Eur. Phys. J. B 60, 89-95 (2007)
https://doi.org/10.1140/epjb/e2007-00324-y

Optimal synchronizability of networks

¹ Department of Physics, BK21 Physics Research Division, and Institute of Basic Science, Sungkyunkwan University, Suwon, 440-746, Korea
² Department of Modern Physics and Nonlinear Science Center, University of Science and Technology of China, Hefei Anhui, 230026, P.R. China
³ School of Environmental and Biological Science and Technology, Dalian University of Technology, Dalian Liaoning, 116024, P.R. China

Corresponding author: ^a beomjun@skku.edu

Received: 16 April 2007
Revised: 4 October 2007
Published online: 29 November 2007

Abstract

We numerically investigate how to enhance synchronizability of coupled identical oscillators in complex networks with research focus on the roles of the high level of clustering for a given heterogeneity in the degree distribution. By using the edge-exchange method with the fixed degree sequence, we first directly maximize synchronizability measured by the eigenratio of the coupling matrix, through the use of the so-called memory tabu search algorithm developed in applied mathematics. The resulting optimal network, which turns out to be weakly disassortative, is observed to exhibit a small modularity. More importantly, it is clearly revealed that the optimally synchronizable network for a given degree sequence shows a very low level of clustering, containing much fewer small-size loops than the original network. We then use the clustering coefficient as an object function to be reduced during the edge exchanges, and find it a very efficient way to enhance synchronizability. We thus conclude that under the condition of a given degree heterogeneity, the clustering plays a very important role in the network synchronization.