Synchronization of dynamical networks with nonidentical nodes: Criteria and control

This paper presents a framework for global synchronization of dynamical networks with nonidentical nodes. Several criteria for synchronization are given using free matrices for both cases of synchronizing to a common equilibrium solution of all isolated nodes and synchronizing to the average state trajectory. These criteria can be viewed as generalizations of the master stability function method for local synchronization of networks with identical nodes to the case of nonidentical nodes. The controlled synchronization problem is also studied. The control action, which is subject to certain constraints, is viewed as reorganization of the connection topology of the network. Synchronizability conditions via control are put forward. The synchronizing controllers can be obtained by solving an optimization problem.published_or_final_versio

Community structure in real-world networks from a non-parametrical synchronization-based dynamical approach

This work analyzes the problem of community structure in real-world networks based on the synchronization of nonidentical coupled chaotic R\"{o}ssler oscillators each one characterized by a defined natural frequency, and coupled according to a predefined network topology. The interaction scheme contemplates an uniformly increasing coupling force to simulate a society in which the association between the agents grows in time. To enhance the stability of the correlated states that could emerge from the synchronization process, we propose a parameterless mechanism that adapts the characteristic frequencies of coupled oscillators according to a dynamic connectivity matrix deduced from correlated data. We show that the characteristic frequency vector that results from the adaptation mechanism reveals the underlying community structure present in the network.Comment: 21 pages, 7 figures; Chaos, Solitons & Fractals (2012

Bounded synchronization of a heterogeneous complex switched network

This paper investigates synchronization issues of a heterogeneous complex network with a general switching topology in the sense of boundedness, when no complete synchronization manifold exists. Several sufficient conditions are established with the Lyapunov method and the differential analysis of convergence to determine the existence and estimate the convergence domain for the local and global bounded synchronization of a heterogeneous complex network. By using the consensus convergence of a switched linear system associated with the switching topology, explicit bounds of the maximum deviation between nodes are obtained in the form of a scalar inequality involving the property of the consensus convergence, the homogeneous and heterogeneous dynamics of individual nodes for the local and global cases. These analytical results are simple yet generic, which can be used to explore synchronization issues of various complex networks. Finally, a numerical simulation illustrates their effectiveness.postprin

Modelling and control for bounded synchronization in multi-terminal VSC-HVDC transmission networks

© 20xx IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.The extension and size of the power grid is expected to increase in the near future. Managing such a system presents challenging control problems that, so far, have been approached with classical control techniques. However, large scale systems of interconnected nodes fall within the framework of the emerging field of complex networks. This paper models multi-terminal VSC-HVDC systems as a complex dynamical network, and derives conditions ensuring bounded synchronization of its trajectories for a family of controllers. The obtained results are validated via numerical simulations.Postprint (author's final draft

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

Enhancing synchronizability of weighted dynamical networks using betweenness centrality

By considering the eigenratio of the Laplacian of the connection graph as synchronizability measure, we propose a procedure for weighting dynamical networks to enhance theirsynchronizability. The method is based on node and edge betweenness centrality measures and is tested on artificially const ructed scale-free, Watts-Strogatz and random networks as well as on some real-world graphs. It is also numerically shown that the same procedure could be used to enhance the phase synchronizability of networks of nonidentical oscillators