Optimal weighted networks of phase oscillators for synchronization.

The phase order parameter of oscillators on a network is optimized using two different sets of constraints. First, the maximization is achieved by adjusting the coupling strengths among the oscillators without changing the total coupling strength and the natural frequencies of the oscillators. This optimization reveals that a stronger weight tends to be assigned to a connection between two oscillators with greatly different natural frequencies. Second, we vary both coupling strengths and natural frequencies while maximizing the phase order and minimizing the penalty function which prevents the natural frequencies of the oscillators from taking the same value. This optimization reveals that a large total coupling strength makes oscillators take two natural frequencies (two-group state), whereas a small total coupling strength facilitates the convergence of natural frequencies to one single value (one-group state). Small and large penalty parameters make the optimized network take the one- and two-group states, respectively. This phase transition is observed in all-to-all, lattice, and scale-free networks although the clustering coefficient of the strongest links in the optimized network reflects the difference of the underlying network topologies.