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Systemic risk in a unifying framework for cascading processes on networks

  • J. LorenzEmail author
  • S. Battiston
  • F. Schweitzer
Topical issue on The Physics Approach to Risk: Agent-Based Models and Networks

Abstract

We introduce a general framework for models of cascade and contagion processes on networks, to identify their commonalities and differences. In particular, models of social and financial cascades, as well as the fiber bundle model, the voter model, and models of epidemic spreading are recovered as special cases. To unify their description, we define the net fragility of a node, which is the difference between its fragility and the threshold that determines its failure. Nodes fail if their net fragility grows above zero and their failure increases the fragility of neighbouring nodes, thus possibly triggering a cascade. In this framework, we identify three classes depending on the way the fragility of a node is increased by the failure of a neighbour. At the microscopic level, we illustrate with specific examples how the failure spreading pattern varies with the node triggering the cascade, depending on its position in the network and its degree. At the macroscopic level, systemic risk is measured as the final fraction of failed nodes, X*, and for each of the three classes we derive a recursive equation to compute its value. The phase diagram of X* as a function of the initial conditions, thus allows for a prediction of the systemic risk as well as a comparison of the three different model classes. We could identify which model class leads to a first-order phase transition in systemic risk, i.e. situations where small changes in the initial conditions determine a global failure. Eventually, we generalize our framework to encompass stochastic contagion models. This indicates the potential for further generalizations.

PACS

64.60.aq Networks 89.65.Gh Economics; econophysics, financial markets, business and management 87.23.Ge Dynamics of social systems 62.20.M- Structural failure of materials 

References

  1. 1.
    S. Morris, H. Shin, Brookings Panel on Economic Activity, Fall, 229–274 (2008)Google Scholar
  2. 2.
    M. Brunnermeier, Journal of Economic Perspectives 23(1), 77–100 (2008)CrossRefGoogle Scholar
  3. 3.
    J. Lorenz, S. Battiston, Netw. Heterog. Media 3, 185 (2008)zbMATHMathSciNetGoogle Scholar
  4. 4.
    F. Schweitzer, G. Fagiolo, D. Sornette, F. Vega-Redondo, A. Vespignani, D.R. White, Science 325, 422 (2009)ADSGoogle Scholar
  5. 5.
    B. Kahng, G. Batrouni, S. Redner, L. de Arcangelis, H. Herrmann, Phys. Rev. B 37, 7625 (1988)CrossRefADSGoogle Scholar
  6. 6.
    D. Kim, B. Kim, H. Jeong, Phys. Rev. Lett. 94, 25501 (2005)CrossRefADSGoogle Scholar
  7. 7.
    D. Sornette, J. Andersen, The European Physical Journal B-Condensed Matter and Complex Systems 1, 353 (1998)CrossRefADSGoogle Scholar
  8. 8.
    Y. Moreno, J. Gomez, A. Pacheco, Europhys. Lett. 58, 630 (2002)CrossRefADSGoogle Scholar
  9. 9.
    P. Crucitti, V. Latora, M. Marchiori, Phys. Rev. E 69, 045104 (2004)CrossRefADSGoogle Scholar
  10. 10.
    B. Carreras, V. Lynch, I. Dobson, D. Newman, Chaos: An Interdisciplinary Journal of Nonlinear Science 14, 643 (2004)CrossRefGoogle Scholar
  11. 11.
    K. Goh, D. Lee, B. Kahng, D. Kim, Phys. Rev. Lett. 91, 148701 (2003)CrossRefADSGoogle Scholar
  12. 12.
    G. Bianconi, M. Marsili, Phys. Rev. E 70, 35105 (2004)CrossRefADSGoogle Scholar
  13. 13.
    S. Battiston, D.D. Gatti, M. Gallegati, B.C.N. Greenwald, J.E. Stiglitz, Journal of Economic Dynamics and Control 31, 2061 (2007)zbMATHCrossRefGoogle Scholar
  14. 14.
    D. Watts, Proceedings of the National Academy of Sciences 99, 5766 (2002)zbMATHCrossRefMathSciNetADSGoogle Scholar
  15. 15.
    A. Motter, Phys. Rev. Lett. 93, 98701 (2004)CrossRefADSGoogle Scholar
  16. 16.
    A. Vespignani, S. Zapperi, Phys. Rev. E 57, 6345 (1998)CrossRefMathSciNetADSGoogle Scholar
  17. 17.
    F. Caruso, V. Latora, A. Pluchino, A. Rapisarda, B. Tadić, The European Physical Journal B-Condensed Matter 50, 243 (2006)CrossRefADSGoogle Scholar
  18. 18.
    D. Sornette, forthcoming on International Journal of Terraspace Science and Engineering, e-print arXiv:0907.4290 (2009)Google Scholar
  19. 19.
    R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001)CrossRefADSGoogle Scholar
  20. 20.
    A. Vespignani, R. Pastor-Satorras, Phys. Rev. E 65, 035108 (2002)CrossRefADSGoogle Scholar
  21. 21.
    M. Boguná, R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 90, 28701 (2003)CrossRefADSGoogle Scholar
  22. 22.
    J. Gómez-Gardeñes, V. Latora, Y. Moreno, E. Profumo, Proceedings of the National Academy of Sciences 105, 1399 (2008)CrossRefADSGoogle Scholar
  23. 23.
    H. Stark, C. Tessone, F. Schweitzer, Phys. Rev. Lett. 101, 18701 (2008)CrossRefADSGoogle Scholar
  24. 24.
    F. Schweitzer, L. Behera, Eur. Phys. J. B 67, 301 (2009)CrossRefMathSciNetADSGoogle Scholar
  25. 25.
    P. Dodds, J. Payne, Phys. Rev. E 79, 066115 (2009)CrossRefADSGoogle Scholar
  26. 26.
    P. Dodds, D. Watts, Phys. Rev. Lett. 92, 218701 (2004)CrossRefADSGoogle Scholar
  27. 27.
    M. Avellaneda, J. Zhu, Risk 14, 125 (2001)Google Scholar
  28. 28.
    M.D. König, S. Battiston, M. Napoletano, F. Schweitzer, Networks and Heterogeneous Media 3, 201 (2008)zbMATHGoogle Scholar
  29. 29.
    M. Granovetter, American Journal of Sociology 83, 1420 (1978)CrossRefGoogle Scholar
  30. 30.
    S. Battiston, D.D. Gatti, M. Gallegati, B.C.N. Greenwald, J.E. Stiglitz (2009)Google Scholar
  31. 31.
    F. Kun, S. Zapperi, H. Herrmann, The European Physical Journal B-Condensed Matter and Complex Systems 17, 269 (2000)CrossRefADSGoogle Scholar
  32. 32.
    R. Kinney, P. Crucitti, R. Albert, V. Latora, The European Physical Journal B-Condensed Matter and Complex Systems 46, 101 (2005)CrossRefADSGoogle Scholar
  33. 33.
    L. Eisenberg, T. Noe, Management Science 47, 236 (2001)CrossRefGoogle Scholar
  34. 34.
    M. Jackson, B. Rogers, The BE Journal of Theoretical Economics 7, 1 (2007)MathSciNetGoogle Scholar
  35. 35.
    F. Schweitzer, R. Mach, PLoS ONE 3 (2008)Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Chair of Systems Design, ETH ZürichZürichSwitzerland

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