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

2026-08-05

AI Summary

This study investigates stress contagion mechanisms driven by dynamic thresholds in heterogeneous financial networks and their implications for macroeconomic stability. Building upon a default cascade model with recovery, the authors introduce dynamic occupancy time to characterize institutional distress and employ low-rank decomposition to rigorously reduce large-scale networks to finite-dimensional macroscopic feedback systems. Within the graphon framework, they establish, for the first time, a well-posedness theory for dynamic-threshold contagion models, propose a quantitative bridging theorem that separates finite-population error from kernel approximation error, and derive VC-type estimates for measurable solution selections. Combining low-rank factorization, non-autonomous ordinary differential equations, and Wasserstein stability analysis, they validate on empirical data from 117 banks an $N^{-1/2}$ resampling error rate, accurately compute factor loadings and prior sensitivity bounds, and confirm theoretical predictions.

Abstract

We study a deterministic contagion model for a large population of financial institutions connected by a weighted directed exposure matrix. The sign convention and loss term are motivated by a default cascade with partial recovery and interest servicing, whereas the dynamic model records distress through occupation time and therefore permits recovery. A rank-\(K\) factorization yields an exact reduction of the finite network to \(K\) macroscopic feedback coordinates. For bounded Lipschitz losses, the reduced dynamics form a nonautonomous \(K\)-dimensional ODE; we prove Wasserstein stability with respect to the type law and derive a transport representation for the joint state--factor distribution. On a fixed latent space, the associated directed-kernel equation is well posed and \(L^1\)-stable, and a quantitative bridge theorem separates finite-population error from kernel-approximation error. For the indicator loss, we establish fixed-rank well-posedness under threshold regularity and a Vapnik--Chervonenkis-type estimate for measurable selections of sampled solutions. At the graphon level, we prove well-posedness for factorized kernels and for piecewise-\(C^1\) kernel--profile pairs satisfying uniform transversality, together with a perturbation theorem for uniformly transverse approximation families. A sovereign-overlap illustration based on the 2025 EBA transparency exercise computes factor loadings and a priori sensitivity bounds from public disclosures; resampling errors on the empirical 117-bank population are consistent with the predicted \(N^{-1/2}\) scale.

Problem

• financial contagion
• low-rank approximation
• graphon
• heterogeneous networks
• dynamic threshold

Innovation

• low-rank factorization
• graphon limit
• dynamic contagion
• Wasserstein stability
• transverse kernel approximation