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

The transmission of financial distress depends on the structure of exposures, the distribution of institutional buffers, and the duration of stress. We develop a dynamic model in which losses flow through weighted directed networks while counterparties are distressed, with recovery and repeated threshold crossings incorporated into the same system. A representation with \(K\) exposure factors closes the \(N\)-institution dynamics through \(K\) feedback coordinates, preserving heterogeneous sender and receiver roles. For bounded Lipschitz losses, Wasserstein stability and aligned \(L^1\) kernel estimates connect finite populations, factor models, and directed-kernel limits, with separate bounds for sampling and exposure approximation. For the hard threshold, every bounded nonnegative kernel admits a greatest cumulative-distress solution selected by vanishing positive-side regularization. The mass of institutions near the threshold governs stability: an Osgood condition along one reference path yields uniqueness, quantitative perturbation bounds, and convergence under sampled latent labels. Rank-one examples establish sharpness among uniqueness criteria based on threshold-layer mass, and a branchwise condition verifies the regularity directly from the initial profile and kernel. Numerical experiments examine how network structure and threshold concentration affect the dynamics. An application to disclosed EBA sovereign holdings constructs common-exposure factors and quantifies sensitivity to portfolio perturbations.