We study the synchronized state in a population of network-coupled, heterogeneous oscillators. In particular, we show that the steady-state solution of the linearized dynamics may be written as a geometric series whose subsequent terms represent different spatial scales of the network. Specifically, each additional term incorporates contributions from wider network neighborhoods. We prove that this geometric expansion converges for arbitrary frequency distributions and for both undirected and directed networks provided that the adjacency matrix is primitive. We also show that the error in the truncated series grows geometrically with the second largest eigenvalue of the normalized adjacency matrix, analogously to the rate of convergence to the stationary distribution of a random walk. Last, we derive a local approximation for the synchronized state by truncating the spatial series, at the first neighborhood term, to illustrate the practical advantages of our approach.
REFERENCES
1.
A.
Prindle
, P.
Samayoa
, I.
Razinkov
, T.
Danino
, L. S.
Tsimring
, and J.
Hasty
, “A sensing array of radically coupled genetic ‘biopixels’
,” Nature
481
, 39
–44
(2012
). 2.
D.
Witthaut
and M.
Timme
, “Braesss paradox in oscillator networks, desynchronization and power outage
,” New J. Phys.
14
, 083036
(2012
). 3.
P. S.
Skardal
, D.
Taylor
, and J.
Sun
, “Optimal synchronization of complex networks
,” Phys. Rev. Lett.
113
, 144101
(2014
). 4.
P. S.
Skardal
, D.
Taylor
, and J.
Sun
, “Optimal synchronization of directed complex networks
,” Chaos
26
, 094807
(2016
). 5.
D.
Taylor
, P. S.
Skardal
, and J.
Sun
, “Synchronization of heterogeneous oscillators under network modifications: Perturbation and optimization of the synchrony alignment function
,” SIAM J. Appl. Math.
76
, 1984
–2008
(2016
). 6.
P. S.
Skardal
and A.
Arenas
, “Control of coupled oscillator networks with application to microgrid technologies
,” Sci. Adv.
1
, e1500339
(2015
). 7.
P. S.
Skardal
and A.
Arenas
, “On controlling networks of limit-cycle oscillators
,” Chaos
26
, 094812
(2016
). 8.
F.
Dörfler
, M.
Chertkov
, and F.
Bullo
, “Synchronization in complex oscillator networks and smart grids
,” Proc. Natl. Acad. Sci. U.S.A.
110
, 2005
–2010
(2013
). 9.
L. M.
Pecora
and T. L.
Carroll
, “Master stability functions for synchronized coupled systems
,” Phys. Rev. Lett.
80
, 2109
–2112
(1998
). 10.
D. J.
Watts
and S. H.
Strogatz
, “Collective dynamics of ’small-world’ networks
,” Nature
393
, 440
–442
(1998
). 11.
A.
Arenas
, A.
Díaz-Guilera
, J.
Kurths
, Y.
Moreno
, and C.
Zhou
, “Synchronization in complex networks
,” Phys. Rep.
469
, 93
–153
(2008
). 12.
A.
Pikovsky
, M.
Rosenblum
, and J.
Kurths
, Synchronization: A Universal Concept in Nonlinear Sciences
(Cambridge University Press
, 2003
), Vol. 12.13.
Y.
Kuramoto
, Chemical Oscillations, Waves, and Turbulence
(Courier Corporation
, 2003
).14.
M.
Newman
, Networks: An Introduction
(Oxford University Press, Inc.
, New York
, 2010
).15.
A.
Ben-Israel
and T.
Greville
, Generalized Inverses: Theory and Applications
(Wiley
, New York
, 1974
).16.
P. S.
Skardal
, D.
Taylor
, J.
Sun
, and A.
Arenas
, “Collective frequency variation in network synchronization and reverse pagerank
,” Phys. Rev. E
93
, 042314
(2016
). 17.
A.
Arenas
, A.
Díaz-Guilera
, and C. J.
Pérez-Vicente
, “Synchronization reveals topological scales in complex networks
,” Phys. Rev. Lett.
96
, 114102
(2006
). 18.
M.
Timme
, “Revealing network connectivity from response dynamics
,” Phys. Rev. Lett.
