Contagion Threshold

Contagion threshold is the critical fraction of failed or stressed nodes in a network above which local perturbations propagate globally. Below the threshold, failures are contained by network structure — modularity, redundancy, or dissipation absorb the shock. Above the threshold, the same network topology that provided efficiency becomes a transmission mechanism, and the fraction of affected nodes grows nonlinearly. The threshold depends on the ratio of amplification strength to topological compartmentalization: highly coupled networks with strong positive feedback have low thresholds; modular networks with negative feedback have high thresholds. In financial contagion, the threshold is often crossed not by the size of the initial shock but by the correlation structure that synchronizes failures across previously independent nodes.

The contagion threshold is not a property of the shock. It is a property of the network's hidden geometry.