Synchronization processes in complex networks

Arenas, Àlex; Dı́az-Guilera, Albert; Pérez-Vicente, Conrad J.

doi:10.1016/j.physd.2006.09.029

Cited by 147 publications

(93 citation statements)

References 42 publications

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“…Researchers have found that an appropriate description of such systems involves an understanding of both the dynamics of the individual oscillators and the connection topology of the network. Investigators studying the latter have found that many complex networks have a modular structure involving motifs [1], communities [2,3], layers [4], or clusters [5]. For example, recent work has shown that as many kinds of networks (including isotropic homogeneous networks and a class of scale-free networks) transition to full synchronization, they pass through epochs in which well-defined synchronized communities appear and interact with one another [3].…”

Section: Introductionmentioning

confidence: 99%

“…Investigators studying the latter have found that many complex networks have a modular structure involving motifs [1], communities [2,3], layers [4], or clusters [5]. For example, recent work has shown that as many kinds of networks (including isotropic homogeneous networks and a class of scale-free networks) transition to full synchronization, they pass through epochs in which well-defined synchronized communities appear and interact with one another [3]. Knowledge of this structure, and the dynamical behavior it supports, informs our understanding of biological [6], social [7], and technological networks [8].…”

Section: Introductionmentioning

confidence: 99%

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Synchronization in networks of networks: The onset of coherent collective behavior in systems of interacting populations of heterogeneous oscillators

et al. 2008

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The onset of synchronization in networks of networks is investigated. Specifically, we consider networks of interacting phase oscillators in which the set of oscillators is composed of several distinct populations. The oscillators in a given population are heterogeneous in that their natural frequencies are drawn from a given distribution, and each population has its own such distribution. The coupling among the oscillators is global, however, we permit the coupling strengths between the members of different populations to be separately specified. We determine the critical condition for the onset of coherent collective behavior, and develop the illustrative case in which the oscillator frequencies are drawn from a set of (possibly different) Cauchy-Lorentz distributions. One motivation is drawn from neurobiology, in which the collective dynamics of several interacting populations of oscillators (such as excitatory and inhibitory neurons and glia) are of interest.In recent years, there has been considerable interest in networks of interacting systems. Researchers have found that an appropriate description of such systems involves an understanding of both the dynamics of the individual oscillators and the connection topology of the network. Investigators studying the latter have found that many complex networks have a modular structure involving motifs [1], communities [2,3], layers [4], or clusters [5]. For example, recent work has shown that as many kinds of networks (including isotropic homogeneous networks and a class of scale-free networks) transition to full synchronization, they pass through epochs in which well-defined synchronized communities appear and interact with one another [3]. Knowledge of this structure, and the dynamical behavior it supports, informs our understanding of biological [6], social [7], and technological networks [8].Here we consider the onset of coherent collective behavior in similarly structured systems for which the dynamics of the individual oscillators is simple. In seminal work, Kuramoto analyzed a mathematical model that illuminates the mechanisms by which synchronization arises in a large set of globally-coupled phase oscillators [9]. An important feature of Kuramoto's model is that the oscillators are heterogeneous in their frequencies. And, although these mathematical results assume global coupling, they have been fruitfully applied to further our understanding of systems of fireflies, arrays of Josephson junctions, electrochemical oscillators, and many other cases [10]. In this work, we study systems of several interacting Kuramoto systems, i.e., networks of interacting populations of phase oscillators. Our motivation is drawn not only from the applications listed above (e.g., an amusing application might be interacting populations of fireflies, where each population inhabits a separate tree), but also from other biological systems. Rhythms arising from coupled cell populations are seen in many of the body's organs (including the heart, the pancreas, and the kidney...

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Synchronization in networks of networks: The onset of coherent collective behavior in systems of interacting populations of heterogeneous oscillators

et al. 2008

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“…We define an indicator function h as 2) so that the switch in the system occurs when h(z) = 0. We shall refer to the line v = a as the switching manifold, call F L (F R ) the left (right) half-system of (2.1) and call the region of the plane where…”

Section: Piece-wise Linear Modelsmentioning

confidence: 99%

“…This is expected since the associated network eigenvectors have components that sum to zero, describing perturbations that act to push the oscillations apart pairwise. To quantify this, we introduce the mean field variables (E(v), E(w)), where 2) and trace the mean field signal for a typical large network simulation in Figure 5. Here, it can be seen that the mean field trajectory does not settle down, and moreover is repelled away from the synchronous solution (also plotted).…”

Section: Where G(a; T) Is Given In (26) Andmentioning

confidence: 99%

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Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function

Coombes

Thul

2016

Eur. J. Appl. Math

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The master stability function is a powerful tool for determining synchrony in high-dimensional networks of coupled limit cycle oscillators. In part, this approach relies on the analysis of a low-dimensional variational equation around a periodic orbit. For smooth dynamical systems, this orbit is not generically available in closed form. However, many models in physics, engineering and biology admit to non-smooth piece-wise linear caricatures, for which it is possible to construct periodic orbits without recourse to numerical evolution of trajectories. A classic example is the McKean model of an excitable system that has been extensively studied in the mathematical neuroscience community. Understandably, the master stability function cannot be immediately applied to networks of such non-smooth elements. Here, we show how to extend the master stability function to non-smooth planar piece-wise linear systems, and in the process demonstrate that considerable insight into network dynamics can be obtained. In illustration, we highlight an inverse period-doubling route to synchrony, under variation in coupling strength, in globally linearly coupled networks for which the node dynamics is poised near a homoclinic bifurcation. Moreover, for a star graph, we establish a mechanism for achieving so-called remote synchronisation (where the hub oscillator does not synchronise with the rest of the network), even when all the oscillators are identical. We contrast this with node dynamics close to a non-smooth Andronov-Hopf bifurcation and also a saddle node bifurcation of limit cycles, for which no such bifurcation of synchrony occurs.

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Graph and Network Theory

Estrada

2015

Digital Encyclopedia of Applied Physics

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This Chapter introduces the basic concepts of graphs and networks and their applications in physics. It explains the connections of graph theory with condensed matter and statistical physics by studying the tight‐binding, Hubbard and Potts models. The use of graphs for solving problems in quantum field theory is illustrated by studying the Feynman graphs. Other physical scenarios in which graph theory is presented here are the study of electrical networks and vibrational systems. The Chapter also introduces the concepts of random graphs and complex networks. In the last topic, the most relevant structural concepts for complex networks are studied and a few dynamical processes such as consensus, synchronization and epidemics spreading, are discussed.

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Synchronization processes in complex networks

Cited by 147 publications

References 42 publications

Synchronization in networks of networks: The onset of coherent collective behavior in systems of interacting populations of heterogeneous oscillators

Synchronization in networks of networks: The onset of coherent collective behavior in systems of interacting populations of heterogeneous oscillators

Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function

Graph and Network Theory

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