Low-rank and graphon limits for dynamic threshold distress contagion in heterogeneous financial networks

Abstract

We study distress contagion in financial networks with weighted directed exposures. Losses accumulate while counterparties are below a threshold, allowing institutions to recover. A representation with \(K\) exposure factors reduces the \(N\)-institution dynamics exactly to \(K\) feedback coordinates. For bounded Lipschitz losses, Wasserstein stability of the reduced system and aligned \(L^1\) stability of the directed-kernel equation give error bounds separating population sampling from kernel approximation. For the hard threshold, every bounded nonnegative kernel has a greatest cumulative-distress solution, selected by vanishing positive-side regularization. An Osgood condition on the mass near the threshold along one reference path yields uniqueness, stability, deterministic approximation bounds, and convergence under sampled latent labels. Rank-one examples show that this condition is sharp for uniqueness criteria based only on threshold-layer mass. A branchwise condition verifies the required regularity from the initial profile and kernel. Numerical examples examine low-rank reduction, approximation error, and solution selection; an application to disclosed EBA sovereign holdings constructs exposure factors and evaluates sensitivity bounds.