13. 13.
    C. D. Brummitt, R. M. D’Souza, and E. A. Leicht. Suppressing cascades of load in interdependent networks. Proc. Natl. Acad. Sci. U.S.A., 109(12):E680–E689, Feb. 2012.Google Scholar
  14. 14.
    C. D. Brummitt, P. D. H. Hines, I. Dobson, C. Moore, and R. M. D’Souza. A transdisciplinary science for 21st-century electric power grids. Forthcoming, 2013.Google Scholar
  15. 15.
    S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, and S. Havlin. Catastrophic cascade of failures in interdependent networks. Nature, 464:1025–1028, 2010.Google Scholar
  16. 16.
    M. Chediak and L. M. Cold snap causes gas shortages across u.s. southwest. Bloomberg News, (, Feb, 2011
  17. 17.
    J. de Arcangelis and H. J. Herrmann. Self-organized criticality on small world networks. Physica A, 308:545–549, 2002.Google Scholar
  18. 18.
    I. Dobson, B. A. Carreras, V. E. Lynch, and D. E. Newman. Complex systems analysis of series of blackouts: Cascading failure, critical points, and self-organization. Chaos, 17(026103), 2007.Google Scholar
  19. 19.
    I. Dobson, B. A. Carreras, and D. E. Newman. A branching process approximation to cascading load-dependent system failure. In Thirty-seventh Hawaii International Conference on System Sciences, 2004.Google Scholar
  20. 20.
    R. M. D’Souza, C. Borgs, J. T. Chayes, N. Berger, and R. D. Kleinberg. Emergence of tempered preferential attachment from optimization. Proc. Natn. Acad. Sci. USA, 104(15):6112–6117, 2007.Google Scholar
  21. 21.
    L. Dueñas-Osorio and S. M. Vemuru. Cascading failures in complex infrastructures. Structural Safety, 31(2):157–167, 2009.Google Scholar
  22. 22.
    B. Dupoyet, H. R. Fiebig, and D. P. Musgrove. Replicating financial market dynamics with a simple self-organized critical lattice model. Physica A, 390(18–19):3120–3135, Sept. 2011.Google Scholar
  23. 23.
    M. J. Eppstein and P. D. H. Hines. A “Random Chemistry” Algorithm for Identifying Collections of Multiple Contingencies That Initiate Cascading Failure. Power Systems, IEEE Transactions on, 27(3):1698–1705, 2012.Google Scholar
  24. 24.
    Federal Energy Regulatory Commission. Arizona-Southern California Outages on September 8, 2011, Apr. 2012.Google Scholar
  25. 25.
    S. Fortunato. Community detection in graphs. Physics Reports, 486:75–174, 2010.Google Scholar
  26. 26.
    X. Gabaix, P. Gopikrishnan, V. Plerou, and H. E. Stanley. A theory of power-law distributions in financial market fluctuations. Nature, 423(6937):267–270, 2003.Google Scholar
  27. 27.
    K. Goh, D. Lee, B. Kahng, and D. Kim. Cascading toppling dynamics on scale-free networks. Physica A: Statistical Mechanics and its Applications, 346(1–2):93–103, 2005.Google Scholar
  28. 28.
    K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim. Sandpile on Scale-Free Networks. Physical Review Letters, 91(14):148701, Oct. 2003.Google Scholar
  29. 29.
    T. H. Grubesic and A. Murray. Vital nodes, interconnected infrastructures and the geographies of network survivability. Annals of the Association of American Geographers, 96(1):64–83, 2006.Google Scholar
  30. 30.
    A. G. Haldane and R. M. May. Systemic risk in banking ecosystems. Nature, 469:351–355, Jan 2011.Google Scholar
  31. 31.
    M. Hanlon. How we could all be victims of the volcano... and why we must hope for rain to get rid of the ash. Daily Mail, April 2010.
  32. 32.
    S. Hergarten. Landslides, sandpiles, and self-organized criticality. Natural Hazards and Earth System Sciences, 3:505–514, 2003.Google Scholar
  33. 33.
