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Global synchronization in Matrix-Weighted networks

A. Gallo,Yu Tian,R. Lambiotte,Timoteo Carletti

Published 2025 in Communications Physics

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. This work explores the emergence and stability of global synchronization in Matrix Weighted Networks through the study of coupled Stuart-Landau oscillators. It provides a theoretical basis for the analysis of multidimensional dynamical processes in complex networked systems.

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