Synchronization of Heterogeneous Oscillators Under Network Modifications: Perturbation and Optimization of the Synchrony Alignment Function

Taylor, Dane; Skardal, Per Sebastian; Sun, Jie

doi:10.1137/16m1075181

Cited by 28 publications

(25 citation statements)

References 64 publications

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“…The first term is the SAF, J(µ, L), with the expected frequencies as the first argument, rather than the particular frequencies ω. This provides theoretical support for why one can optimize the SAF using approximated frequencies, even when the exact frequencies are unknown [see also figure 6.2(c) in [43] for a related numerical experiment]. One can optimize J(µ, L) under a variety of system constraints on L and/or µ using the same techniques that were explored in previous research on the SAF [36,37,43].…”

Section: Discussionmentioning

confidence: 67%

“…In this paper, we have investigated the behavior of the synchrony alignment function, or SAF, (see subsection 2.2 for a definition), which is a framework for evaluating how the interplay of structural network properties and internal dynamics (i.e., natural frequencies) shape the overall synchronization properties of heterogeneous oscillator networks. Extending previous research of the SAF [36,37,38,37,43,37] that treats the oscillators' natural frequencies as deterministic parameters, in section 3 we considered the oscillators' natural frequencies as random variables and derived analytical expressions for the expectation and variance of the SAF. Theorems 3.2 and 3.7 and their corollaries characterize the expectation and variance of the SAF for when the frequencies are uncertain-they are drawn from a distribution with given mean and variance.…”

Section: Discussionmentioning

confidence: 99%

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Synchronization of Network-Coupled Oscillators with Uncertain Dynamics

Skardal¹,

Taylor²,

Sun³

2019

SIAM J. Appl. Math.

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Synchronization of network-coupled dynamical units is important to a variety of natural and engineered processes including circadian rhythms, cardiac function, neural processing, and power grids. Despite this ubiquity, it remains poorly understood how complex network structures and heterogeneous local dynamics combine to either promote or inhibit synchronization. Moreover, for most real-world applications it is impossible to obtain the exact specifications of the system, and there is a lack of theory for how uncertainty affects synchronization. We address this open problem by studying the Synchrony Alignment Function (SAF), which is an objective measure for the synchronization properties of a network of heterogeneous oscillators with given natural frequencies.We extend the SAF framework to analyze network-coupled oscillators with heterogeneous natural frequencies that are drawn as a multivariate random vector. Using probability theory for quadratic forms, we obtain expressions for the expectation and variance of the SAF for given network structures. We conclude with numerical experiments that illustrate how the incorporation of uncertainty yields a more robust theoretical framework for enhancing synchronization, and we provide new perspectives for why synchronization is generically promoted by network properties including degree-frequency correlations, link directedness, and link weight delocalization.

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Section: Discussionmentioning

confidence: 67%

Section: Discussionmentioning

confidence: 99%

Synchronization of Network-Coupled Oscillators with Uncertain Dynamics

Skardal¹,

Taylor²,

Sun³

2019

SIAM J. Appl. Math.

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“…However, the regime for which this approximation is valid (i.e., how small needs to be) generally depends on L, , and the perturbation ∆L. Accuracy typically requires λ i (0)/λ i to be small [80].…”

Section: Perturbation Of Vne and Laplacian Matricesmentioning

confidence: 99%

Network-Ensemble Comparisons with Stochastic Rewiring and Von Neumann Entropy

Li¹,

Mucha²,

Taylor³

2018

SIAM J. Appl. Math.

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Assessing whether a given network is typical or atypical for a random-network ensemble (i.e., network-ensemble comparison) has widespread applications ranging from null-model selection and hypothesis testing to clustering and classifying networks. We develop a framework for network-ensemble comparison by subjecting the network to stochastic rewiring. We study two rewiring processes—uniform and degree-preserved rewiring—which yield random-network ensembles that converge to the Erdős-Rényi and configuration-model ensembles, respectively. We study convergence through von Neumann entropy (VNE)—a network summary statistic measuring information content based on the spectra of a Laplacian matrix—and develop a perturbation analysis for the expected effect of rewiring on VNE. Our analysis yields an estimate for how many rewires are required for a given network to resemble a typical network from an ensemble, offering a computationally efficient quantity for network-ensemble comparison that does not require simulation of the corresponding rewiring process.

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“…( syn g means threshold) [5]. Therefore, to study the complete synchronization behavior in the complex systems, finding the different pathways that could reduce R remain a primary goal for the researchers [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. However, modification (1) increases the coupling cost of synchronization whereas the modification (2) decreases the robustness by increasing the centralization in a network.…”

Section: Introductionmentioning

confidence: 99%

Enhancement of network synchronizability via two oscillatory system

Singh

2018

J. Phys. Commun.

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Network's synchronization destabilizes at large coupling strength when its nodes exchange information by scalar coupling (few coordinates) instead of using vector coupling (all coordinates). This issue is commonly tackled by modifying the network topology like a ring network to Small World or Scale Free networks which increases the coupling cost and decreases the robustness. In the present work, we show that without any structural alteration even a large ring network in the nearest neighborhood configuration using only a single coordinate in the coupling, could be made synchronizable. The primary condition is that the node dynamics should be given by a pair of oscillators (say, two oscillatory system TOS) rather than by a conventional way of single oscillator (say, single oscillatory system). It has been found that TOS not only stabilizes the chaotic synchronization but also the hyperchaotic synchronization manifold (a major challenge in the field of secure communication wherein multi parameter BK method is needed). The frameworks of drive-response system and master stability function have been used to study the TOS effect by varying TOS parameters with and without feedback (feedback means quorum sensing conditions). The TOS effect has been found numerically both in the chaotic (Rössler, Chua and Lorenz) and hyperchaotic (electrical circuit) systems. However, since threshold also increases as a side effect of TOS, the extent of β enhancement depends on the choice of oscillator model like larger for Rössler, intermediate for Chua and smaller for Lorenz. dsyn g ) limit the size of a synchronizable network, e.g. the synchronized state of a ring network having x 1 -coupled chaotic Rössler becomes unstable with the increase in N from 18 to 19 at dsyn g = 1.5. To tackle this issue of limited values of dsyn g , the network modification methods are generally used such as (1) adding the additional edges between the nodes deterministically (Pristine World) or/and stochastically (Small World) [5], (2) modifying the ring network topology to a synchronizable topology like a unidirectional tree network or a star network (a hub of Scale Free network) [6][7][8]. Practically these modifications could be considered as the distribution of overload among the nodes and theoretically these alterations imply the minimization of eigen-ratio (R) of the Laplacian/ coupling matrix (largest eigenvalue to the smallest nonzero eigenvalue). The concept of R minimization comes from the theory of master stability function (MSF) [9] which says that a complex network of size N is

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Synchronization of Heterogeneous Oscillators Under Network Modifications: Perturbation and Optimization of the Synchrony Alignment Function

Cited by 28 publications

References 64 publications

Synchronization of Network-Coupled Oscillators with Uncertain Dynamics

Synchronization of Network-Coupled Oscillators with Uncertain Dynamics

Network-Ensemble Comparisons with Stochastic Rewiring and Von Neumann Entropy

Enhancement of network synchronizability via two oscillatory system

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