Synchronization of Interconnected Networks: The Role of Connector Nodes

Abstract

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. [5,6] or interaction between modules [7,8], can be obtained, sometimes with counter-intuitive results. Similarly, while synchronization in complex networks has been widely studied [9], very few works have investigated synchronization in N oN s. Huang et al. [10] showed that when two networks interact through random connections an exact balance between the weight of internal links in a network and the weight of links between networks results in greater synchronization between the two networks. It has also been shown that for multiple interacting networks, random connections between distant networks increase the synchronization of the complete N oN [11].Real networks exhibit high heterogeneity of the node degree, with hubs (i.e., high-degree) and peripheral (i.e., lowdegree) nodes [12]. What happens if connector links between the networks, termed inter-links, are not randomly created, but are instead chosen according to a particular connection strategy? Carlson et al. [13] analyzed the influence that lowdegree nodes may have on the collective dynamics of networks. Wang et al. [14] showed that when two neuron clusters get connected, both the heterogeneity of the network and the degree (i.e. number of connections) of the connector nodes, i.e. the nodes reached by inter-links, influence the coherent behavior of the whole system. A recent study demonstrated that the proper selection of connector nodes has strong implications on structural (centrality) and dynamical properties (spreading or population dynamics) occurring in a N oN [15].In this Letter, we study in a systematic way how connector nodes between a group of networks with heterogeneous topology affect synchronization and stability of the resulting N oN , and provide general rules for electing in a non-random fashion the connector nodes that maximize the synchronizability.The stability of the synchronized state of a group of coupled identical dynamical units is given by the corresponding Master Stability Function (MSF) [16]. For a given dynamical system and coupling form, the stability of synchronization depends on the second lowest eigenvalue λ 2 , usually called the spectral gap or algebraic connectivity, and the largest ei...

Value of peripheral nodes in controlling multilayer scale-free networks

“…Assuming that it is less costly to access peripheral nodes as compared to central nodes, the PP strategy would also be the most cost-efficient strategy. We emphasize that this finding differs from [14] where the CC strategy was preferred. This was found for an undirected network, on which a synchronization dynamics was investigated, whereas we consider a directed network, on which we assume a linear dynamics.…”

“…It was shown recently that structural network properties can change even in an abrupt way [13]. Furthermore, interconnecting links can significantly affect the way dynamic processes evolve in multilayer networks [14][15][16][17].…”

Reaction-diffusion processes on interconnected scale-free networks

“…This makes our knowledge about isolated networks relevant for the interconnected case. However, there exist network properties that are strongly affected by interconnectivity [4][5][6][7][8][9][10][11] In addition, interconnecting links may be different (with respect to their function) than the normal links within the networks. In the example at hand, via interconnecting links we provide power to the communication network, and we control how the load is distributed in the power network.…”

Exact coupling threshold for structural transition reveals diversified behaviors in interconnected networks

“…Interconnected networks are mathematical representations of systems where two or more simple networks, possibly with different functionalities, are coupled to each other. The omnipresence of interconnected networks has spurred a variety of research [4][5][6][7], with particular interest in dynamical processes such as percolation [8,9], epidemic spreading [10][11][12][13], and diffusion [14,15].…”