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Systemic Risk in a Unifying Framework for Cascading Processes on Networks

  • Jan Lorenz
  • Stefano Battiston
  • Frank Schweitzer

    ()

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, $X^\ast$, and for each of the three classes we derive a recursive equation to compute its value. The phase diagram of $X^\ast$ 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 lead to a first-order phase transition in systemic risk, i.e. situations where small changes in the initial conditions may lead to a global failure. Eventually, we generalize our framework to encompass stochastic contagion models. This indicates the potential for further generalizations.

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Paper provided by ETH Zurich, Chair of Systems Design in its series Working Papers with number CCSS-09-011.

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Handle: RePEc:stz:wpaper:ccss-09-011
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  1. F. Schweitzer & L. Behera, 2009. "Nonlinear voter models: the transition from invasion to coexistence," The European Physical Journal B - Condensed Matter and Complex Systems, Springer, vol. 67(3), pages 301-318, February.
  2. R. Kinney & P. Crucitti & R. Albert & V. Latora, 2005. "Modeling cascading failures in the North American power grid," The European Physical Journal B - Condensed Matter and Complex Systems, Springer, vol. 46(1), pages 101-107, 07.
  3. Battiston, Stefano & Delli Gatti, Domenico & Gallegati, Mauro & Greenwald, Bruce & Stiglitz, Joseph E., 2012. "Liaisons dangereuses: Increasing connectivity, risk sharing, and systemic risk," Journal of Economic Dynamics and Control, Elsevier, vol. 36(8), pages 1121-1141.
  4. Battiston, Stefano & Delli Gatti, Domenico & Gallegati, Mauro & Greenwald, Bruce & Stiglitz, Joseph E., 2007. "Credit chains and bankruptcy propagation in production networks," Journal of Economic Dynamics and Control, Elsevier, vol. 31(6), pages 2061-2084, June.
  5. Jackson Matthew O. & Rogers Brian W., 2007. "Relating Network Structure to Diffusion Properties through Stochastic Dominance," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 7(1), pages 1-16, February.
  6. F. Caruso & V. Latora & A. Pluchino & A. Rapisarda & B. Tadić, 2006. "Olami-Feder-Christensen model on different networks," The European Physical Journal B - Condensed Matter and Complex Systems, Springer, vol. 50(1), pages 243-247, 03.
  7. Larry Eisenberg & Thomas H. Noe, 2001. "Systemic Risk in Financial Systems," Management Science, INFORMS, vol. 47(2), pages 236-249, February.
  8. J. Lorenz & S. Battiston & F. Schweitzer, 2009. "Systemic risk in a unifying framework for cascading processes on networks," The European Physical Journal B - Condensed Matter and Complex Systems, Springer, vol. 71(4), pages 441-460, October.
  9. Markus K. Brunnermeier, 2008. "Deciphering the Liquidity and Credit Crunch 2007-08," NBER Working Papers 14612, National Bureau of Economic Research, Inc.
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