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 Itô{} 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 Barabási--Albert networks confirm that hub dominance magnifies nonlinear gradients and compresses stability margins, producing heavy-tailed survival distributions and occasional explosive behaviour.