Spectral Contagion: Network Fragility through the Latent Lens

We formalize financial contagion as a Grade-2 hazard model on networks and show how the cascade threshold is tied, under explicit spectral proportionalities, to a concentration index: the Latent Number \(\rho\) of the interbank network. The hazard function \(h(D) = \beta D^2 - \alpha D\) admits a critical default fraction \(D_c = \alpha/\beta\), and the bridge layer identifies \(\beta \propto \lambda_1\) (spectral radius of the exposure matrix) up to a positive base scale. Under explicit proportionalities linking \(\beta\) to a concentration index, \(D_c \propto 1/\rho\): more concentrated networks (higher \(\rho\)) have lower contagion thresholds and are more fragile. We derive machine-checked consequences that contagion amplifies priced variance along a \((1+\rho)\) factor and tighten several regulatory readings (buffers versus spectral stress; deconcentration raises \(D_c\) in the bridge parameterization). Verification: 37 theorems checked across two companion scripts—the Grade-2 base file records eight modeling primitives (threshold identity, hazard map, counterparty-counting layer); the bridge file is axiom-free at the domain layer.