Rich-club network topology to minimize synchronization cost due to phase difference among frequency-synchronized oscillators

TL;DR

This paper introduces a new measure of synchronization cost based on phase differences among oscillators and demonstrates that rich-club network topologies minimize this cost, enhancing network efficiency.

Contribution

It defines a novel synchronization cost metric and shows that rich-club network structures minimize this cost using the Kuramoto model and analytical insights.

Findings

Rich-club networks minimize synchronization cost.

Optimal networks have bimodal degree distributions.

Rich-club topology enhances synchronization efficiency.

Abstract

Functions of some networks, such as power grids and large-scale brain networks, rely on not only frequency synchronization, but also phase synchronization. Nevertheless, even after the oscillators reach to frequency-synchronized status, phase difference among oscillators often shows non-zero constant values. Such phase difference potentially results in inefficient transfer of power or information among oscillators, and avoid proper and efficient functioning of the network. In the present study, we newly define synchronization cost by the phase difference among the frequency-synchronized oscillators, and investigate the optimal network structure with the minimum synchronization cost through rewiring-based optimization. By using the Kuramoto model, we demonstrate that the cost is minimized in a network topology with rich-club organization, which comprises the densely-connected center nodes and peripheral nodes connecting with the center module. We also show that the network topology is characterized by its bimodal degree distribution, which is quantified by Wolfson's polarization index. Furthermore, we provide analytical interpretation on why the rich-club network topology is related to the small amount of synchronization cost.