Weighted networks are more synchronizable: how and why

Most real-world networks display not only a heterogeneous distribution of degrees, but also a heterogeneous distribution of weights in the strengths of the connections. Each of these heterogeneities alone has been shown to suppress synchronization in random networks of dynamical systems. Here we review our recent findings that complete synchronization is significantly enhanced and becomes independent of both distributions when the distribution of weights is suitably combined with the distribution of degrees. We also present new results addressing the optimality of our findings and extending our analysis to phase synchronization in networks of non-identical dynamical units.Comment: Proceedings of the CNET 2004 (29 August - 2 September 2004

Network skeleton for synchronization: Identifying redundant connections

Synchronization is an important dynamical process on complex networks with wide applications. In this paper, we design a greedy link removal algorithm and find that many links in networks are actually redundant for synchronization, i.e. the synchronizability of the network is hardly affected if these links are removed. Our analysis shows that homogeneous networks generally have more redundant links than heterogeneous networks. We denote the reduced network with the minimum number of links to preserve synchronizability (eigenratio of the Laplacian matrix) of the original network as the synchronization backbone. Simulating the Kuramoto model, we confirm that the network synchronizability is effectively preserved in the backbone. Moreover, the topological properties of the original network and backbone are compared in detail

Heterogeneous delays making parents synchronized: A coupled maps on Cayley tree model

We study the phase synchronized clusters in the diffusively coupled maps on the Cayley tree networks for heterogeneous delay values. Cayley tree networks comprise of two parts: the inner nodes and the boundary nodes. We find that heterogeneous delays lead to various cluster states, such as; (a) cluster state consisting of inner nodes and boundary nodes, and (b) cluster state consisting of only boundary nodes. The former state may comprise of nodes from all the generations forming self-organized cluster or nodes from few generations yielding driven clusters depending upon on the parity of heterogeneous delay values. Furthermore, heterogeneity in delays leads to the lag synchronization between the siblings lying on the boundary by destroying the exact synchronization among them. The time lag being equal to the difference in the delay values. The Lyapunov function analysis sheds light on the destruction of the exact synchrony among the last generation nodes. To the end we discuss the relevance of our results with respect to their applications in the family business as well as in understanding the occurrence of genetic diseases.Comment: 9 pages, 11 figure

Synchronization of heterogeneous oscillators under network modifications: Perturbation and optimization of the synchrony alignment function

Synchronization is central to many complex systems in engineering physics (e.g., the power-grid, Josephson junction circuits, and electro-chemical oscillators) and biology (e.g., neuronal, circadian, and cardiac rhythms). Despite these widespread applications---for which proper functionality depends sensitively on the extent of synchronization---there remains a lack of understanding for how systems evolve and adapt to enhance or inhibit synchronization. We study how network modifications affect the synchronization properties of network-coupled dynamical systems that have heterogeneous node dynamics (e.g., phase oscillators with non-identical frequencies), which is often the case for real-world systems. Our approach relies on a synchrony alignment function (SAF) that quantifies the interplay between heterogeneity of the network and of the oscillators and provides an objective measure for a system's ability to synchronize. We conduct a spectral perturbation analysis of the SAF for structural network modifications including the addition and removal of edges, which subsequently ranks the edges according to their importance to synchronization. Based on this analysis, we develop gradient-descent algorithms to efficiently solve optimization problems that aim to maximize phase synchronization via network modifications. We support these and other results with numerical experiments.Comment: 25 pages, 6 figure

On the onset of synchronization of Kuramoto oscillators in scale-free networks

Despite the great attention devoted to the study of phase oscillators on complex networks in the last two decades, it remains unclear whether scale-free networks exhibit a nonzero critical coupling strength for the onset of synchronization in the thermodynamic limit. Here, we systematically compare predictions from the heterogeneous degree mean-field (HMF) and the quenched mean-field (QMF) approaches to extensive numerical simulations on large networks. We provide compelling evidence that the critical coupling vanishes as the number of oscillators increases for scale-free networks characterized by a power-law degree distribution with an exponent $2 < \gamma \leq 3$, in line with what has been observed for other dynamical processes in such networks. For $\gamma > 3$, we show that the critical coupling remains finite, in agreement with HMF calculations and highlight phenomenological differences between critical properties of phase oscillators and epidemic models on scale-free networks. Finally, we also discuss at length a key choice when studying synchronization phenomena in complex networks, namely, how to normalize the coupling between oscillators

Synchronization in complex networks

Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.Comment: Final version published in Physics Reports. More information available at http://synchronets.googlepages.com