Consensus Analysis in Hierarchical Networked Systems

This paper introduces evolving hierarchical networks based on hierarchical product. We discuss two important consensus behavior indexes in both un-weighted and weighted cases in those networked systems: (1) convergence speed in system without delay and noise; (2) delay robustness in system with communication time-delay. We analyze the approximative behaviors of these two consensus indexes by determining the second smallest and largest eigenvalue of Laplacian matrix, respectively, as well as how the scale of the weight factor affects the consensus performance. Moreover, the method applied in this paper is also applicable to many classic networks. Finally, two numerical simulations are given to verify the advantages of communication topology like hierarchical networks and the validity of the theoretical results.