Dimension reduction of dynamical systems on networks with leading and non-leading eigenvectors of adjacency matrices

TL;DR

This paper investigates dimension reduction of networked dynamical systems using eigenvectors of the adjacency matrix, revealing that non-leading eigenvectors can sometimes outperform the leading one in approximating system behavior.

Contribution

It introduces a theoretical framework for selecting eigenvectors of the adjacency matrix for effective dimension reduction, including non-leading eigenvectors, and compares their performance.

Findings

Non-leading eigenvectors can outperform the leading eigenvector in certain cases.

The optimal eigenvector minimizes the approximation error.

Using the leading eigenvector is practically preferable despite theoretical advantages of non-leading eigenvectors.

Abstract

Dimension reduction techniques for dynamical systems on networks are considered to promote our understanding of the original high-dimensional dynamics. One strategy of dimension reduction is to derive a low-dimensional dynamical system whose behavior approximates the observables of the original dynamical system that are weighted linear summations of the state variables at the different nodes. Recently proposed methods use the leading eigenvector of the adjacency matrix of the network as the mixture weights to obtain such observables. In the present study, we explore performances of this type of one-dimensional reductions of dynamical systems on networks when we use non-leading eigenvectors of the adjacency matrix as the mixture weights. Our theory predicts that non-leading eigenvectors can be more efficient than the leading eigenvector and enables us to select the eigenvector minimizing the error. We numerically verify that the optimal non-leading eigenvector outperforms the leading eigenvector for some dynamical systems and networks. We also argue that, despite our theory, it is practically better to use the leading eigenvector as the mixture weights to avoid misplacing the bifurcation point too distantly and to be resistant against dynamical noise.