Global Synchronization in Matrix-Weighted Networks

TL;DR

This paper explores how global synchronization occurs in Matrix-Weighted Networks, revealing that structural coherence is essential for synchronization in complex multidimensional systems modeled by oscillators and chaotic systems.

Contribution

It introduces a generalized Master Stability Function for MWNs and establishes coherence as a necessary condition for global synchronization.

Findings

Coherence is necessary for global synchronization in MWNs.

Derived a generalized MSF for matrix-weighted networks.

Applied to oscillators and chaotic systems, confirming theoretical predictions.

Abstract

Synchronization phenomena in complex systems are fundamental to understanding collective behavior across disciplines. While classical approaches model such systems by using scalar-weighted networks and simple diffusive couplings, many real-world interactions are inherently multidimensional and transformative. To address this limitation, Matrix-Weighted Networks (MWNs) have been introduced as a versatile framework where edges are associated with matrix weights that encode both interaction strength and directional transformation. In this work, we investigate the emergence and stability of global synchronization (GS) in MWNs by studying coupled Stuart-Landau (SL) oscillators, an archetypal model of nonlinear dynamics near a Hopf bifurcation. Besides the SL, we considered a generalization of regular oscillators to higher dimensions and also the Lorenz model as a prototype of chaotic oscillators. We derive a generalized Master Stability Function (MSF) tailored to MWNs and establish necessary and sufficient conditions for GS to occur. Central to our analysis is the concept of coherence, a structural property of MWNs ensuring path-independent transformations. Our results show that coherence is necessary to have global synchronization and provides a theoretical foundation for analyzing multidimensional dynamical processes in complex networked systems.