Synchronization in complex networks of phase oscillators: A survey <s> Introduction <s> The Kuramoto model describes a large population of coupled limit-cycle oscillators whose natural frequencies are drawn from some prescribed distribution. If the coupling strength exceeds a certain threshold, the system exhibits a phase transition: some of the oscillators spontaneously synchronize, while others remain incoherent. The mathematical analysis of this bifurcation has proved both problematic and fascinating. We review 25 years of research on the Kuramoto model, highlighting the false turns as well as the successes, but mainly following the trail leading from Kuramoto’s work to Crawford’s recent contributions. It is a lovely winding road, with excursions through mathematical biology, statistical physics, kinetic theory, bifurcation theory, and plasma physics. © 2000 Elsevier Science B.V. All rights reserved. <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Introduction <s> The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, researchers have made important steps toward understanding the qualitatively new critical phenomena in complex networks. We review the results, concepts, and methods of this rapidly developing field. Here we mostly consider two closely related classes of these critical phenomena, namely structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. We also discuss systems where a network and interacting agents on it influence each other. We overview a wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, k-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks. We also discuss strong finite size effects in these systems and highlight open problems and perspectives. <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Introduction <s> Synchronization in complex networks Alex Arenas, 1, 2, 3 Albert D´ iaz-Guilera, 4, 2 Jurgen Kurths, 5 Yamir Moreno, 2, 6 and Changsong Zhou 7 Departament d’Enginyeria Inform` tica i Matem` tiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain a a Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza 50009, Spain Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Departament de F´ isica Fonamental, Universitat de Barcelona, 08028 Barcelona, Spain Institute of Physics, University of Potsdam PF 601553, 14415 Potsdam, Germany Department of Theoretical Physics, University of Zaragoza, Zaragoza 50009, Spain Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong (Dated: February 6, 2008) 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 de- voted 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 nu- merical 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. PACS numbers: 05.45.Xt,89.75.Fb,89.75.Hc Contents I. Introduction II. Complex networks in a nutshell III. Coupled phase oscillator models on complex networks A. Phase oscillators 1. The Kuramoto model 2. Kuramoto model on complex networks 3. Onset of synchronization in complex networks 4. Path towards synchronization in complex networks 5. Kuramoto model on structured or modular networks 6. Synchronization by pacemakers B. Pulse-coupled models C. Coupled maps IV. Stability of the synchronized state in complex networks A. Master Stability Function formalism 1. Linear Stability and Master Stability Function 2. Measures of synchronizability 3. Synchronizability of typical network models 4. Synchronizability and structural characteristics of networks 5. Graph theoretical bounds to synchronizability 6. Synchronizability of weighted networks 7. Universal parameters controlling the synchronizability B. Design of synchronizable networks 1. Weighted couplings for enhancing synchronizability 2. Topological modiﬁcation for enhancing synchronizability 3. Optimization of synchronizability C. Beyond the Master Stability Function formalism V. Applications A. Biological systems and neuroscience 1. Genetic networks 2. Circadian rhythms 3. Ecology 4. Neuronal networks 5. Cortical networks B. Computer science and engineering 1. Parallel/Distributed computation 2. Data mining 3. Consensus problems 4. Communication networks 5. Material and traﬃc ﬂow 6. Power-grids C. Social sciences and economy 1. Opinion formation 2. Finance 3. World Trade Web D. Perspectives VI. Conclusions Acknowledgments References I. INTRODUCTION Synchronization, as an emerging phenomenon of a pop- ulation of dynamically interacting units, has fascinated humans from ancestral times. No matter whether the phenomenon is spontaneous or induced, synchronization captivates our minds and becomes one of the most in- teresting scientiﬁc problems. Synchronization processes are ubiquitous in nature and play a very important role in many diﬀerent contexts as biology, ecology, climatol- ogy, sociology, technology, or even in arts ((Osipov et al., 2007; Pikovsky et al., 2001). It is known that synchrony <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Introduction <s> The emergence of synchronization in a network of coupled oscillators is a pervasive topic in various scientific disciplines ranging from biology, physics, and chemistry to social networks and engineering applications. A coupled oscillator network is characterized by a population of heterogeneous oscillators and a graph describing the interaction among the oscillators. These two ingredients give rise to a rich dynamic behavior that keeps on fascinating the scientific community. In this article, we present a tutorial introduction to coupled oscillator networks, we review the vast literature on theory and applications, and we present a collection of different synchronization notions, conditions, and analysis approaches. We focus on the canonical phase oscillator models occurring in countless real-world synchronization phenomena, and present their rich phenomenology. We review a set of applications relevant to control scientists. We explore different approaches to phase and frequency synchronization, and we present a collection of synchronization conditions and performance estimates. For all results we present self-contained proofs that illustrate a sample of different analysis methods in a tutorial style. <s> BIB004
Synchronization in networks of coupled oscillators is a pervasive topic in various scientific disciplines ranging from biology, physics, and chemistry to social networks and technological applications. A coupled oscillator network is characterized by a population of heterogeneous oscillators and a graph describing the interaction among the oscillators. These two ingredients give rise to a rich dynamic behavior that keeps on fascinating the scientific community. Within the rich modeling phenomenology on synchronization among coupled oscillators, this article focuses on the widely adapted model of a continuous-time and periodic limit-cycle oscillator network with continuous, bidirectional, and antisymmetric coupling. We consider a system of n oscillators, each characterized by a phase angle θ i ∈ S 1 and a natural rotation frequency ω i ∈ R. The dynamics of each isolated oscillator are thusθ i = ω i This material is based in part upon work supported by UCLA startup funds and NSF grants IIS-0904501 and CPS-1135819. A preliminary short version of this document appeared as BIB004 . Email addresses: dorfler@seas.ucla.edu (Florian Dörfler), bullo@engineering.ucsb.edu (Francesco Bullo). for i ∈ {1, . . . , n}. The interaction topology and coupling strength among the oscillators are modeled by a connected, undirected, and weighted graph G = (V, E, A) with nodes V = {1, . . . , n}, edges E ⊂ V ×V, and positive weights a ij = a ji > 0 for each undirected edge {i, j} ∈ E. The interaction between neighboring oscillators is assumed to be additive, anti-symmetric, diffusive, 1 and proportional to the coupling strengths a ij . In this case, the simplest 2π-periodic interaction function between neighboring oscillators {i, j} ∈ E is a ij sin(θ i − θ j ), and the overall model of coupled phase oscillators reads aṡ a ij sin(θ i − θ j ) , i ∈ {1, . . . , n} . (1) Despite its apparent simplicity, this coupled oscillator model gives rise to rich dynamic behavior, and it is encountered in many scientific disciplines ranging from natural and life sciences to engineering. This article surveys recent results and applications of the coupled oscillator model (1) and of its variations. The motivations for this survey are manifold. Recent 1 The interaction between two oscillators is diffusive if its strength depends on the corresponding phase difference; such interactions arise for example in the discretization of the Laplace operator in diffusive partial differential equations. years have witnessed much theoretical progress and novel applications, which are not covered in existing surveys BIB001  BIB003 BIB002 published in the physics literature. Indeed, control scientists have shown an increasing interest in complex networks of coupled oscillators and have recently contributed many novel approaches and results. Much of this interest has focused on (i) synchronization rather than more complex dynamic phenomena, (ii) finite numbers of oscillators with a non-trivial interaction topology, and (iii) connections with graph theory and multi-agent systems. It is therefore timely to provide a comprehensive review in a unified control-theoretical language of the best known results in this area. With this aim, this survey provides a systems and control perspective to coupled oscillator networks, focusing on quantitative results and control-relevant applications in sciences and technology.