ON SYNCHRONIZATION IN DYNAMICAL HYPER-NETWORKS

The paper’s Theorem 3.2 proves global synchronization for the discrete hyper-network u(n+1)=g(u(n))+εCf(u(n))Γ under the Lipschitz condition kg+ε‖C‖kf‖Γ^T‖<1, and in the special case f=g, Γ=I it yields ‖I+εC‖<1/kf; the proof proceeds by setting e(n)=u(n)−v(n) for a synchronized comparator v(n) and applying Lipschitz and operator norm bounds to obtain a linear error recursion that decays geometrically . The candidate solution reproduces exactly this argument (same constants and structure), differing only in presentation (casting the update as a global contraction map and appealing to invariance of the synchronized subspace via C=−LG with zero row-sum) which is consistent with the paper’s construction C=AG−DG=−LG and its use of the operator norm bound ‖TΓ‖=‖Γ^T‖ . Hence both are correct with substantially the same proof.