  The behavior of weakly coupled self-sustained oscillators can often be well
described by phase equations. Here we use the paradigm of Kuramoto phase
oscillators which are coupled in a network to calculate first and second order
corrections to the frequency of the fully synchronized state for nonidentical
oscillators. The topology of the underlying coupling network is reflected in
the eigenvalues and eigenvectors of the network Laplacian which influence the
synchronization frequency in a particular way. They characterize the importance
of nodes in a network and the relations between them. Expected values for the
synchronization frequency are obtained for oscillators with quenched random
frequencies on a class of scale-free random networks and for a
Erd\H{o}s-R\'enyi random network. We briefly discuss an application of the
perturbation theory in the second order to network structural analysis.
