Synchronization of phase oscillators in large complex networks

It has been shown in recent years that many real world networks have a complex structure (e.g., scale-free networks). The effect of a complex interaction network on the dyamics of coupled dynamical systems is, therefore, of interest. An important aspect of the dynamics is the synchronization of coupled oscillators. I will present a generalization of the classical Kuramoto model of all-to-all coupled oscillators to the case of a general topology of the network of interactions. We find that for a large class of networks, there is still a transition from incoherence to coherent behavior at a critical coupling strength that depends on the largest eigenvalue of the adjacency matrix of the network. I will discuss the application of our theory to study the effect of heterogeneity in the degree distribution and degree-degree correlations in the network. Finally, I will comment on generalizations to more realistic dynamical systems.