Fractional catastrophe geometry of rural market percolation: a Koopman-Spectral analysis of socioeconomic complexity in off-season price fluctuations
DOI:
https://doi.org/10.21839/jaar.2026.v11.10051Keywords:
Catastrophe theory, Fractional diffusion, Information geometry, Koopman operator, Percolation/network contagionAbstract
This article presents a six-layer mathematical framework to model rural off‑season price dynamics as a multi-scale complex system. Layer I formulates stochastic fractional reaction-diffusion-advection PDEs with anisotropic diffusion and multiplicative weather-income noise to capture anomalous spatial transport. Layer II couples regional prices via fractional Brownian motions, yielding fractional Liouville dynamics that encode long‑range temporal dependence. Layer III employs cusp catastrophe theory to characterize abrupt price bifurcations and stream‑tube flow in control‑parameter space. Layer IV uses information geometry (Fisher–Rao metric, KL divergence, heat kernel) to quantify statistical divergence and market proximity on a Riemannian manifold. Layer V applies Koopman operator spectral theory and EDMD to extract modal decompositions and intrinsic mixing timescales. Layer VI frames market connectivity with random geometric graphs and percolation theory, relating giant‑component emergence, critical exponents, and contagion thresholds to systemic risk. Integrating fractional transport, critical transitions, geometric information measures, operator modes, and network percolation explains persistent volatility and systemic contagion in off‑season markets and suggests targeted interventions, namely storage capacity, income stabilization, and connectivity regulation, to mitigate catastrophic price events. Extensions include variable‑order fractional operators, Lévy flight jumps, deep‑Koopman representations, and nonperturbative percolation analysis.
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