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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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File URL: http://arxiv.org/pdf/0907.5325
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Paper provided by arXiv.org in its series Papers with number 0907.5325.

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Date of creation: Jul 2009
Date of revision: Jan 2010
Publication status: Published in European Physical Journal B, vol 71, no 4 (2009) pp. 441-460
Handle: RePEc:arx:papers:0907.5325
Contact details of provider: Web page: http://arxiv.org/

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  1. Stefano Battiston & Domenico Delli Gatti & Mauro Gallegati & Bruce C. Greenwald & Joseph E. Stiglitz, 2009. "Liaisons Dangereuses: Increasing Connectivity, Risk Sharing, and Systemic Risk," NBER Working Papers 15611, National Bureau of Economic Research, Inc.
  2. 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.
  3. Markus K. Brunnermeier, 2008. "Deciphering the Liquidity and Credit Crunch 2007-08," NBER Working Papers 14612, National Bureau of Economic Research, Inc.
  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. Jan Lorenz & Stefano Battiston & Frank Schweitzer, 2009. "Systemic Risk in a Unifying Framework for Cascading Processes on Networks," Papers 0907.5325, arXiv.org, revised Jan 2010.
  6. 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.
  7. 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.
  8. 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.
  9. Larry Eisenberg & Thomas H. Noe, 2001. "Systemic Risk in Financial Systems," Management Science, INFORMS, vol. 47(2), pages 236-249, February.
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