Clustering and the synchronization of oscillator networks

Abstract

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 withPoisson and scale-free degree distributions. For both types of network, 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 clusterenhanced synchronization of hubs may be relevant to the brain which is scale-free and highly clustered.

Laplacian spectra as a diagnostic tool for network structure and dynamics

“…Since these modes are associated with the divisions among communities, this suggests that the frequency clusters noted at moderately high β in [30] are identical to topological communities. The degrees of freedom that inhibit full synchronization are clearly different from the ones responsible for the advanced partial synchronization of SFN's.…”

“…In [30] it was shown that nodes in different parts of the degree distribution can play different dynamical roles: for example, in the case of the SFN it is the hubs (highdegree nodes) that synchronize first. A complementary way of viewing the synchronization dynamics is to examine it in terms of appropriately chosen collective degrees of freedom rather than the behavior of individual oscillators.…”

“…The normal coordinates φ α are the appropriate ones for describing the relaxation to equilibrium of a strongly synchronized system [30] [18]. In addition, we define the observed frequencies (rotation numbers) of the oscillators as the time averages…”

“…[30][25] [26] [12] We show that projections onto the Laplacian eigenbasis are useful in understanding these effects, and that specific dynamical behaviors of the networks are associated with specific sets of modes in the Laplacian spectrum. This is true even in strongly nonlinear regimes of partial synchronization, despite the fact that the Laplacian is most naturally applied to linear problems near full synchronization.…”

“…We next examine the effect of clustering, which is known to have strong effects on the synchronizability of networks. [30] We applied a stochastic rewiring algorithm [31][30]to increase the clustering coefficient of the PN and SFN while leaving the degree distributions unchanged. Clustering (also called transitivity) refers to the tendency of two nodes which share a common neighbor to have an increased likelihood of also being directly connected to each other (compared to two nodes that do not share a neighbor) [6].…”