Synchronization of Interconnected Networks: The Role of Connector Nodes

In this Letter we identify the general rules that determine the synchronization properties of interconnected networks. We study analytically, numerically, and experimentally how the degree of the nodes through which two networks are connected influences the ability of the whole system to synchronize. We show that connecting the high-degree (low-degree) nodes of each network turns out to be the most (least) effective strategy to achieve synchronization. We find the functional relation between synchronizability and size for a given network of networks, and report the existence of the optimal connector link weights for the different interconnection strategies. Finally, we perform an electronic experiment with two coupled star networks and conclude that the analytical results are indeed valid in the presence of noise and parameter mismatches.

Figure 1

(a) Schematic representation of the interconnection of two heterogeneous networks. HH corresponds to a strategy connecting high-degree nodes and LL to the connection between low-degree nodes. (b) Synchronization error $\epsilon (t)$ of two interconnected Barabási-Albert networks of $N=200$ Rössler oscillators at three different stages: isolated, interconnected following a LL strategy, and replacing the LL connection with a HH one. Equations of the Rössler system are given in [27] and the parameters used in the simulations are $a=b=0.2$ and $c=5.7$. $\epsilon (t)$ is obtained as the average across all pairs of oscillators of the pairwise distance in three dimensional phase space, $2/N(N-1)\sum _{i<j}\parallel {\mathbf{x}}_i-{\mathbf{x}}_j\parallel$, where ${\mathbf{x}}_i$ is the state vector of oscillator $i$ and $\parallel ·\parallel$ denotes the (Euclidean) norm. (c) ${\lambda }_2$ of the NON obtained from connecting two $N=200$ Barabási-Albert networks with one interlink, in all possible configurations. The node numbers are ordered according to the node degree and, when coinciding, the eigenvector centrality. (d) Eigenratio $r={\lambda }_N/{\lambda }_2$ for the same case as (c).

Figure 2

Synchronizability for two networks connected by a single interlink of weight $a$. (a) ${\lambda }_2$ and ${\lambda }_N$ for two star networks of 6 nodes each. (b),(c),(d) Eigenratio $r$ for (b) two star networks ($N=6$), (c) two scale-free networks ($N=500$), and (d) two Erdős-Rényi random networks ($N=500$). Three connecting strategies are shown: HH (black), HL (red), and LL (green). The minima of the curves (arrows) correspond to maximum synchronizability [34]. Plots (a)–(b) were obtained analytically and (c)–(d) numerically.

Figure 3

Dependence of synchronizability of class II and III systems on the size of the networks $N$. Averaged second eigenvalue ${\lambda }_2$ [(a) and (b)] and $r$ [(c) and (d)] over 30 realizations of two Erdős-Rényi networks and two scale-free networks of average degree $\overline{k}=12$. See Section S2 of [29] for details.

Figure 4

Experimental verification of the phenomenology presented here. (a) and (b) show the regions of complete synchronization of two star networks of type III Rössler systems coupled by a HH strategy. Neither HL (LH) nor LL strategies lead to synchronization, as predicted by the theory and confirmed by the experiments (not shown here). (c) and (d) depict class II Rössler systems. Regions correspond to (1) no synchronization, (2) complete synchronization with the HH strategy, and (3) complete synchronization with the HH and the LL strategies. Results are theoretical [(a) and (c)] and experimental [(b) and (d)]. The zeroes of the MSF are ${\nu }_1=0.107$ and ${\nu }_2=2.863$ for class III and ${\nu }_c=0.0651$ for class II.