Synchronization is Enhanced in Weighted Complex Networks

The propensity for synchronization of complex networks with directed and weighted links is considered. We show that a weighting procedure based upon the global structure of network pathways enhances complete synchronization of identical dynamical units in scale-free networks. Furthermore, we numerically show that very similar conditions hold also for phase synchronization of nonidentical chaotic oscillators.

Figure 1

Schematic representation of the weighting procedure. The link thickness is proportional to the link load, while the node colors reflect the node degrees (white, black, and gray correspond to nodes with low, high, and medium degrees, respectively). In general, nodes with low degrees (like node $b$) may have links with very high loads, and nodes with high degrees (like nodes $a$ and $c$) may have links with low loads.

Figure 2

(a) ${\lambda }_N/{\lambda }_2$ (in logarithmic scale) for SF networks vs the dimensionless parameter space ($\alpha ,B$). (b) $\Gamma$ (see text for definition) vs ($\alpha ,B$). In all cases $m=2$, and the graphs refer to averaging over 10 realizations of networks with $N=1000$. The domain with $\Gamma <0$ is outlined by the black contours drawn on the figure.

Figure 3

(a) ${\lambda }_N/{\lambda }_2$ (in logarithmic scale) for random graphs vs $\alpha$. (b) ${\Gamma }_c$ (see text for definition) vs ($\alpha ,B$). Random networks have $⟨k⟩=2m=4$, identical to that of SF networks. In the domain where ${\Gamma }_c<0$ SF networks synchronize better than random networks. Same stipulations as in the caption of Fig. 2.

Figure 4

The phase order parameter $\Psi$ (see text for definition) vs $\sigma$ for nonidentical chaotic Rössler oscillators. All parameters are specified in the text. Data refer to averages over 10 different realizations of SF networks with $B=10$, $m=2$, and $N=1000$. The inset is a zoom of the dashed region, with the corresponding error bars.