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.Comment: 4 figures + one appendix figur
