Optimal Tree for Both Synchronizability and Converging Time

Abstract

It has been proved that the spanning tree from a given network has the optimal synchronizability, which means the index $R=\lambda_{N}/\lambda_{2}$ reaches the minimum 1. Although the optimal synchronizability is corresponding to the minimal critical overall coupling strength to reach synchronization, it does not guarantee a shorter converging time from disorder initial configuration to synchronized state. In this letter, we find that it is the depth of the tree that affects the converging time. In addition, we present a simple and universal way to get such an effective oriented tree in a given network to reduce the converging time significantly by minimizing the depth of the tree. The shortest spanning tree has both the maximal synchronizability and efficiency.