Perturbation analysis of complete synchronization in networks of phase oscillators

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 Erdös-Rényi random network. We briefly discuss an application of the perturbation theory in the second order to network structural analysis.

Figure 1

(Color online) Frequency shift in undirected, random networks of size $N=400$ with Poissonian degree distribution and given mean degree $k$. The mean degree must be larger than $k_{cr}=2$ for an infinite random tree network. The frequency shift could be measured precisely by enforcing a zero mean frequency $\overline{\omega }=0$. Random frequencies were drawn from a uniform distribution of standard deviation $\sigma ={10}^{-1}$. From an ensemble of ten realizations we show the mean network synchronization frequency divided by variance ${\sigma }^2$ and nonisochronicity $\gamma$ (diamonds, Newton Method, $\gamma =1.0$), the predicted second order term $⟨{\Omega }^{\left(2\right)}⟩$ from equation Eq. (24) and the spectrum of the network Laplacian (circles), the asymptotic line $k^{-1}$ (dashed line) and the line ${\left(k-2\right)}^{-1}$ (solid line), which describes the actual behavior of the frequency shift even for small mean degrees close to $k_{cr}=2$.