  Complex systems are ubiquitous in nature and engineering, but their analysis
and control are hampered by their high dimensionality and the influence of
various factors on their dynamics. Dimensionality reduction aims 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 assume
deterministic systems, while many real-world systems exhibit stochasticity.
Here, we develop a general analytical framework for dimensionality reduction of
stochastic complex dynamical networks that can map a high-dimensional system
with stochastic terms to a low-dimensional effective system with a single
effective state variable and few effective parameters. The effective parameters
are those that determine the network's dynamical behavior and are associated
with specific system states. The effective equation is a low-dimensional
representation of the original stochastic complex dynamical network that
preserves its essential dynamical features. The framework also allows us to
analyze the dynamic behavior and potential convergence of the stochastic
complex dynamical network by using the standard deviation of the effective
equation.
