We study the effect of structured higher-order interactions on the collective behavior of coupled phase oscillators. By combining a hypergraph generative model with dimensionality reduction techniques, we obtain a reduced system of differential equations for the system’s order parameters. We illustrate our framework with the example of a hypergraph with hyperedges of sizes 2 (links) and 3 (triangles). For this case, we obtain a set of two coupled nonlinear algebraic equations for the order parameters. For strong values of coupling via triangles, the system exhibits bistability and explosive synchronization transitions. We find conditions that lead to bistability in terms of hypergraph properties and validate our predictions with numerical simulations. Our results provide a general framework to study the synchronization of phase oscillators in hypergraphs, and they can be extended to hypergraphs with hyperedges of arbitrary sizes, dynamic-structural correlations, and other features.
REFERENCES
1.
G.
Petri
, P.
Expert
, F.
Turkheimer
, R.
Carhart-Harris
, D.
Nutt
, P. J.
Hellyer
, and F.
Vaccarino
, “Homological scaffolds of brain functional networks
,” J. R. Soc. Interface
11
, 20140873
(2014
). 2.
M. G.
Kitzbichler
, M. L.
Smith
, S. R.
Christensen
, and E.
Bullmore
, “Broadband criticality of human brain network synchronization
,” PLoS Comput. Biol.
5
, e1000314
(2009
). 3.
B.
Zhu
, J.
Schachenmayer
, M.
Xu
, F.
Herrera
, J. G.
Restrepo
, M. J.
Holland
, and A. M.
Rey
, “Synchronization of interacting quantum dipoles
,” New J. Phys.
17
, 083063
(2015
). 4.
M.
Rohden
, A.
Sorge
, M.
Timme
, and D.
Witthaut
, “Self-organized synchronization in decentralized power grids
,” Phys. Rev. Lett.
109
, 064101
(2012
). 5.
Y.
Fujino
, B. M.
Pacheco
, S.-I.
Nakamura
, and P.
Warnitchai
, “Synchronization of human walking observed during lateral vibration of a congested pedestrian bridge
,” Earthq. Eng. Struct. Dyn.
22
, 741
–758
(1993
). 6.
S. H.
Strogatz
, D. M.
Abrams
, A.
McRobie
, B.
Eckhardt
, and E.
Ott
, “Crowd synchrony on the millennium bridge
,” Nature
438
, 43
–44
(2005
). 7.
C.
Bick
, P.
Ashwin
, and A.
Rodriquez
, “Chaos in generically coupled phase oscillator networks with non-pairwise interaction
,” Chaos
26
, 8
(2016
). 8.
I.
Leon
and D.
Pazo
, “Phase reduction beyond the first order: The case of mean-field complex Ginzburg-Laundau equation
,” Phys. Rev. E
100
, 15
(2019
). 9.
A. E.
Sizemore
, C.
Giusti
, A.
Kahn
, J. M.
Vettel
, R. F.
Betzel
, and D. S.
Bassett
, “Cliques and cavities in the human connectome
,” J. Comput. Neurosci.
44
, 115
–145
(2018
). 10.
P. S.
Skardal
and A.
Arenas
, “Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching
,” Commun. Phys.
3
, 6
(2020
). 11.
C.
Giusti
, R.
Ghrist
, and D. S.
Bassett
, “Two’s company, three (or more) is a simplex
,” J. Comput. Neurosci.
41
, 1
–14
(2016
). 12.
P. S.
Skardal
and A.
Arenas
, “Abrupt desynchronization and extensive multistability in globally coupled oscillator simplexes
,” Phys. Rev. Lett.
122
, 6
(2019
). 13.
A. P.
Millán
, J. J.
Torres
, and G.
Bianconi
, “Explosive higher-order Kuramoto dynamics on simplicial complexes
,” Phys. Rev. Lett.
124
, 218301
(2020
). 14.
R.
Ghorbanchian
, J. G.
Restrepo
, J. J.
Torres
, and G.
Bianconi
, “Higher-order simplicial synchronization of coupled topological signals
,” Commun. Phys.
4
, 1
–13
(2021
). 15.
L.
Calmon
, J. G.
Restrepo
, J. J.
Torres
, and G.
Bianconi
, “Topological synchronization: Explosive transition and rhythmic phase,” arXiv:2107.05107 (2021).16.
G.
Osipov
, J.
Kurths
, and C.
Zhou
, Synchronization in Oscillatory Networks
(Springer
, 1982
).17.
A.
Arenas
, A.
Díaz-Guilera
, J.
Kurths
, Y.
Moreno
, and C.
Zhou
, “Synchronization in complex networks
,” Phys. Rep.
469
, 93
–153
(2008
). 18.
J.
Buck
and E.
Buck
, “Biology of synchronous flashing of fireflies
,” Nature
211
, 562
–564
(1966
). 19.
R.
Sarfati
, J. C.
Hayes
, É.
Sarfati
, and O.
Peleg
, “Spatio-temporal reconstruction of emergent flash synchronization in firefly swarms via stereoscopic 360-degree cameras
,” J. R. Soc. Interface
17
, 20200179
(2020
). 20.
Y.
Penn
, M.
Segal
, and E.
Moses
, “Network synchronization in hippocampal neurons
,” Proc. Natl. Acad. Sci. U.S.A.
113
, 6
(2016
). 21.
M.
Rohden
, M. T. A.
Sorge
, and D.
Witthaut
, “Self-organized synchronization in decentralized power grids
,” Phys. Rev. Lett.
