Synchronization of higher-dimensional Kuramoto oscillators on networks: from scalar to matrix-weighted couplings

The Kuramoto model is the paradigmatic model to study synchronization in coupled oscillator systems. In its classical formulation, the oscillators move on the unit circle, each characterized by a scalar phase and a natural frequency, by interacting through a sinusoidal coupling. In this work, we propose a d-dimensional generalization in which oscillators are represented as unit vectors on the (d-1)-sphere and interact through a matrix-weighted network (MWN), a recently introduced framework where links are endowed with a matrix weight instead of a scalar one. We derive necessary conditions for global synchronization via a Master Stability Function approach: the existence of a synchronous solution requires identical frequency matrices across nodes and, in the MWN case, a coherence condition on the network structure. Through a suitable change of variables, the stability analysis reduces the full Nd-dimensional problem to a family of d-dimensional eigenvalue problems, each one parametrized by the eigenvalue of a suitable scalar weighted Laplacian, showing that the synchronous solution is locally stable for any positive coupling strength K on any connected network. Analytical results are complemented by numerical simulations.

💡 Research Summary

The paper extends the classic Kuramoto model, which describes synchronization of phase oscillators on the unit circle, to a d‑dimensional setting where each oscillator is a unit vector on the (d‑1)‑sphere S^{d‑1}. The intrinsic dynamics of each oscillator are governed by an antisymmetric matrix Ω_i (d×d), representing a constant rotation in the d‑dimensional space. Interaction among oscillators is mediated by a network. First, the authors consider the usual scalar‑weighted adjacency matrix A and define a mean‑field term Z_i(x)=K/N∑j A{ij}x_j. By projecting the coupling onto the tangent space of the sphere, the dynamics preserve the unit‑norm constraint.

A key result is that a globally synchronous solution (all x_i(t)=s(t) for some trajectory s) can exist only if all intrinsic frequency matrices are identical, Ω_i≡Ω. Under this assumption, a rotating‑frame transformation ξ_i=e^{‑Ωt}x_i removes the Ω term, reducing the system to a pure consensus dynamics on the sphere. Linearizing around the synchronous trajectory yields the variational equation δ̇x = (I_N⊗Ω – K L⊗I_d)δx, where L is the graph Laplacian. By diagonalizing L, the N·d‑dimensional problem decouples into N independent d‑dimensional blocks: δ̂̇x_α = (Ω – KΛ_α I_d)δ̂x_α, with Λ_α the Laplacian eigenvalues (Λ_1=0, Λ_{α>1}>0). Because Ω is antisymmetric, its eigenvalues are purely imaginary, so the real part of each block’s eigenvalues is –KΛ_α for α>1, which is strictly negative for any positive coupling K. Consequently, the synchronous state is locally asymptotically stable on any connected graph for all K>0.

The second major contribution is the incorporation of matrix‑weighted networks (MWNs). Each edge (i,j) carries a weight matrix W_{ij}=w_{ij}R_{ij}, where w_{ij}>0 is a scalar strength and R_{ij}∈O(d) is an orthogonal rotation. The authors impose a “coherence” condition: the product of R_{ij} around any directed cycle equals the identity. This condition is equivalent to the existence of node‑wise orthogonal matrices Q_i such that R_{ij}=Q_i^T Q_j. Defining O_{1i}=∏{path(1→i)}R{ij}, the change of variables y_i=O_{1i}x_i eliminates all edge rotations, yielding a dynamics identical in form to the scalar‑weighted case: ẏ_i = Ω y_i + ζ_i(y) – ⟨ζ_i(y),y_i⟩y_i, with ζ_i(y)=K/N∑j w{ij} y_j. Thus the MWN model reduces to a scalar‑weighted Kuramoto model on the same underlying graph, and the previous Master Stability Function (MSF) analysis applies directly. The same stability conclusion holds: for any connected MWN and any K>0, the synchronous trajectory is locally stable, provided the intrinsic frequency matrices are identical and the coherence condition is satisfied.

The paper validates the theory with numerical simulations for d=3 and d=4 on random Erdős‑Rényi graphs, assigning random rotation matrices to edges while respecting coherence. The order parameter r=|∑_i x_i|/N quickly approaches 1 for any positive K, confirming the absence of a finite critical coupling. When the Ω_i are heterogeneous, synchronization fails, illustrating the necessity of identical intrinsic rotations.

In summary, the authors deliver a comprehensive analytical framework for high‑dimensional Kuramoto oscillators on both scalar‑ and matrix‑weighted networks. By leveraging the Master Stability Function and the spectral properties of the graph Laplacian, they reduce the N·d‑dimensional stability problem to a family of d‑dimensional eigenvalue problems, establishing that global synchronization is guaranteed for any positive coupling on connected networks, as long as the intrinsic rotations are uniform and the MWN satisfies a coherence condition. This work broadens the applicability of Kuramoto‑type models to systems where interactions involve multidimensional linear transformations, such as coordinated robotics, neural population dynamics, and power‑grid synchronization with vector‑valued states.