Global Synchronization in Matrix-Weighted Networks

The candidate solution mirrors the paper’s construction and stability analysis: (i) it uses coherence to define a node-wise rotation gauge S that transforms the MWN Laplacian to L̄ = SLS^T and identifies the generalized synchronous orbit x_s(t) = S^T(1_n ⊗ s(t)), provided f and h are equivariant (paper’s O1i-invariance), exactly as in the Methods/result sections of the paper ; (ii) it derives the Master Stability equation δẏ̂_α = [Jf(s) − Λ(α)Jh(s)] δŷ_α by projecting perturbations on an orthonormal eigenbasis of L̄, again matching the paper’s Eq. (14) and subsequent MSF criterion that λ(Λ(α)) < 0 for all α ensures GS, whereas any α > 1 with positive MSF destroys GS . The candidate’s summary conditions (coherence, invariance of f and h, and negative MSF for all Λ(α)) align with the paper’s own three-point summary and empirical illustration that non-coherence invalidates the synchronized solution even when the MSF is non-positive . Minor differences: the model’s necessity argument in (a) takes an extra step (edge-wise cancellation) that needs justification; the paper asserts necessity and demonstrates it by counterexample when coherence fails, and positions the three conditions as necessary for GS .