The Coherence and Robustness Analysis for a Family of Unbalanced Networks

Abstract

This study focuses on the coherence in a class of unbalanced networks, emphasizing the investigation of the impact of leader selection and network parameters on coherence. We discuss three different categories of consensus algorithms: leaderless algorithms of first and second orders, and first-order leader-follower algorithm. Firstly, we derive exact solutions for leaderless network coherence in the first and second orders. Secondly, we design various leader allocation schemes for the first-order leader-follower networks, obtaining expressions correlated with the network scale $N$, the double-ring cardinalities $p$ and $q$, and the number of leaders $k$. Thirdly, we conduct numerical simulations and robustness analyses to explore the impact of leader quantity and topology structure on network coherence. Specifically, the conclusions are as follows: (1) When $N$ decreases, an increase in $k$ demonstrates improved network coherence. (2) The smaller the difference between $p$ and $q$, the better the coherence. (3) The smaller the sum of $p$ and $q$, the stronger the robustness. (4) Uniform distribution of $k_{1}$ and $k_{2}$ leads to optimal coherence. Finally, the Laplacian energy and Kirchhoff index of the network are analyzed.