The Graph-Embedded Hazard Model (GEHM): Stochastic Network Survival Dynamics on Economic Graphs

TL;DR

This paper introduces a novel nonlinear stochastic PDE-SDE framework for modeling survival dynamics on economic networks, revealing how network topology influences stability and risk of explosive failures.

Contribution

It develops a new mathematical model coupling graph-based diffusion with stochastic drift, providing theoretical analysis and numerical validation for survival behavior on economic graphs.

Findings

Hub dominance increases nonlinear gradients and reduces stability margins.

Heavy-tailed survival distributions emerge in scale-free networks.

Explosive behavior can occur under certain network conditions.

Abstract

This paper develops a nonlinear evolution framework for modelling survival dynamics on weighted economic networks by coupling a graph-based $p$-Laplacian diffusion operator with a stochastic structural drift. The resulting finite-dimensional PDE--SDE system captures how node-level survival reacts to nonlinear diffusion pressures while an aggregate complexity factor evolves according to an It\^o{} process. Using accretive operator theory, nonlinear semigroup methods, and stochastic analysis, we establish existence and uniqueness of mild solutions, derive topology-dependent energy dissipation inequalities, and characterise the stability threshold separating dissipative, critical, amplifying, and explosive regimes. Numerical experiments on Barab\'asi--Albert networks confirm that hub dominance magnifies nonlinear gradients and compresses stability margins, producing heavy-tailed survival distributions and occasional explosive behaviour.