109
, 5
(2012
). 22.
F.
Dörfler
, M.
Chertkov
, and F.
Bullo
, “Synchronization in complex oscillator networks and smart grids
,” Proc. Natl. Acad. Sci. U.S.A.
110
, 6
(2012
).23.
Y.
Kuramoto
, “Self-entrainment of a population of coupled non-linear oscillators,” in International Symposium on Mathematical Problems in Theoretical Physics
(Springer, 1975), pp. 420–422. 24.
J. A.
Acebrón
, L. L.
Bonilla
, C. J. P.
Vicente
, F.
Ritort
, and R.
Spigler
, “The Kuramoto model: A simple paradigm for synchronization phenomena
,” Rev. Mod. Phys.
77
, 137
(2005
). 25.
F. A.
Rodrigues
, T. K. D.
Peron
, P.
Ji
, and J.
Kurths
, “The Kuramoto model in complex networks
,” Phys. Rep.
610
, 1
–98
(2016
). 26.
J.
Gómez-Gardeñes
, S.
Gómez
, A.
Arenas
, and Y.
Moreno
, “Explosive synchronization transitions in scale-free networks
,” Phys. Rev. Lett.
106
, 128701
(2011
). 27.
C. R.
Laing
, “Chimera states in heterogeneous networks
,” Chaos
19
, 013113
(2009
). 28.
E.
Ott
and T. M.
Antonsen
, “Low dimensional behavior of large systems of globally coupled oscillators
,” Chaos
18
, 6
(2008
). 29.
K.
Miller
, M.
Jordan
, and T.
Griffiths
, “Nonparametric latent feature models for link prediction
,” Adv. Neural Inf. Process. Syst.
22
, 1
–9
(2009
).30.
N. W.
Landry
and J. G.
Restrepo
, “The effect of heterogeneity in the hypergraph contagion models
,” Chaos
30
, 13
(2020
). 31.
G.
Bianconi
and C.
Rahmede
, “Network geometry with flavor: From complexity to quantum geometry
,” Phys. Rev. E
93
, 032315
(2016
). 32.
O. T.
Courtney
and G.
Bianconi
, “Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes
,” Phys. Rev. E
93
, 062311
(2016
). 33.
K.
Kovalenko
, I.
Sendiña-Nadal
, N.
Khalil
, A.
Dainiak
, D.
Musatov
, A. M.
Raigorodskii
, K.
Alfaro-Bittner
, B.
Barzel
, and S.
Boccaletti
, “Growing scale-free simplices
,” Commun. Phys.
4
, 1
–9
(2021
). 34.
C.
Bick
, E.
Gross
, H. A.
Harrington
, and M. T.
Schaub
, “What are higher-order networks?,” arXiv:2104.11329v3 (2021).35.
R.
Pastor-Satorras
, C.
Castellano
, P.
Van Mieghem
, and A.
Vespignani
, “Epidemic processes in complex networks
,” Rev. Mod. Phys.
87
, 925
(2015
). 36.
S. N.
Dorogovtsev
, A. V.
Goltsev
, and J. F.
Mendes
, “Critical phenomena in complex networks
,” Rev. Mod. Phys.
80
, 1275
(2008
). 37.
M.
Boguná
, C.
Castellano
, and R.
Pastor-Satorras
, “Langevin approach for the dynamics of the contact process on annealed scale-free networks
,” Phys. Rev. E
79
, 036110
(2009
). 38.
G.
Poux-Médard
, R.
Pastor-Satorras
, and C.
Castellano
, “Influential spreaders for recurrent epidemics on networks
,” Phys. Rev. Res.
2
, 023332
(2020
). 39.
J. G.
Restrepo
and E.
Ott
, “Mean-field theory of assortative networks of phase oscillators
,” Europhys. Lett.
107
, 60006
(2014
). 40.
C.
Bick
, M.
Goodfellow
, C. R.
Laing
, and E. A.
Martens
, “Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: A review
,” J. Math. Neurosci.
10
, 9
(2020
). 41.
A.
Pikovsky
and M.
Rosenblum
, “Dynamics of heterogeneous oscillator ensembles in terms of collective variables
,” Physica D
240
, 872
–881
(2011
). 42.
E. J.
Hildebrand
, M. A.
Buice
, and C. C.
Chow
, “Kinetic theory of coupled oscillators
,” Phys. Rev. Lett.
98
, 054101
(2007
). 43.
J. G.
Restrepo
, E.
Ott
, and B.
Hunt
, “Onset of synchronization in large networks of coupled oscillators
,” Phys. Rev. E
71
, 14
(2005
). 44.
T.
Ichinomiya
, “Frequency synchronization in a random oscillator network
,” Phys. Rev. E
70
, 1
–11
(2004
). 45.
C.
Kuehn
and C.
Bick
, “A universal route to explosive phenomena
,” Sci. Adv.
7
, eabe3824
(2021
). 46.
I.
Iacopini
, G.
Petri
, A.
Barrat
, and V.
Latora
, “Simplicial models of social contagion
,” Nat. Commun.
10
, 1
–9
(2019
). 47.
G.
Burgio
, A.
Arenas
, S.
Gómez
, and J. T.
Matamalas
, “Network clique cover approximation to analyze complex contagions through group interactions
,” Commun. Phys.
4
, 1
–10
(2021
). 48.
Y.
Zhang
, M.
Lucas
, and F.
Battiston
, “Do higher-order interactions promote synchronization?,” arXiv:2203.03060 (2022).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
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