Synchronization waves in geometric networks

We report synchronization of networked excitable nodes embedded in a metric space, where the connectivity properties are mostly determined by the distance between units. Such a high clustered structure, combined with the lack of long-range connections, prevents full synchronization and yields instead the emergence of synchronization waves. We show that this regime is optimal for information transmission through the system, as it enhances the options of reconstructing the topology from the dynamics. Measurements of topological and functional centralities reveal that the wave-synchronization state allows detection of the most structurally relevant nodes from a single observation of the dynamics, without any a priori information on the model equations ruling the evolution of the ensemble.

Figure 1

(a) Global synchronization $S$, (b) local synchronization $S_{\rho }$, and (c) wave synchronization $S_w=S_{\rho }-S$ (see text for definitions) in the parameter space $(d,\sigma )$. In all cases, $\rho =10$ and $N=500$. The color code is reported in the bar at the right of the three plots. (d) Successive snapshots of the activity of the network, for $\langle k\rangle =15$, $d=0.2$ and $\sigma =10$. Empty (full) circles correspond to inactive (spiking) nodes. The underlying network has been hidden for a better visualization.

Figure 2

Synchronization route as a function of the coupling strength $d$ for $\sigma =10$ (blue dashed lines) and $\sigma =1000$ (black continuous lines). (a) Global synchronization $S$ and (b) local synchronization $S_{\rho }$. (c) Number of synchronized connected components for $\sigma =10$, $\sigma =1000$, and scale-free network (red dotted lines).

Figure 3

(a), (c) Temporal evolution of the centrality correlation $C_c\left(\tau \right)$ (black solid line) and global synchronization at each bin $S\left(\tau \right)$ (blue dash-dotted line). (b), (d) Raster plots of the network activity. See text for the definition of all quantities. Parameters are $N=500$, $\langle k\rangle =15$, $d=0.2$, and [(a), (b)] $\sigma =10$ and [(c), (d)] $\sigma =30$.

Figure 4

Sample nets showing the 50 nodes with the highest topological ($\circ$) and functional ($•$) centrality for (a) $\sigma =10$ and (b) $\sigma =25$.