# Network effects in default clustering for large systems

## TL;DR

This paper models how defaults spread in large interconnected systems using graph theory, proving a law of large numbers and identifying key components with high contagion impact through spectral analysis.

## Contribution

It introduces a law of large numbers for default clustering in large systems and uses spectral decomposition to identify influential components.

## Key findings

- Law of large numbers for default measures
- Identification of high-impact components via eigenvalues
- Numerical validation of theoretical results

## Abstract

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. We prove a law of large numbers for the empirical measure capturing the evolution of the different components in the pool and from this we extract important information for quantities such as the loss rate in the overall pool as well as the mean impact on a given component from system wide defaults. 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 pool. Numerical simulations demonstrate the theoretical findings.