Systemic Risk in a Unifying Framework for Cascading Processes on Networks

European Physical Journal B, Vol. 71, No. 4, pp. 441-460, 2009

CCSS Working Paper No. 09-011

45 Pages Posted: 27 Apr 2010

See all articles by Jan Lorenz

Jan Lorenz

Jacobs University Bremen

Stefano Battiston

University of Zurich - Department of Banking and Finance; Swiss Finance Institute

Frank Schweitzer

ETH Zürich

Date Written: December 12, 2009


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 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.

Keywords: Networks, Economics Econophysics, Financial Markets, Business And Management, Dynamics of Social Systems, Structural Failure of Materials

Suggested Citation

Lorenz, Jan and Battiston, Stefano and Schweitzer, Frank, Systemic Risk in a Unifying Framework for Cascading Processes on Networks (December 12, 2009). CCSS Working Paper No. 09-011. Available at SSRN:

Jan Lorenz

Jacobs University Bremen ( email )

Campus Ring 1
Bremen, 28759

Stefano Battiston

University of Zurich - Department of Banking and Finance ( email )

Andreasstrasse 15
Zürich, 8050

Swiss Finance Institute

c/o University of Geneva
40, Bd du Pont-d'Arve
CH-1211 Geneva 4

Frank Schweitzer (Contact Author)

ETH Zürich ( email )

Weinbergstrasse 56/58
WEV Room G 211
Zurich, CH-8032
+41 44 632 83 50 (Phone)


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