通过谱子流形对复杂网络进行非线性模型降维
Nonlinear Model Reduction of Complex Networks via Spectral Submanifolds
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中文总结 AI 辅助
研究针对复杂网络高维非线性动力学分析预测难题，提出基于谱子流形（SSM）的降维框架及其全局扩展（gSSM），该方法能准确预测，SSM可作临界点预测器，在建模临界转变上优于经典方法，是适用于多领域的强大工具。
中文摘要 AI 辅助
复杂网络系统在生物学、工程学和社会科学中普遍存在，但其高维非线性动力学给分析和预测带来重大挑战。一种数学上严格的简化途径是在称为谱子流形（SSM）的低维光滑不变流形上表示系统行为。本文提出了一个全面的SSM降维框架及其全局扩展（gSSM）用于大规模非线性网络的降维。我们的方法在合成网络和真实网络上都能产生准确的全局和节点级预测。SSM是一个强大的临界点预测器，即使在低截断阶（如$O(2)$）时也能可靠地识别持续活动的开始，更高阶和gSSM能捕捉开始后的振幅和饱和度。在所有实现中，SSM/gSSM在微观和宏观尺度上建模临界转变时始终优于经典谱方法和平均场方法，确立了基于SSM的降维作为一种强大、可解释的工具，广泛适用于流行病学、生态学和工程网络。
英文摘要
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.
