Generalized dimension reduction approach for heterogeneous networked systems with time-delay

TL;DR

This paper introduces a generalized dimension reduction method for heterogeneous networked systems with time delays, enabling accurate prediction of system dynamics and critical points in complex networks.

Contribution

It develops a novel dimension reduction technique that extends existing methods to heterogeneous networks with multiple communities and time delays.

Findings

Accurately predicts system states and tipping points.

Effectively approximates system evolution in complex networks.

Identifies critical points in networks with time delay.

Abstract

Networks of interconnected agents are essential to study complex networked systems' state evolution, stability, resilience, and control. Nevertheless, the high dimensionality and nonlinear dynamics are vital factors preventing us from theoretically analyzing them. Recently, the dimension-reduction approaches reduced the system's size by mapping the original system to a one-dimensional system such that only one effective representative can capture its macroscopic dynamics. However, the approaches dramatically fail as the network becomes heterogeneous and has multiple community structures. Here, we bridge the gap by developing a generalized dimension reduction approach, which enables us to map the original system to a $m$-dimensional system that consists of $m$ interacting components. Notably, by validating it on various dynamical models, this approach accurately predicts the original system state and the tipping point, if any. Furthermore, the numerical results demonstrate that this approach approximates the system evolution and identifies the critical points for complex networks with time delay.