Synchrony-optimized networks of non-identical Kuramoto oscillators

Brede, Markus

doi:10.1016/j.physleta.2007.11.069

Cited by 95 publications

(115 citation statements)

References 19 publications

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“…Notice that, interestingly, this conditions does not depend on the value of λ. It is clear also that optimal synchronization for the Kuramoto model demands a negative correlation of the natural frequencies of adjacent oscillators and a positive correlation between the degrees k i and the values of |ω i |, which are precisely the results obtained previously by different and more intricate numerical approaches [10][11][12][13].…”

Section: Optimal Synchronization In the Kuramoto Modelsupporting

confidence: 83%

Optimal synchronization of Kuramoto oscillators: A dimensional reduction approach

Pinto

Saa

2015

Phys. Rev. E

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A recently proposed dimensional reduction approach for studying synchronization in the Kuramoto model is employed to build optimal network topologies to favor or to suppress synchronization. The approach is based in the introduction of a collective coordinate for the time evolution of the phase locked oscillators, in the spirit of the Ott-Antonsen ansatz. We show that the optimal synchronization of a Kuramoto network demands the maximization of the quadratic function ω T Lω, where ω stands for the vector of the natural frequencies of the oscillators, and L for the network Laplacian matrix. Many recently obtained numerical results can be re-obtained analytically and in a simpler way from our maximization condition. A computationally efficient hill climb rewiring algorithm is proposed to generate networks with optimal synchronization properties. Our approach can be easily adapted to the case of the Kuramoto models with both attractive and repulsive interactions, and again many recent numerical results can be rederived in a simpler and clearer analytical manner.

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Section: Optimal Synchronization In the Kuramoto Modelsupporting

confidence: 83%

Optimal synchronization of Kuramoto oscillators: A dimensional reduction approach

Pinto

Saa

2015

Phys. Rev. E

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“…Experiments are run until no configuration has been accepted for L 2 trials, with L the number of links of the network. Results in the optimisation are obtained for initial conditions θ i = 0, but as observed in [13,14] we find that the optimised configurations are independent of this choice.…”

Section: B Optimisation Experimentsmentioning

confidence: 54%

“…The degree of synchronisation is measured by the order parameter r which we time-average after discarding a transient. Similar to [13,14] we use the following optimisation scheme: (i) start with some random initial condition, (ii) pick a node i at random and add a phase shift randomly selected from [−π, π] to its frustration parameter to obtain a modified configuration, (iii) calculate the average order parameter for the modified configuration (for initial conditions set to zero), and (vi) accept the new configuration if it gives a larger average order parameter than the previous one and reject otherwise. If a modified configuration is rejected, we revert to the last previously accepted configuration and proceed with step (ii).…”

Section: B Optimisation Experimentsmentioning

confidence: 96%

“…Inspired by Watts and Strogatz' introduction of small world networks [12], the literature has strongly focussed on analysing the model on sparse complex networks [4,5]. A particular interest in this line of thinking has been to understand conditions under which synchronisation can be improved, e.g., determining the types of correlations of native frequencies along the links of a network that can strongly influence overall synchronisation behaviour [13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

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Frustration tuning and perfect phase synchronization in the Kuramoto-Sakaguchi model

Brede

Kalloniatis

2016

Phys. Rev. E

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We present an analysis of conditions under which the dynamics of a frustrated Kuramoto -or Kuramoto-Sakaguchi -model on sparse networks can be tuned to enhance synchronisation. Using numerical optimisation techniques, linear stability and dimensional reduction analysis, a simple tuning scheme for setting node specific frustration parameters as functions of native frequencies and degrees is developed. Finite size scaling analysis reveals that even partial application of the tuning rule can significantly reduce the critical coupling for the onset of synchronisation. In the second part of the paper, a co-dynamics is proposed which allows a dynamic tuning of frustration parameters simultaneously with the ordinary Kuramoto dynamics. We find that such co-dynamics enhance synchronisation when operating on slow time scales, and impede synchronisation when operating on fast time scales relative to the Kuramoto dynamics.

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“…These observations agree with those of Refs. [27,28], where similar positive and negative correlations were found to promote global synchronization. We finally note that such degree-frequency correlations may help explain the increased sharpness of transitions shown in Figs.…”

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confidence: 71%

Optimal Synchronization of Complex Networks

2014

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We study optimal synchronization in networks of heterogeneous phase oscillators. Our main result is the derivation of a synchrony alignment function that encodes the interplay between network structure and oscillators' frequencies and can be readily optimized. We highlight its utility in two general problems: constrained frequency allocation and network design. In general, we find that synchronization is promoted by strong alignments between frequencies and the dominant Laplacian eigenvectors, as well as a matching between the heterogeneity of frequencies and network structure. PACS numbers: 05.45.Xt, 89.75.Hc A central goal of complexity theory is to understand the emergence of collective behavior in large ensembles of interacting dynamical systems. Synchronization of networkcoupled oscillators has served as a paradigm for understanding emergence [1][2][3][4], where examples arise in nature (e.g., flashing of fireflies [5] and cardiac pacemaker cells [6]), engineering (e.g., power grid [7] and bridge oscillations [8]), and at their intersection (e.g., synthetic cell engineering [9]). We consider the dynamics of N network-coupled phase oscillators θ i for i = 1, . . . , N , whose evolution is governed bẏHere ω i is the natural frequency of oscillator i, K > 0 is the coupling strength, [A ij ] is a symmetric network adjacency matrix, and H is a 2π-periodic coupling function [10]. We treat H(θ) with full generality so long as H ′ (0) > 0. The choices H(θ) = sin(θ) and H(θ) = sin(θ − α) with the phase-lag parameter α ∈ (−π/2, π/2) yield the classical Kuramoto [10] and Sakaguchi-Kuramoto models [11].Considerable research has shown that the underlying structure of a network plays a crucial role in determining synchronization [12][13][14][15][16][17][18][19][20][21], yet the precise relationship between the dynamical and structural properties of a network and its synchronization remains not fully understood. One unanswered question is, given an objective measure of synchronization, how can synchronization be optimized? One application lies in synchronizing the power grid [22], where sources and loads can be modeled as oscillators with different frequencies. To this end, we ask: what structural and/or dynamical properties should be present to optimize synchronization?We measure the degree of synchronization of an ensemble of oscillators using the Kuramoto order parameterHere re iψ denotes the phases' centroid on the complex unit circle, with the magnitude r ranging from 0 (incoherence) to 1 (perfect synchronization) [10]. In general, the question of optimization (maximizing r) is challenging due to the fact that the macroscopic dynamics depend on both the natural frequencies and the network structure. To quantify the interplay between node dynamics and network structure, we derive directly from Eqs. (1) and (2) a synchrony alignment function which is an objective measure of synchronization and can be used to systematically optimize a network's synchronization. We highlight this result by addressing two classes of op...

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Synchrony-optimized networks of non-identical Kuramoto oscillators

Cited by 95 publications

References 19 publications

Optimal synchronization of Kuramoto oscillators: A dimensional reduction approach

Optimal synchronization of Kuramoto oscillators: A dimensional reduction approach

Frustration tuning and perfect phase synchronization in the Kuramoto-Sakaguchi model

Optimal Synchronization of Complex Networks

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