Global Synchronization in Matrix-Weighted Networks

TL;DR

The paper extends synchronization analysis from scalar-weighted networks to Matrix-Weighted Networks (MWNs), where edges carry multidimensional linear transformations. By introducing a coherence condition on edge transformations and a generalized Master Stability Function (MSF), the authors derive necessary and sufficient criteria for global synchronization (GS) of multidimensional node dynamics, and demonstrate the theory on Stuart–Landau oscillators, higher-dimensional systems, and chaotic Lorenz models. A key insight is that a frame change via a coherence-based similarity transform reveals a generalized synchronization manifold and reduces stability analysis to the spectrum of a transformed supra-Laplacian. The framework connects to connection Laplacians, equivariant dynamics, and multilayer/multidimensional network theories, with potential applications in neuroscience, social dynamics, and graph-based learning. Overall, the work provides a rigorous, transferable method for predicting and engineering GS in complex, multidimensional networks.

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.

Figures (12)

• Figure 1: Examples of coherent MWNs. In panel a), we show a triangular network whose matrix weights are denoted by $\mathbf{R}_{ij}$, notice that the matrix associated to each reciprocal link is the transpose of the direct one (shown in light gray). Moreover $\mathbf{R}_{13}=\mathbf{R}_{12}\mathbf{R}_{23}$ is the condition to have coherence. In panel b), we report a MWN obtained by joining two triangles; for the sake of simplicity, we only show directed weighted matrices, the reciprocal ones are assumed to be given by the transpose as in panel a. Moreover, we assume the following conditions hold true to ensure the coherence: $\mathbf{R}_{13}=\mathbf{R}_{12}\mathbf{R}_{23}$ and $\mathbf{R}_{15}=\mathbf{R}_{14}\mathbf{R}_{45}$. In panel c), we propose a MWN obtained by gluing a triangle and a square; once again, the matrix weights of the reciprocal edges are given by the transpose of the original matrix weight. The coherence condition is given by: $\mathbf{R}_{12}\mathbf{R}_{23}=\mathbf{R}_{13}$ and $\mathbf{R}_{24}\mathbf{R}_{45}=\mathbf{R}_{23}\mathbf{R}_{35}$.
• Figure 2: Global Synchronization of SL coupled via the $\mathrm{MWN_a}$ shown in Fig. \ref{['fig:CohNet1']}a. By visual inspection of the Master Stability Function presented in panel (a), one can clearly appreciate that the latter is negative for all $\Lambda^{(\alpha)}>0$ (red dots), ensuring thus the onset of GS as testified by the order parameters reaching the constant value $\sqrt{\sigma_{\Re}/\beta_{\Re}}=1$ (panel (b)) and the time evolution of $u_j(t)$ for short and long times (panel (c)). The remaining model parameters are given by $\sigma = 1.0 - i 0.5$, $\beta = 1.0+i 1.1$, $m = 3$ and $\mu = 0.1-i 5.5$. The scalar weights have been drawn from $\mathrm{U}[0,0.8]$ and the rotation matrices have been set to $\mathbf{R}_{12}=\mathbf{R}_{23}= \cos 2\pi/3-\sin 2\pi/3\sin 2\pi/3\cos 2\pi/3$ and $\mathbf{R}_{13}= \cos 2\pi/3\sin 2\pi/3-\sin 2\pi/3\cos 2\pi/3$.
• Figure 3: Absence of Global Synchronization of SL coupled via the $\mathrm{MWN_a}$ shown in Fig. \ref{['fig:CohNet1']}a. The Master Stability Function (panel (a)) assumes positive values, hence the order parameter (panel (b)) cannot converge to $\sqrt{\sigma_{\Re}/\beta_{\Re}}=1$ and $u_j(t)$ oscillates out of phase for short and long times (panel (c)). The remaining model parameters and rotation matrices have been set to the same values of Fig. \ref{['fig:NumResEx1Sync']} but $w_{ij}$, now drawn from $\mathrm{U}[0,0.1]$.
• Figure 4: Impact of the coherence assumption on GS of SL coupled via the $\mathrm{MWN_b}$ shown in Fig. \ref{['fig:CohNet1']}b. Panels (a), (b) and (c) report simulations obtained under the assumption of coherent network, i.e., we set $\theta_1=2\pi/3$, $\theta_2=2\pi/5$ and $\theta=\pi/5$ in Eqs. \ref{['eq:O1jb']}- \ref{['eq:O1jb4']}; the Master Stability Function (panel (a)) is non-positive and the system converges to synchronization, $R(t)$ stabilizes at $\sqrt{\sigma_{\Re}/\beta_{\Re}}=1$ (panel (b)) and the oscillators are in phase after a transient time (panel (c)). In the panels (d), (e) and (f), the coherence condition does not hold true because we set $\theta=2\pi/5$ in Eq. \ref{['eq:O1jb4']}, and the system is not capable to synchronize, the order parameter converges to a small value (panel (e)) and two oscillators are out of phase (panel (f)). The model parameters are given by $\sigma = 1.0 - i0.5$, $\beta = 1.0+i1.1$, $m = 3$, $\mu = 0.1-i5.5$ and the scalar weights have been drawn from $\mathrm{U}[0,0.8]$.
• Figure 5: Synchronization of the $3$D-system \ref{['eq:3Dmodnety']} on the $\mathrm{MWN_a}$ shown in Fig. \ref{['fig:CohNet1']}a with rotation matrices given by \ref{['eq:rotmat3nodes']} and \ref{['eq:rotmat3nodesb']} and scalar weights $w_{ij}=1$. Panel (a): the Master Stability Function; panel (b): $\xi_{1,j}(t)$, i.e., the first component of $\vec{y}_j$, and panel (c): $y_{3,j}(t)$, for $j=1,2,3$, for short and long times. The matrices, $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$ are defined with $a = 2\pi/5$, $b = 2\pi/3$, $c = 2\pi/7$, $\lambda_A = 1$, $\lambda_B = -3$, $\lambda_C = 2$, $\mu_A = 1$, $\mu_B = 5$ and $\mu_C = 1$. The coupling parameters is fixed to $\epsilon = 0.1$.

Theorems & Definitions (2)

• Remark 1
• Remark 2