Synchronization in complex networks of phase oscillators: A survey <s> History, Related Applications, and Theoretical Developments: <s> A series of lectures on the role of nonlinear processes in physics, mathematics, electrical engineering, physiology, and communication theory.From the preface:"For some time I have been interested in a group of phenomena depending upon random processes. One the one hand, I have recorded the random shot effect as a suitable input for testing nonlinear circuits. On the other hand, for some of the work that Professor W. A. Rosenblith and I have been doing concerning the nature of the electroencephalogram, and in particular of the alpha rhythm, it has occurred to me to use the model of a system of random nonlinear oscillators excited by a random input...At the beginning we had contemplated a series of only four or five lectures. My ideas developed pari passu with the course, and by the end of the term we found ourselves with a set of fifteen lectures. The last few of these were devoted to the application of my ideas to problems in the statistical mechanics of gases. This work is both new and tentative, and I found that I had to supplement my course by the writing over of these with the help of Professer Y. W. Lee. " <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> History, Related Applications, and Theoretical Developments: <s> Abstract The impressive variety of biological rhythms leaves no doubt that autonomously periodic processes contribute to the coordination of life-processes. The question here raised is, “What modes of temporal organization—if any—could result from weak interactions in a population of innately oscillatory devices (e.g. electronic oscillators, secretory cells, spontaneously active neurons, or individual animals)?” By mathematical analysis, electronic experiments, and digital computer simulation, answers are found as functions of the nature of the periodic processes and the kind of interactions involved. For populations of “generalized relaxation oscillators”, threshold conditions are discovered for mutual synchronization in any of a variety of modes. It is proposed that self-entraining communities of this sort may exist within individual metazoan animals and plants as the basis of the observed diurnal coordination of their physiological process. <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> History, Related Applications, and Theoretical Developments: <s> Improved mobility control in secondary and tertiary recovery processes is obtained by injecting at least two slugs into the reservoir, one slug designed to effect a permeability reduction and the other slug containing an agent which more effectively increases the fluid viscosity. For example, an aqueous slug containing a partially hydrolyzed, high molecular weight polyacrylamide can be injected (this slug efficiently reduces the relative permeability to the flow of the aqueous slug through the reservoir), followed by an aqueous slug containing a biopolymer, e.g., a linear polysaccharide having a molecular weight of about one million (this agent more effectively increases the viscosity of the water and has a lesser effect on permeability reduction). This combination of slugs gives more efficient and economical mobility control in flooding subterranean oil-bearing formations to recover oil therefrom. <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> History, Related Applications, and Theoretical Developments: <s> This book describes examples and applications of synchronization, and gives the problem a working mathematical formulation. It is intended for a wide range of specialists who may encounter nonlinear vibrations problems: mathematicians, engineers, physicists, chemists, and biologists, as well as students in these fields. <s> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> History, Related Applications, and Theoretical Developments: <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> BIB005 </s> Synchronization in complex networks of phase oscillators: A survey <s> History, Related Applications, and Theoretical Developments: <s> Preface 1. Introduction Part I. Synchronization Without Formulae: 2. Basic notions: the self-sustained oscillator and its phase 3. Synchronization of a periodic oscillator by external force 4. Synchronization of two and many oscillators 5. Synchronization of chaotic systems 6. Detecting synchronization in experiments Part II. Phase Locking and Frequency Entrainment: 7. Synchronization of periodic oscillators by periodic external action 8. Mutual synchronization of two interacting periodic oscillators 9. Synchronization in the presence of noise 10. Phase synchronization of chaotic systems 11. Synchronization in oscillatory media 12. Populations of globally coupled oscillators Part III. Synchronization of Chaotic Systems: 13. Complete synchronization I: basic concepts 14. Complete synchronization II: generalizations and complex systems 15. Synchronization of complex dynamics by external forces Appendix 1. Discovery of synchronization by Christiaan Huygens Appendix 2. Instantaneous phase and frequency of a signal References Index. <s> BIB006 </s> Synchronization in complex networks of phase oscillators: A survey <s> History, Related Applications, and Theoretical Developments: <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> BIB007 </s> Synchronization in complex networks of phase oscillators: A survey <s> History, Related Applications, and Theoretical Developments: <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> BIB008
A brief history of synchronization: The scientific interest in synchronization of coupled oscillators can be traced back to the work by  on "an odd kind of sympathy" between coupled pendulum clocks, locking phenomena in circuits and radio technology    , mutual influence of organ pipes  , the analysis of brain waves and self-organizing systems  BIB001 , and it still fascinates the scientific community nowadays   . We refer to BIB006 and BIB004 for a detailed historical account of synchronization studies. A variation of the considered coupled oscillator model (1) was first proposed by BIB002 . Winfree considered general (not necessarily sinusoidal) interactions among the oscillators. He discovered a phase transition from incoherent behavior with dispersed phases to synchrony with aligned frequencies and coherent (i.e., nearby) phases. Winfree found that this phase transition depends on the trade-off between the heterogeneity of the oscillator population and the strength of the mutual coupling, which he could formulate by parametric thresholds. However, Winfree's model was too general to be analytically tractable. Inspired by these works, BIB003 simplified Winfree's model and arrived at the coupled oscillator dynamics (1) with a complete interaction graph and uniform weights a ij = K/n: sin(θ i −θ j ) , i ∈ {1, . . . , n} . In an ingenious analysis, BIB003  showed that synchronization occurs in the model (3) if the coupling gain K exceeds a certain threshold K critical function of the distribution of the natural frequencies ω i . The dynamics (3) are nowadays known as the Kuramoto model of coupled oscillators, and Kuramoto's original work initiated a broad stream of research. A compelling historical perspective is offered by BIB005 . We also recommend the surveys by  , BIB007 , and BIB008 .