Enhancing complex-network synchronization

– Heterogeneity in the degree (connectivity) distribution has been shown to suppress synchronization in networks of symmetrically coupled oscillators with uniform coupling strength (unweighted coupling). Here we uncover a condition for enhanced synchronization in weighted networks with asymmetric coupling. We show that, in the optimum regime, synchronizability is solely determined by the average degree and does not depend on the system size and the details of the degree distribution. In scale-free networks, where the average degree may increase with heterogeneity, synchronizability is drastically enhanced and may become positively correlated with heterogeneity, while the overall cost involved in the network coupling is significantly reduced as compared to the case of unweighted coupling. Networks of dynamical elements serve as natural models for a variety of systems, with examples ranging from cell biology to epidemiology to the Internet [1]. Many of these complex networks display common structural features, such as the small-world [2] and scale-free properties [3]. Small-world networks (SWNs) exhibit short average distance between nodes and high clustering [2], while scale-free networks (SFNs) are characterized by an algebraic, highly heterogeneous distribution of degrees (number of links per node) [3]. The interplay between structure and dynamics has attracted a great deal of attention, especially in connection with the problem of synchronization of coupled oscillators [4–10]. The ability of a network to synchronize is generally enhanced in both SWNs and random SFNs as compared to regular lattices [11]. This enhancement was previously believed to be due to the decrease of the average distance between oscillators. Recently, it was shown that random networks with strong heterogeneity in the degree distribution, such as SFNs, are much more difficult to synchronize than random homogeneous networks [7], even though the former display smaller average path length [12]. This suggests that, although structurally advantageous [13], the scale-free property may be dynamically detrimental. Here we present a solution to this problem. A basic assumption of most previous works is that the oscillators are coupled symmetrically and with the same coupling strength. Under the assumption of symmetric coupling, the maximum synchronizability may be indeed achieved when the coupling strength is uniform [14]. (∗) E-mail: [email protected] (∗∗) E-mail: [email protected]