Clustering and the synchronization of oscillator networks

By manipulating the clustering coefficient of a network without changing its degree distribution, we examine the effect of clustering on the synchronization of phase oscillators on networks with Poisson and scale-free degree distributions. For both types of networks, increased clustering hinders global synchronization as the network splits into dynamical clusters that oscillate at different frequencies. Surprisingly, in scale-free networks, clustering promotes the synchronization of the most connected nodes (hubs) even though it inhibits global synchronization. As a result, they show an additional, advanced transition instead of a single synchronization threshold. This cluster-enhanced synchronization of hubs may be relevant to the brain that is scale-free and highly clustered.

Figure 1

(Color online) Order parameter $m$ vs coupling strength $\lambda$, for different values of the clustering coefficient $\gamma$. (a) The Poisson degree distribution. (b),(c) The scale-free degree distribution. (c) A close-up of the transition region showing that increased clustering leads to an advanced (lower-$\lambda$) transition.

Figure 2

Poisson networks: Scatter plots of average oscillation frequency $\Omega$ (vertical axis) vs intrinsic frequency $\omega$ (horizontal). At zero coupling, the points all lie on the line $\Omega =\omega$. With increasing coupling, more oscillators line up at synchronized frequencies. For high clustering, several bands form at different frequencies, rather than one frequency as in the low-$\gamma$ case.

Figure 3

Scale-free networks: Scatter plots of $\Omega$ vs $\omega$. Synchronized subsets (horizontal bands) begin to form at weaker coupling for the more clustered networks. Frequency clustering is evident at high $\gamma$.

Figure 4

Scale-free networks: Average frequency $\Omega$ vs degree $k$ at several values of clustering $\gamma$ and coupling strength $\lambda$. The degree (horizontal axis) is plotted on a logarithmic scale. Synchronization begins with the hubs (nodes with the highest $k$) and progresses downward to those with lower $k$.