Probability Seminar Guido Lagos (University of Santiago, Chile)

Systemic risk & reliability in networks: a new asymptotic

Systemic risk in networks studies the risk of collapse of the entire system due to failure, shocks or actions of components of the network. This area has received particular attention in the last decade or so due to the need to better understand recent financial and economical crises. In this context, it is of interest to analyze, from a probabilistic perspective, the instant at which the whole system is considered to fail, in the case where there are failures that simultaneously affect several components of the network.

In this work we take a new approach in the field of reliability and systemic risk and study the following question: Are there asymptotical regimes for the probabilistic behavior of the network that allow to approximate the system and ultimately say something about the its reliability? Here, in broad terms, we refer to "asymptotic regime" as an equilibrium distribution or state of the system as the size of the network and time grow together in a "balanced" way. We answer the previous question in the positive, in the case where the dependency model for the failures is a particular exponential Marshall-Olkin copula and when the whole system is considered to fail when the last component fails — or more generally when the k-th component (out of a total of n components) fails.

Our main contribution is a collection of results giving a detailed analysis of a non-trivial scaling regime for the probability of the network being
working at a certain time, as the time and size of the network scales. These approximations allow to estimate and give confidence bounds for the
failure probabilities of the system. Moreover, the dependency model we study only needs to specify a reduced number of parameters, and our
asymptotic results shed light on the effects of choosing such parameters.

Based on the paper Joint work with Javiera Barrera (Universidad Adolfo Ibáñez).

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