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Systemic Risk in a Unifying Framework for Cascading Processes on Networks

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  • Jan Lorenz
  • Stefano Battiston
  • Frank Schweitzer

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, $X^\ast$, 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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Bibliographic Info

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

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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. Jan Lorenz & Stefano Battiston & Frank Schweitzer, . "Systemic Risk in a Unifying Framework for Cascading Processes on Networks," Working Papers CCSS-09-011, ETH Zurich, Chair of Systems Design.
  3. Markus K. Brunnermeier, 2008. "Deciphering the Liquidity and Credit Crunch 2007-08," NBER Working Papers 14612, National Bureau of Economic Research, Inc.
  4. Larry Eisenberg & Thomas H. Noe, 2001. "Systemic Risk in Financial Systems," Management Science, INFORMS, vol. 47(2), pages 236-249, February.
  5. 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.
  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. 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.
  9. 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.
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Citations

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Cited by:
  1. Frank Schweitzer & Giorgio Fagiolo & Didier Sornette & Fernando Vega-Redondo & Douglas R. White, . "Economic Networks: What do we know and what do we need to know?," Working Papers CCSS-09-010, ETH Zurich, Chair of Systems Design.
  2. F. Schweitzer & D. Garcia, 2010. "An agent-based model of collective emotions in online communities," The European Physical Journal B - Condensed Matter and Complex Systems, Springer, vol. 77(4), pages 533-545, October.
  3. Teruyoshi Kobayashi, 2013. "A model of financial contagion with variable asset returns may be replaced with a simple threshold model of cascades," Discussion Papers 1315, Graduate School of Economics, Kobe University.
  4. 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.
  5. Teruyoshi Kobayashi & Kohei Hasui, 2013. "Efficient immunization strategies to prevent financial contagion," Papers 1308.0652, arXiv.org, revised Dec 2013.
  6. Jean-Philippe Bouchaud, 2012. "Crises and collective socio-economic phenomena: simple models and challenges," Papers 1209.0453, arXiv.org, revised Dec 2012.
  7. Claudio J. Tessone & Antonios Garas & Beniamino Guerra & Frank Schweitzer, . "How big is too big? Critical Shocks for Systemic Failure Cascades," Working Papers ETH-RC-12-015, ETH Zurich, Chair of Systems Design.
  8. Sandoval, Leonidas Junior, 2013. "To lag or not to lag? How to compare indices of stock markets that operate at different times," Insper Working Papers wpe_319, Insper Working Paper, Insper Instituto de Ensino e Pesquisa.
  9. Andreas Joseph & Stephan Joseph & Guanrong Chen, 2013. "Cross-border Portfolio Investment Networks and Indicators for Financial Crises," Papers 1306.0215, arXiv.org, revised Jan 2014.
  10. Josselin Garnier & George Papanicolaou & Tzu-Wei Yang, 2012. "Large deviations for a mean field model of systemic risk," Papers 1204.3536, arXiv.org, revised Aug 2012.

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