Synchronizability determined by coupling strengths and topology on complex networks

We investigate in depth the synchronization of coupled oscillators on top of complex networks with different degrees of heterogeneity within the context of the Kuramoto model. In a previous paper [Phys. Rev. Lett. 98, 034101 (2007)], we unveiled how for fixed coupling strengths local patterns of synchronization emerge differently in homogeneous and heterogeneous complex networks. Here, we provide more evidence on this phenomenon, extending the previous work to networks that interpolate between homogeneous and heterogeneous topologies. We also introduce details of the path towards synchronization for the evolution of clustering in the synchronized patterns. Finally, we investigate the synchronization of networks with modular structure and conclude that, in these cases, local synchronization is first attained at the most internal level of organization of modules, progressively evolving to the outer levels as the coupling constant is increased. The present work introduces parameters that are proved to be useful for the characterization of synchronization phenomena in complex networks.

Figure 1

Global synchronization curves $r\left(\lambda \right)$ for different network topologies labeled by $\alpha$ ($\alpha =0$ corresponds to the BA limit and $\alpha =1$ to ER graphs). The inset shows the region where the onset of synchronization takes place. The network sizes are $N={10}^4$ and $⟨k⟩=6$ $(N_l=3\times {10}^4)$ and were generated using the model introduced in [45].

Figure 2

Evolution of the control parameters $r$ and $r_{\mathit{\text{link}}}$ as a function of the coupling strength for networks generated with the model introduced in [45], corresponding to $\alpha =0.0$ (SF), 0.25, 0.5, 0.75, and 1.0 (ER). The size of the networks is $N={10}^3$ and their average degree is $⟨k⟩=6$. The exponent of the SF networks increases from $\gamma =3$ $(\alpha =0)$.

Figure 3

Evolution of the number of synchronized clusters, $N_c$, and the synchronized giant component (GC) size as a function of $r_{\mathit{\text{link}}}$ for the the different topologies considered. Small values of $r_{\mathit{\text{link}}}$ correspond to values of $\lambda$ for which $r\approx 0$. Despite $r$ being vanishing and hence no global synchronization is yet attained, a significant number of clusters show up. This indicates that for any $\lambda >0$ the system self-organizes towards macroscopic synchronization. The network parameters are as in Fig. 2.

Figure 4

(Color online) Giant synchronized components for several values of $\lambda$ in the two limiting cases of the different topologies studied (ER and SF). The size of the underlying networks is small ($N=100$ nodes), in order to have a sizable picture of the system. Note that for the SF case links and nodes are incorporated together to the GC, while for the ER network, what is added are links between nodes already belonging to the GC.

Figure 5

Evolution of the ratio between the clustering coefficient of the giant synchronized cluster, $⟨c_{\mathit{sync}}⟩$, and that of the substrate network, $⟨c_{\mathit{\text{network}}}⟩$, as a function of $\lambda$ for the two limiting cases of BA and ER networks. Network parameters are those used in Fig. 2.

Figure 6

(Color online) The plot shows the fraction of links that a node with degree $k$ belonging to the synchronized cluster shares with other nodes of the same synchronized cluster. This fraction $k_{\mathit{int}}∕k$ is plotted as a function of ${\log }_{10}\left(k\right)$ and $\lambda$. The figure shows how the hubs progressively incorporate their neighbors to the synchronized component as $\lambda$ grows. The network is SF with parameters as those used in Fig. 2 and $\alpha =0$.

Figure 7

(Color online) Evolution of $r$ (top) and $r_{\mathit{\text{link}}}$ (bottom) as a function of $\lambda$ for structured modular networks. The networks are synthetically built with an a priori community structure. The network size is 256 nodes and the number of links is 4608. We prescribe 16 compartments that will represent our first community organizational level and four compartments, each one embedding four different compartments of the above first level, that define the second organizational level of the network. Each node has 18 links distributed between its first community level, the second, and the whole network at random. The network 13-4 has 13 internal connections in its first hierarchical level, 4 external connections in its second hierarchical level, and 1 connection with any other community in the network. The generation of the 15-2 structure is equivalent. The curves show that although 13-4 has always a better global synchronization, 15-2 has better local synchronization as shown by $r_{\mathit{\text{link}}}$.

Figure 8

(Color online) Size of the largest synchronized cluster GC and number of clusters, $N_c$, for the same networks of Fig. 7. See the text for details.

Figure 9

(Color online) Number of links $\left(N_{\mathit{\text{links}}}\right)$ that connect synchronized nodes in each of the three levels in the community hierarchy of the networks (1 means inner layer). The numbers are normalized by the total number of links at each level in each network.

Figure 10

(Color online) We represent the degree of synchronization between pairs of connected nodes for several values of the coupling $\lambda$ in a 13-4 modular network (with two organizational levels) of $N=256$ nodes. The color code denotes the value of the averaged (over different initial conditions) filtered matrix $⟨{\mathcal{D}}_{ij}⟩∊[0,1]$. The values of the coupling are (from left to right and top to bottom) $\lambda =0.011,0.026,0.032,0.035,0.038,0.046,0.210$ (corresponding to full synchronization). The pictures show that the order of synchronization is given by the organizational levels. The first community level is the first one to get synchronized; subsequently, second-level nodes attain synchronization for a larger value of $\lambda$; and finally, the full synchronized state is reached when outer links have $⟨D_{ij}⟩=1$.