98
, 224101
(2007
). 19.
M.
De Domenico
, “Diffusion geometry unravels the emergence of functional clusters in collective phenomena
,” Phys. Rev. Lett.
118
, 168301
(2017
). 20.
D.
Manik
, M.
Rohden
, X.
Ronellenfitsch
, H.
Zhang
, S.
Hallerberg
, D.
Witthaut
, and M.
Timme
, “Network susceptibilities: Theory and applications
,” Phys. Rev. E
95
, 012319
(2017
). 21.
T.
Coletta
and P.
Jacquod
, “Linear stability and the Braess paradox in coupled-oscillator networks and electric power grids
,” Phys. Rev. E
93
, 032222
(2016
). 22.
P.
Van Mieghem
, K.
Devriendt
, and H.
Cetinay
, “Pseudoinverse of the Laplacian and best spreader node in a network
,” Phys. Rev. E
96
, 032311
(2017
). 23.
H.
Ronellenfitsch
, J.
Dunkel
, and M.
Wilczek
, “Optimal noise-canceling networks
,” Phys. Rev. Lett.
121
, 208301
(2018
). 24.
M. E. J.
Newman
, “A measure of betweenness centrality based on random walks
,” Soc. Netw.
27
, 39
–54
(2005
). 25.
F.
Chung
and S. T.
Yau
, “Discrete Green’s functions
,” J. Comb. Theory Ser. A
91
, 191
–214
(2000
). 26.
E.
Estrada
and N.
Hatano
, “Communicability in complex networks
,” Phys. Rev. E
77
, 036111
(2008
). 27.
C. R.
MacCluer
, “The many proofs and applications of Perron’s theorem
,” SIAM Rev.
42
, 487
–498
(2000
). 28.
P. N.
McGraw
and M.
Menzinger
, “Analysis of nonlinear synchronization dynamics of oscillator networks by Laplacian spectral methods
,” Phys. Rev. E
75
, 027104
(2007
). 29.
D. I.
Shuman
, S. K.
Narang
, P.
Frossard
, A.
Ortega
, and P.
Vandergheynst
, “The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains
,” IEEE Signal Process. Mag.
30
, 83
–98
(2013
). 30.
N.
Masuda
, M. A.
Porter
, and R.
Lambiotte
, “Random walks and diffusion on networks
,” Phys. Rep.
716-717
, 1
–58
(2017
). 31.
P.
Holme
and B.
Kim
, “Growing scale-free networks with tunable clustering
,” Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys.
65
, 026107
(2002
). 32.
33.
P. S.
Skardal
, J.
Sun
, D.
Taylor
, and J. G.
Restrepo
, “Effects of degree-frequency correlations on network synchronization: Universality and full phase-locking
,” Europhys. Lett.
101
, 20001
(2013
). 34.
J.
Gómez-Gardeñes
, S.
Gómez
, A.
Arenas
, and Y.
Moreno
, “Explosive synchronization transitions in scale-free networks
,” Phys. Rev. Lett.
106
, 128701
(2011
). 35.
P.
Skardal
, D.
Taylor
, and J.
Sun
, “Synchronization of network-coupled oscillators with uncertain dynamics
,” SIAM J. Appl. Math.
79
, 2409
–2433
(2019
). 36.
A. E.
Motter
and M.
Timme
, “Antagonistic phenomena in network dynamics
,” Annu. Rev. Condens. Matter Phys.
9
, 463
–484
(2018
). 37.
T.
Nishikawa
and A.
Motter
, “Synchronization is optimal in nondiagonalizable networks
,” Phys. Rev. E
73
, 065106
(2006
). 38.
L.
Arola-Fernández
, A.
Díaz-Guilera
, and A.
Arenas
, “Synchronization invariance under network structural transformations
,” Phys. Rev. E
97
, 060301
(2018
). 39.
L.
Arola-Fernández
, G.
Mosquera-Doñate
, B.
Steinegger
, and A.
Arenas
, “Uncertainty propagation in complex networks: From noisy links to critical properties
,” Chaos
30
, 023129
(2020
). © 2021 Author(s). Published under an exclusive license by AIP Publishing.
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