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

We study a recoverable dynamic distress model for financial institutions connected by a weighted directed exposure network. Counterparty distress affects losses through cumulative occupation time, so institutions may subsequently recover. A \(K\)-factor representation of exposures, interpreted as a small number of dominant transmission channels, reduces the \(N\)-institution system exactly to \(K\) macroscopic feedback coordinates. For bounded Lipschitz losses, we prove Wasserstein stability of the reduced dynamics and aligned \(L^1\) stability of the corresponding directed-kernel equation, obtaining quantitative error bounds that separate finite-population and kernel-approximation effects. We then address the economically natural hard-threshold rule, whose discontinuity creates solution-selection and stability problems. For every bounded nonnegative directed kernel, we construct a canonical greatest cumulative-distress solution and show that it is selected by vanishing positive-side regularization. An Osgood condition controlling the mass near the distress threshold along a reference solution yields uniqueness, one-sided stability, finite-network and low-rank approximation guarantees, and convergence under sampled latent labels. Nonnegative rank-one examples show that this condition is sharp for uniqueness among criteria based only on threshold-layer mass. Numerical experiments illustrate the reduction and selection mechanisms. Using public data from the 2025 EBA transparency exercise, we construct sovereign-exposure factors and evaluate the associated sensitivity bounds.