The wheels: an infinite family of bi-connected planar synchronizing graphs

Canale, Eduardo; Monzón, Pablo; Robledo, Franco

doi:10.1109/iciea.2010.5515313

Cited by 4 publications

(2 citation statements)

References 11 publications

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“…Later on, we made some improvements: we proved that the AGS property depends only in the block of the graphs [3], we also proved that every connected graph is the induced graph of a synchronized one, and that any graph with at least one cycle is homeomorphic to a non synchronized one [4]. Besides, we proved some other less general results, for instance that the wheels synchronize [5] as well as the complete k-partite graphs [2]. Lastly, in [14], Taylor made a big progress proving that there is a non trivial upper bound for the ratio of the minimum degree over the number of nodes to assure the synchronizability of a graph.…”

Section: Introductionmentioning

confidence: 94%

Exotic equilibria of Harary graphs and a new minimum degree lower bound for synchronization

Canale¹,

Monzón

2015

Chaos

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This work is concerned with stability of equilibria in the homogeneous (equal frequencies) Kuramoto model of weakly coupled oscillators. In 2012 [R. Taylor, J. Phys. A: Math. Theor. 45, 1-15 (2012)], a sufficient condition for almost global synchronization was found in terms of the minimum degree-order ratio of the graph. In this work, a new lower bound for this ratio is given. The improvement is achieved by a concrete infinite sequence of regular graphs. Besides, non standard unstable equilibria of the graphs studied in Wiley et al. [Chaos 16, 015103 (2006)] are shown to exist as conjectured in that work.

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Section: Introductionmentioning

confidence: 94%

Exotic equilibria of Harary graphs and a new minimum degree lower bound for synchronization

Canale¹,

Monzón

2015

Chaos

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“…Thus complete graphs can have only one single stable fixed point [18] as can all dense graphs with degree at least 0.9395(n − 1) [24]. Also trees can have only one stable fixed point [4] though the nature of all fixed points and the relationship to the coupling constant is well understood [7]. On the other hand ring graphs do have multiple stable fixed points the nature of which is also well understood [19].…”

Section: Non-zero Stable Fixed Pointsmentioning

confidence: 99%

Finding Non-Zero Stable Fixed Points of the Weighted Kuramoto model is NP-hard

Taylor¹

2015

Preprint

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The Kuramoto model when considered over the full space of phase angles [0, 2π) can have multiple stable fixed points which form basins of attraction in the solution space. In this paper we illustrate the fundamentally complex relationship between the network topology and the solution space by showing that determining the possibility of multiple stable fixed points from the network topology is NP-hard for the weighted Kuramoto Model. In the case of the unweighted model this problem is shown to be at least as difficult as a number partition problem, which we conjecture to be NP-hard. We conclude that it is unlikely that stable fixed points of the Kuramoto model can be characterized in terms of easily computable network invariants.

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Synchronization in complex networks of phase oscillators: A survey

2014

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The emergence of synchronization in a network of coupled oscillators is a fascinating subject of multidisciplinary research. This survey reviews the vast literature on the theory and the applications of complex oscillator networks. We focus on phase oscillator models that are widespread in real-world synchronization phenomena, that generalize the celebrated Kuramoto model, and that feature a rich phenomenology. We review the history and the countless applications of this model throughout science and engineering. We justify the importance of the widespread coupled oscillator model as a locally canonical model and describe some selected applications relevant to control scientists, including vehicle coordination, electric power networks, and clock synchronization. We introduce the reader to several synchronization notions and performance estimates. We propose analysis approaches to phase and frequency synchronization, phase balancing, pattern formation, and partial synchronization. We present the sharpest known results about synchronization in networks of homogeneous and heterogeneous oscillators, with complete or sparse interconnection topologies, and in finite-dimensional and infinite-dimensional settings. We conclude by summarizing the limitations of existing analysis methods and by highlighting some directions for future research.

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The wheels: an infinite family of bi-connected planar synchronizing graphs

Cited by 4 publications

References 11 publications

Exotic equilibria of Harary graphs and a new minimum degree lower bound for synchronization

Exotic equilibria of Harary graphs and a new minimum degree lower bound for synchronization

Finding Non-Zero Stable Fixed Points of the Weighted Kuramoto model is NP-hard

Synchronization in complex networks of phase oscillators: A survey

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