    P. Hines, E. Cotilla-Sanchez, and S. Blumsack. Do topological models provide good information about electricity infrastructure vulnerability? Chaos, 20(033122), Jan 2010.Google Scholar
  34. 34.
    C. Joyce. Building power lines creates a web of problems. NPR, April 2009.
  35. 35.
    J. Lahtinen, J. Kertész, and K. Kaski. Sandpiles on Watts-Strogatz type small-worlds. Physica A, 349:535–547, 2005.Google Scholar
  36. 36.
    D. Lee, K. Goh, B. Kahng, and D. Kim. Sandpile avalanche dynamics on scale-free networks. Physica A: Statistical Mechanics and its Applications, 338(1–2):84–91, 2004.Google Scholar
  37. 37.
    K.-M. Lee, K.-I. Goh, and I. M. Kim. Sandpiles on multiplex networks. Journal of the Korean Physical Society, 60(4):641–647, Feb. 2012.Google Scholar
  38. 38.
    K.-M. Lee, J. Y. Kim, W.-K. Cho, K. Goh, and I. Kim. Correlated multiplexity and connectivity of multiplex random networks. New Journal of Physics, 14(3):033027, 2012.Google Scholar
  39. 39.
    E. A. Leicht and R. M. D’Souza. Random graph models of interconnected networks. Forthcoming.Google Scholar
  40. 40.
    E. A. Leicht and R. M. D’Souza. Percolation on interacting, networks. arXiv:0907.0894, July 2009.Google Scholar
  41. 41.
    S. Lise and M. Paczuski. Nonconservative earthquake model of self-organized criticality on a random graph. Physical Review Letters, 88:228301, 2002.Google Scholar
  42. 42.
    R. G. Little. Controlling cascading failure: Understanding the vulnerabilities of interconnected infrastructures. Journal of Urban Technology, 9(1):109–123, 2002.Google Scholar
  43. 43.
    Y.-Y. Liu, J.-J. Slotine, and A.-Á. Barabási. Controllability of complex networks. Nature, 473(7346):167–173, May 2011.Google Scholar
  44. 44.
    T. Lo, K. Chan, P. Hui, and N. Johnson. Theory of enhanced performance emerging in a sparsely connected competitive population. Physical Review E, 71(5):050101, May 2005.Google Scholar
  45. 45.
    B. D. Malamud, G. Morein, and D. L. Turcotte. Forest Fires: An Example of Self-Organized Critical Behavior. Science, 281(5384):1840–1842, sep 1998.Google Scholar
  46. 46.
    C. Minoiu and J. Reyes. A Network Analysis of Global Banking: 1978–2010. Journal of Financial Stability, 2013. in press.Google Scholar
  47. 47.
    M. Molloy and B. Reed. A critical point for random graphs with a given degree sequence. Random Structures and Algorithms, 6:161–180, 1995.Google Scholar
  48. 48.
    R. G. Morris and M. Barthelemy. Transport on Coupled Spatial Networks. Physical Review Letters, 109(12):128703, Sept. 2012.Google Scholar
  49. 49.
    P. J. Mucha, T. Richardson, K. Macon, M. A. Porter, and J.-P. Onnela. Community structure in time-dependent, multiscale, and multiplex networks. Science, 328(5980):876–878, 2010.Google Scholar
  50. 50.
    D. P. Nedic, I. Dobson, D. S. Kirschen, B. A. Carreras, and V. E. Lynch. Criticality in a cascading failure blackout model. International Journal of Electrical Power & Energy Systems, 28(9):627–633, 2006.Google Scholar
  51. 51.
    D. E. Newman, B. Nkei, B. A. Carreras, I. Dobson, V. E. Lynch, and P. Gradney. Risk assessment in complex interacting infrastructure systems. In Thirty-eight Hawaii International Conference on System Sciences, 2005.Google Scholar
  52. 52.
    M. E. J. Newman, S. H. Strogatz, and D. J. Watts. Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E, 64(2):026118, 2001.Google Scholar
  53. 53.
    P.-A. Noël, C. D. Brummitt, and R. M. D’Souza. Controlling self-organizing dynamics using self-organizing models. Forthcoming, 2013.Google Scholar
  54. 54.
