Network effects and default clustering for large systems: typical and atypical events
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We consider a large collection of dynamically interacting components defined on a weighted directed graph determining the impact of default of one component to another one. The empirical measure captures the evolution of different components in the system and from this we extract important information for quantities such as the loss rate in the overall pool and the mean impact on a given component from system wide defaults. We prove a law of large numbers and large deviations on the empirical measure of the survival distribution in the system. In addition, we study the typical and atypical behavior of statistics of interest under the combined effect of default cluster and network structure. A singular value decomposition of the adjacency matrix of the graph allows to coarse-grain the system by focusing on the highest eigenvalues which also correspond to the components with the highest contagion impact on the system. Numerical simulations demonstrate the theoretical findings in both the law of large number and in the large deviations principle.