Dimensionality reduction of discrete-time dynamical systems

TL;DR

This paper introduces an analytical framework to reduce high-dimensional discrete-time dynamical systems to a low-dimensional manifold, enabling better understanding and prediction of system transitions in complex networks.

Contribution

It provides the first analytical method for collapsing discrete-time systems into a low-dimensional space based on effective parameters, improving analysis of complex network dynamics.

Findings

Framework accurately predicts low-dimensional collapse quality

Successfully applied to real-world systems

Identifies transition regions in parameter space

Abstract

One of the outstanding problems in complexity science and dynamical system theory is understanding the dynamic behavior of high-dimensional networked systems and their susceptibility to transitions to undesired states. Because of varied interactions, large number of parameters and different initial conditions, the study is extremely difficult and existing methods can be applied only to continuous-time systems. Here we propose an analytical framework for collapsing N-dimensional discrete-time systems into a S+1-dimensional manifold as a function of S effective parameters with S << N. Specifically, we provide a quantitative prediction of the quality of the low-dimensional collapse. We test our framework on a variety of real-world complex systems showing its good performance and correctly identify the regions in the parameter space corresponding to the system's transitions. Our work offers an analytical tool to reduce dimensionality of discrete-time networked systems that can be applied to a broader set of systems and dynamics.