    W. of the ETH Risk Center. New Views on Extreme Events: Coupled Networks, Dragon Kings and Explosive Percolation. October 25–26th, 2012.Google Scholar
  55. 55.
    S. Panzieri and R. Setola. Failures propagation in critical interdependent infrastructures. Int. J. Modelling, Identification and, Control, 3(1):69–78, 2008.Google Scholar
  56. 56.
    P. Pederson, D. Dudenhoeffer, S. Hartley, and M. Permann. Critical infrastructure interdependency modeling: A survey of u.s. and international research. Idaho National Laboratory, INL/EXT-06-11464, 2006.Google Scholar
  57. 57.
    M. A. Porter, J.-P. Onnela, and P. J. Mucha. Communities in networks. Notices of the American Mathematical Society, 56(9):1082–1097, 2009.Google Scholar
  58. 58.
    E. Quill. When networks network. Science News, 182(6), 2012.Google Scholar
  59. 59.
    M. RE. North America most affected by increase in weather-related natural catastrophes., Oct. 17, 2012
  60. 60.
    S. Rinaldi, J. Peerenboom, and T. Kelly. Identifying, understanding, and analyzing critical infrastructure interdependencies. IEEE Control Systems Magazine, December:11–25, 2001.Google Scholar
  61. 61.
    S. M. Rinaldi. Modeling and simulating critical infrastructures and their interdependencies. In 38th Hawaii International Conference on System Sciences, pages 1–8, Big Island, Hawaii, 2004.Google Scholar
  62. 62.
    M. P. Rombach, M. A. Porter, J. H. Fowler, P. J. Mucha. Core-Periphery Structure in, Networks. arXiv:1202.2684, Feb. 2012.Google Scholar
  63. 63.
    V. Rosato, L. Issacharoff, F. Tiriticco, S. Meloni, S. D. Procellinis, and R. Setola. Modelling interdependent infrastructures using interacting dynamical models. Int. J. Critical Infrastructures, 4(1/2):63–79, 2008.Google Scholar
  64. 64.
    A. Saichev and D. Sornette. Anomalous power law distribution of total lifetimes of branching processes: Application to earthquake aftershock sequences. Physical Review E, 70(4):046123, Oct 2004.Google Scholar
  65. 65.
    P. Sinha-Ray and H. J. Jensen. Forest-fire models as a bridge between different paradigms in self-organized criticality. Physical Review E, 62(3):3216, sep 2000.Google Scholar
  66. 66.
    N. N. Taleb. The Black Swan: The Impact of the Highly Improbable. Random House Inc., New York, NY, 2007.Google Scholar
  67. 67.
    N. N. Taleb. Antifragile: Things That Gain from Disorder. Random House Inc., New York, NY, November 2012.Google Scholar
  68. 68.
    B. Wang, X. Chen, and L. Wang. Probabilistic interconnection between interdependent networks promotes cooperation in the public goods game. arXiv:1208.0468, Nov. 2012.Google Scholar
  69. 69.
    Z. Wang, A. Scaglione, and R. J. Thomas. Generating statistically correct random topologies for testing smart grid communication and control networks. IEEE Transactions on Smart Grid, 1:28–39, 2010.Google Scholar
  70. 70.
    S. Wasserman and K. Faust. Social network analysis: Methods and applications, volume 8. Cambridge university press, 1994.Google Scholar
  71. 71.
    N. Wolchover. Treading softly in a connected world. Simons Science News, (, 2013
  72. 72.
    R. Zimmerman, C. Murillo-Sánchez, and R. Thomas. Matpower: Steady-state operations, planning, and analysis tools for power systems research and education. Power Systems, IEEE Transactions on, 26(1):12–19, feb. 2011.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • R. M. D’Souza
    • 1
    Email author
  • C. D. Brummitt
    • 1
  • E. A. Leicht
    • 2
  1. 1.University of CaliforniaDavisUSA
  2. 2.CABDyN Complexity CenterUniversity of OxfordOxfordUK

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