Optimal Tree for Both Synchronizability and Converging Time

TL;DR

This paper demonstrates that the depth of a spanning tree influences convergence time in synchronization processes and proposes a method to construct trees that optimize both synchronizability and convergence speed.

Contribution

It introduces a universal approach to generate spanning trees with minimal depth, enhancing both synchronizability and convergence efficiency in networks.

Findings

Minimal depth trees reduce convergence time.

Optimal trees balance synchronizability and efficiency.

Proposed method significantly improves synchronization performance.

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.