On Synchronization in Networks of Coupled Oscillators

We study the synchronization of coupled nonlinear oscillators over arbitrary undirected weighted networks. We give an explicit characterization of phase cohesion in these networks in terms of a parameter which we refer to as the coefficient of ergodicity of the coupled oscillator network. Also, we characterize an estimate of the region of attraction of the frequency synchronized solution of phase cohesive oscillators and show that the frequency synchronized equilibrium is exponentially stable. As an implication of these results, we show that almost no oscillators coupled in graphs with diameter greater than two is a phase cohesive system. This in turn implies that phase cohesion happens in networked oscillators only on small-world networks.