The coherence analysis and Laplacian spectrum applications of cycle-based iterative networks

This paper investigates network coherence for cycle-based iterative networks under noisy distribution. First, the Laplacian spectrum of the network is derived, yielding exact analytical expressions for leaderless first- and second-order coherence. Second, with the vertices of the initial cycle graph designated as leaders, closed-form solutions for the corresponding leader-follower coherence are established. Third, a comparative analysis between the two configurations reveals that the presence of leaders markedly accelerates consensus convergence. Finally, additional applications of the Laplacian spectrum are explored, elucidating intrinsic connections between the network's evolving topology and its dynamic behavior. These results collectively provide a theoretical foundation for analyzing and controlling dynamics in complex networks.