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Systemic risk in a unifying framework for cascading processes on networks

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  • J. Lorenz

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  • S. Battiston
  • F. 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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Suggested Citation

  • 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;EDP Sciences, vol. 71(4), pages 441-460, October.
  • Handle: RePEc:spr:eurphb:v:71:y:2009:i:4:p:441-460 DOI: 10.1140/epjb/e2009-00347-4
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    1. 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;EDP Sciences, vol. 71(4), pages 441-460, October.
    2. Markus K. Brunnermeier, 2008. "Deciphering the Liquidity and Credit Crunch 2007-08," NBER Working Papers 14612, National Bureau of Economic Research, Inc.
    3. 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;EDP Sciences, vol. 46(1), pages 101-107, July.
    4. 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.
    5. Larry Eisenberg & Thomas H. Noe, 2001. "Systemic Risk in Financial Systems," Management Science, INFORMS, pages 236-249.
    6. 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;EDP Sciences, vol. 67(3), pages 301-318, February.
    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;EDP Sciences, vol. 50(1), pages 243-247, March.
    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. 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.
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    Citations

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    Cited by:

    1. 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.
    2. Paolo Giudici & Laura Parisi, 2016. "Bail in or Bail out? The Atlante example from a systemic risk perspective," DEM Working Papers Series 124, University of Pavia, Department of Economics and Management.
    3. 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;EDP Sciences, vol. 71(4), pages 441-460, October.
    4. Safarzyńska, Karolina & van den Bergh, Jeroen C.J.M., 2017. "Integrated crisis-energy policy: Macro-evolutionary modelling of technology, finance and energy interactions," Technological Forecasting and Social Change, Elsevier, vol. 114(C), pages 119-137.
    5. Periklis Gogas & Theophilos Papadimitriou & Maria-Artemis Matthaiou, 2014. "A Novel Banking Supervision Method using the Minimum Dominating Set," Working Paper series 29_14, Rimini Centre for Economic Analysis.
    6. Kobayashi, Teruyoshi, 2014. "A model of financial contagion with variable asset returns may be replaced with a simple threshold model of cascades," Economics Letters, Elsevier, vol. 124(1), pages 113-116.
    7. Paolo Giudici & Laura Parisi, 2016. "CoRisk: measuring systemic risk through default probability contagion," DEM Working Papers Series 116, University of Pavia, Department of Economics and Management.
    8. 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.
    9. Sam Langfield & Kimmo Soramäki, 2016. "Interbank Exposure Networks," Computational Economics, Springer;Society for Computational Economics, vol. 47(1), pages 3-17, January.
    10. Teruyoshi Kobayashi & Kohei Hasui, 2013. "Efficient immunization strategies to prevent financial contagion," Papers 1308.0652, arXiv.org, revised Dec 2013.
    11. Sergey Nadtochiy & Mykhaylo Shkolnikov, 2017. "Particle systems with singular interaction through hitting times: application in systemic risk modeling," Papers 1705.00691, arXiv.org.
    12. Gualdi, Stanislao & Tarzia, Marco & Zamponi, Francesco & Bouchaud, Jean-Philippe, 2015. "Tipping points in macroeconomic agent-based models," Journal of Economic Dynamics and Control, Elsevier, vol. 50(C), pages 29-61.
    13. 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;EDP Sciences, vol. 77(4), pages 533-545, October.
    14. Jean-Philippe Bouchaud, 2012. "Crises and collective socio-economic phenomena: simple models and challenges," Papers 1209.0453, arXiv.org, revised Dec 2012.
    15. Leonidas Sandoval Junior, 2014. "Dynamics in two networks based on stocks of the US stock market," Papers 1408.1728, arXiv.org, revised Aug 2014.
    16. repec:wsi:acsxxx:v:12:y:2009:i:04n05:n:s0219525909002337 is not listed on IDEAS
    17. 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.
    18. Matthew Elliott & Benjamin Golub & Matthew O. Jackson, 2014. "Financial Networks and Contagion," American Economic Review, American Economic Association, vol. 104(10), pages 3115-3153, October.
    19. Ellinas, Christos & Allan, Neil & Johansson, Anders, 2016. "Project systemic risk: Application examples of a network model," International Journal of Production Economics, Elsevier, vol. 182(C), pages 50-62.
    20. Claudio J. Tessone & Antonios Garas & Beniamino Guerra & Frank Schweitzer, "undated". "How big is too big? Critical Shocks for Systemic Failure Cascades," Working Papers ETH-RC-12-015, ETH Zurich, Chair of Systems Design.
    21. Rebekka Burkholz & Matt V. Leduc & Antonios Garas & Frank Schweitzer, 2015. "Systemic risk in multiplex networks with asymmetric coupling and threshold feedback," Papers 1506.06664, arXiv.org.
    22. Frank Schweitzer & Giorgio Fagiolo & Didier Sornette & Fernando Vega-Redondo & Douglas R. White, 2009. "Economic Networks: What Do We Know And What Do We Need To Know?," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 12(04n05), pages 407-422.
    23. Paolo Giudici & Laura Parisi, 2015. "Modeling Systemic Risk with Correlated Stochastic Processes," DEM Working Papers Series 110, University of Pavia, Department of Economics and Management.
    24. Paolo Giudici & Peter Sarlin & Alessandro Spelta, 2016. "The multivariate nature of systemic risk: direct and common exposures," DEM Working Papers Series 118, University of Pavia, Department of Economics and Management.

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