A novel analysis on the efficiency of hierarchy among leader-following systems

In a recent NATURE paper, Nagy et al. find a well-defined hierarchy among the individuals of the pigeon flock, which may lead to a rapid decision making in the directional choice dynamics of the flock. Motivated by this interesting discovery, we present a novel analysis on the efficiency of the hierarchical topology among the leader-following systems in this paper. To this end, we first propose a measurement of the convergence rate of leader-following consensus, and then connect the convergence rates with the communication topologies of leader-following systems. It is proved that the hierarchical network organization can achieve the best performance in terms of convergence rates. It is also established that the connections between the leader and the followers have effective impacts on increasing the convergence rates. Extensive numerical results are provided to show the effectiveness of our conclusions.