Inhomogeneity induces relay synchronization in complex networks

Relay synchronization is a collective state, originally found in chains of interacting oscillators, in which uncoupled dynamical units synchronize through the action of mismatched inner nodes that relay the information but do not synchronize with them. It is demonstrated herein that relay synchronization is not limited to such simple motifs, rather it can emerge in larger and arbitrary network topologies. In particular, we show how this phenomenon can be observed in networks of chaotic systems in the presence of some mismatched units, the relay nodes, and how it is actually responsible for an enhancement of synchronization in the network.

Figure 1

Sketch of the considered scale-free network with $N=100, \langle k\rangle =4, k^{*}=7$. The higher-degree nodes are colored in blue (the size of the nodes being proportional to their degree), and the others are in red. The inset zooms on a portion of the graph containing two different sets of nodes that are in relay configurations.

Figure 2

$N\times N$ matrix summarizing the synchronization scenario in the network of Fig. 1 at $\sigma =1$. Color codes are used with the following stipulations: black corresponds to pairs of unsynchronized nodes, orange to those pairs of synchronized nodes directly connected in the network either by physical links or paths, and white to pairs of nodes whose asymptotic behavior is synchronous thanks to the effect of dynamical relay. The vertical and horizontal blue lines denote furthermore the nodes with $c_i=c_0=9$.

Figure 3

Relay synchronization in the SF network of Fig. 1. (a),(b) SPP (upper plots, see text for definition) and synchronization error $e_{ij}$ (lower plots) vs $\sigma$ (a) between nodes 16 and 41 (blue line) and between nodes 41 and 79 (red line), and (b) between nodes 8 and 75 (blue line) and between nodes 75 and 71 (red line). (c) Global synchronization error $E$ vs $\sigma$. (d) Schematic representation of three paths from node 41 to 79. The network edges have been redrawn as continuous (dashed) lines if they connect (do not connect) two nodes that are synchronized. Light blue arrows (light red boxes) indicate synchronization through dynamical relaying (through direct links).

Figure 4

$l_r$ (see text for definition) in the parameter space $(\sigma ,f)$ for SF (a) and ER networks (b). Color bars report the corresponding color codes.

Figure 5

$\langle e\rangle$ (see text for definition) vs $\sigma$ for the isolated subnetwork $\mathrm{\Gamma }$ (black line) and when an increasing number of relay nodes is considered (colored lines, color codes in the inset). Dynamical relays enhances synchronization of $\mathrm{\Gamma }$ by lowering the synchronization threshold.

Figure 6

$\langle e\rangle$ vs $\sigma$ in a SF pristine network, $\mathrm{\Gamma }$, of $N=100$ identical nodes with $c_i=c$ (black line), compared with the case when two relay nodes with $c_j=c_0$ are added to this network with a different degree: in ${\mathrm{\Gamma }}_a$ (blue line) each relay node forms 10 links with the pristine networks, in ${\mathrm{\Gamma }}_b$ (red line) it forms 20 links, in ${\mathrm{\Gamma }}_c$ (green line) 30 links, and in ${\mathrm{\Gamma }}_d$ (purple line) 40 links.