Dimensionality reduction in stochastic complex dynamical networks

Complex systems often exhibit diverse dynamical behaviors in high-dimensional spaces that depend on various factors. Dimensionality reduction is a powerful tool for analyzing and understanding complex systems, aiming to find a low-dimensional representation of the complex system that preserves its essential features and reveals its underlying mechanisms and long-term dynamics. However, most existing methods for dimensionality reduction are limited to deterministic systems and cannot account for the stochastic effects that are ubiquitous in realworld complex networks. Here we develop a general analytical framework for dimensionality reduction of stochastic complex dynamical networks that can capture the essential features and long-term dynamics of the original system in a low-dimensional effective equation. The effective equation is a function of a set of effective parameters that are associated with specific system states and determine the network's dynamical behavior. We show that the standard deviation of the effective equation can be used to analyze the dynamic behavior and possible convergence of the stochastic complex dynamical network. Our framework can be applied to various types of stochastic complex dynamical networks and can reveal the underlying mechanisms and emergent phenomena of these systems.