Journal of Statistical Physics

, Volume 151, Issue 3–4, pp 765–783 | Cite as

How Big Is Too Big? Critical Shocks for Systemic Failure Cascades

  • Claudio J. Tessone
  • Antonios Garas
  • Beniamino Guerra
  • Frank Schweitzer
Article

Abstract

External or internal shocks may lead to the collapse of a system consisting of many agents. If the shock hits only one agent initially and causes it to fail, this can induce a cascade of failures among neighboring agents. Several critical constellations determine whether this cascade affects the system in part or as a whole which, in the second case, leads to systemic risk. We investigate the critical parameters for such cascades in a simple model, where agents are characterized by an individual threshold θi determining their capacity to handle a load αθi with 1−α being their safety margin. If agents fail, they redistribute their load equally to K neighboring agents in a regular network. For three different threshold distributions P(θ), we derive analytical results for the size of the cascade, X(t), which is regarded as a measure of systemic risk, and the time when it stops. We focus on two different regimes, (i) EEE, an external extreme event where the size of the shock is of the order of the total capacity of the network, and (ii) RIE, a random internal event where the size of the shock is of the order of the capacity of an agent. We find that even for large extreme events that exceed the capacity of the network finite cascades are still possible, if a power-law threshold distribution is assumed. On the other hand, even small random fluctuations may lead to full cascades if critical conditions are met. Most importantly, we demonstrate that the size of the “big” shock is not the problem, as the systemic risk only varies slightly for changes in the number of initially failing agents, the safety margin and the threshold distribution, which further gives hints on how to reduce systemic risk.

Keywords

Systemic risk Cascading dynamics 

References

  1. 1.
    Watts, D.J.: A simple model of global cascades on random networks. Proc. Natl. Acad. Sci. USA 99(9), 5766 (2002) MathSciNetADSMATHCrossRefGoogle Scholar
  2. 2.
    Dodds, P.S., Watts, D.J.: Universal behavior in a generalized model of contagion. Phys. Rev. Lett. 92(21), 218701 (2004) ADSCrossRefGoogle Scholar
  3. 3.
    Lorenz, J., Battiston, S., Schweitzer, F.: Systemic risk in a unifying framework for cascading processes on networks. Eur. Phys. J. B 71(4), 441–460 (2009) MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    Dobson, I., Carreras, B., Lynch, V.E., Newman, D.E.: Complex systems analysis of series of blackouts: cascading failure, critical points, and self-organization. Chaos 17(2), 026103 (2007) ADSCrossRefGoogle Scholar
  5. 5.
    Carreras, B.A., Lynch, V.E., Dobson, I., Newman, D.E.: Critical points and transitions in an electric power transmission model for cascading failure blackouts. Chaos 12(4), 985–994 (2002) MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    Motter, A.E., Lai, Y.C.: Cascade-based attacks on complex networks. Phys. Rev. E 66(6), 65102 (2002) ADSCrossRefGoogle Scholar
  7. 7.
    Crucitti, P., Latora, V., Marchiori, M.: Model for cascading failures in complex networks. Phys. Rev. E 69(4), 45104 (2004) ADSCrossRefGoogle Scholar
  8. 8.
    Rossi, D., Mellia, M., Meo, M.: Evidences behind skype outage. In: IEEE International Conference on Communications, Dresden (2009) Google Scholar
  9. 9.
    Battiston, S., Delli Gatti, D., Gallegati, M., Greenwald, B.C., Stiglitz, J.E.: Liaisons dangereuses: increasing connectivity, risk sharing, and systemic risk. J. Econ. Dyn. Control 36, 1121–1141 (2012) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gai, P., Kapadia, S.: Contagion in financial networks. Bank of England working papers (2010) Google Scholar
  11. 11.
    Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74(1), 47 (2002) ADSMATHCrossRefGoogle Scholar
  12. 12.
    Motter, A.E.: Cascade control and defense in complex networks. Phys. Rev. Lett. 93(9), 98701 (2004) ADSCrossRefGoogle Scholar
  13. 13.
    Kinney, R., Crucitti, P., Albert, R., Latora, V.: Modeling cascading failures in the North American power grid. Eur. Phys. J. B 46, 101–107 (2005) ADSCrossRefGoogle Scholar
  14. 14.
    Tessone, C.J., Geipel, M.M., Schweitzer, F.: Sustainable growth in complex networks. Europhys. Lett. 96(5), 58005 (2011) ADSCrossRefGoogle Scholar
  15. 15.
    Bekessy, A., Bekessy, P., Komlos, J.: Asymptotic enumeration of regular matrices. Studia Sci. Math. Hung. 7, 343–353 (1972) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Claudio J. Tessone
    • 1
  • Antonios Garas
    • 1
  • Beniamino Guerra
    • 1
  • Frank Schweitzer
    • 1
  1. 1.Chair of Systems DesignETH ZurichZurichSwitzerland

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