A Unified Sigmoidal Difference Equation for Modeling Multi-Domain Cascade and Criticality Dynamics in Heterogeneous Networks

We introduce a discrete-time dynamical framework — the Logistic General Propagation Equation (LGPE) — to model excitability and cascade dynamics on complex networks. While domain-specific models such as Hopfield networks (neuroscience), metapopulation SIS models (epidemiology), Olami-Feder-Christensen systems (seismology), and Bak–Tang–Wiesenfeld sandpiles (self-organized criticality) describe network propagation under isolated mathematical formalisms, our sigmoidal difference equation highlights shared mathematical analogies among these processes. The LGPE integrates a continuous sigmoidal activation threshold (phi), a physical saturation operator (sigma), local dissipation (gamma), and stochastic noise (eta). Through computational validation across 24 scenarios (8 real-world systems x 3 regimes), we demonstrate that the LGPE reproduces: (1) critical phase transitions governed by the topological threshold lambda_c = <k> / <k^2> * gamma, (2) power-law avalanche distributions P(s) ~ s^-tau with tau ≈ 1.5 at criticality, and (3) excitation-relaxation cycles across all mapped systems. Rather than claiming a literal physical unification, this work proposes a minimalist, differentiable, and degree-normalized model that serves as a useful conceptual benchmark for studying multi-domain network propagation.