External or internal shocks may lead to the collapse of a system consistingof many agents. If the shock hits only one agent initially and causes it tofail, this can induce a cascade of failures among neighoring agents. Severalcritical constellations determine whether this cascade remains finite orreaches the size of the system, i.e. leads to systemic risk. We investigate thecritical parameters for such cascades in a simple model, where agents arecharacterized by an individual threshold \theta_i determining their capacity tohandle a load \alpha\theta_i with 1-\alpha being their safety margin. If agentsfail, they redistribute their load equally to K neighboring agents in a regularnetwork. For three different threshold distributions P(\theta), we deriveanalytical results for the size of the cascade, X(t), which is regarded as ameasure of systemic risk, and the time when it stops. We focus on two differentregimes, (i) EEE, an external extreme event where the size of the shock is ofthe order of the total capacity of the network, and (ii) RIE, a random internalevent where the size of the shock is of the order of the capacity of an agent.We find that even for large extreme events that exceed the capacity of thenetwork finite cascades are still possible, if a power-law thresholddistribution is assumed. On the other hand, even small random fluctuations maylead to full cascades if critical conditions are met. Most importantly, wedemonstrate that the size of the "big" shock is not the problem, as thesystemic risk only varies slightly for changes of 10 to 50 percent of theexternal shock. Systemic risk depends much more on ingredients such as thenetwork topology, the safety margin and the threshold distribution, which giveshints on how to reduce systemic risk.
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