Nonlinear model reduction of complex networks via spectral submanifolds
TL;DR
Across all the realizations, SSM/gSSM consistently outperform classical spectral and mean-field methods in modeling critical transitions at both microscopic and macroscopic scales, establishing SSM-based reduction as a robust, interpretable tool for nonlinear networked systems with broad applicability to epidemiology, ecology, and engineered networks.
Abstract
Complex networked systems are prevalent in biology, engineering, and the social sciences, yet their high-dimensional, nonlinear dynamics pose major challenges for analysis and prediction. A mathematically rigorous route to simplification is to represent system behavior on a low-dimensional, smooth invariant manifold known as a spectral submanifold (SSM). Here we present a comprehensive SSM reduction framework and its globalized extension (gSSM) for dimensionality reduction in large-scale nonlinear networks. Our approach yields accurate global and node-level predictions across synthetic and real networks, including highly heterogeneous topologies and systems with higher-order interactions. Crucially, SSM is a robust tipping-point predictor: even at low truncation order (e.g., $O(2)$) it reliably identifies the onset of sustained activity, while higher orders and gSSM capture post-onset amplitudes and saturation. Consistently, the reduction collapses the full network dynamics to a one-dimensional system, offering clarity and efficiency. Across all the realizations, SSM/gSSM consistently outperform classical spectral and mean-field methods in modeling critical transitions at both microscopic and macroscopic scales, establishing SSM-based reduction as a robust, interpretable tool for nonlinear networked systems with broad applicability to epidemiology, ecology, and engineered networks.