Optimizing synchrony with a minimal coupling strength of coupled phase oscillators on complex networks based on...

Finding the globally optimal network is an unsolved problem in synchrony optimization. In this paper, an efficient edge-adding optimization method based on global information is proposed. The edge-adding scheme is obtained through the eigenvector corresponding to the maximum eigenvalue of the system's Jacobian matrix. With different frequency distributions, we find that the optimized networks have similar features, concluded as three conditions: (i) The deviations of nodes' frequencies from the mean value are linear with the nodes' degrees, (ii) the oscillators form a bipartite network divided according to the frequencies of the oscillators, and (iii) oscillators are only connected to those with sufficiently large frequency differences. An optimal network can be constructed directly based on these three conditions for a given distribution of natural frequencies. We show that the critical coupling strengths of these constructed networks approach the theoretical lower bound. The constructed networks are or at least close to the globally optimal ones for synchrony.