Eigenvector-based analysis of cluster synchronization in general complex networks of coupled chaotic oscillators
Huawei Fan1,2, Ya Wang2, Xingang Wang2()
1. School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710121, China 2. School of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China
Whereas topological symmetries have been recognized as crucially important to the exploration of synchronization patterns in complex networks of coupled dynamical oscillators, the identification of the symmetries in large-size complex networks remains as a challenge. Additionally, even though the topological symmetries of a complex network are known, it is still not clear how the system dynamics is transited among different synchronization patterns with respect to the coupling strength of the oscillators. We propose here the framework of eigenvector-based analysis to identify the synchronization patterns in the general complex networks and, incorporating the conventional method of eigenvalue-based analysis, investigate the emergence and transition of the cluster synchronization states. We are able to argue and demonstrate that, without a prior knowledge of the network symmetries, the method is able to predict not only all the cluster synchronization states observable in the network, but also the critical couplings where the states become stable and the sequence of these states in the process of synchronization transition. The efficacy and generality of the proposed method are verified by different network models of coupled chaotic oscillators, including artificial networks of perfect symmetries and empirical networks of non-perfect symmetries. The new framework paves a way to the investigation of synchronization patterns in large-size, general complex networks.
Kuramoto Y., Chemical Oscillations, Waves, and Turbulence, Springer, Berlin, 1984
2
T. Winfree A., Timing of Biological Clocks, W H Freeman & Co, 1987
3
S. Pikovsky A.G. Rosenblum M.Kurths J., Synchronization: A Universal Concept in Nonlinear Science, Cambridge University Press, Cambridge, 2001
4
Strogatz S., Sync: The Emerging Science of Spontaneous Order, Hyperion, New York, 2003
5
M. Pecora L. , L. Carroll T. . Master stability functions for synchronized coupled systems. Phys. Rev. Lett., 1998, 80(10): 2109 https://doi.org/10.1103/PhysRevLett.80.2109
6
Hu G. , Z. Yang J. , Liu W. . Instability and controllability of linearly coupled oscillators: Eigenvalue analysis. Phys. Rev. E, 1998, 58(4): 4440 https://doi.org/10.1103/PhysRevE.58.4440
7
Huang L. , Chen Q. , C. Lai Y. , M. Pecora L. . Generic behavior of master-stability functions in coupled nonlinear dynamical systems. Phys. Rev. E, 2009, 80(3): 036204 https://doi.org/10.1103/PhysRevE.80.036204
8
A. Acebrón J. , L. Bonilla L. , J. Pérez Vicente C. , Ritort F. , Spigler R. . The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys., 2005, 77(1): 137 https://doi.org/10.1103/RevModPhys.77.137
9
Ott E. , M. Antonsen T. . Low dimensional behavior of large systems of globally coupled oscillators. Chaos, 2008, 18(3): 037113 https://doi.org/10.1063/1.2930766
10
Kaneko K., Theory and Application of Coupled Map Lattice, Wiley, Chichester, 1993
11
J. Watts D. , H. Strogatz S. . Collective dynamics of “small-world” networks. Nature, 1998, 393(6684): 440 https://doi.org/10.1038/30918
E. J. Newman M., Networks: An Introduction, Oxford University Press, 2010
14
Boccaletti S. , Latora V. , Moreno Y. , Chavez M. , U. Hwang D. . Complex networks: Structure and dynamics. Phys. Rep., 2006, 424(4−5): 175 https://doi.org/10.1016/j.physrep.2005.10.009
Wu T. , Zhang X. , Liu Z. . Understanding the mechanisms of brain functions from the angle of synchronization and complex network. Front. Phys., 2022, 17(3): 31504 https://doi.org/10.1007/s11467-022-1161-6
Nishikawa T. , E. Motter A. , C. Lai Y. , C. Hoppensteadt F. . Heterogeneity in oscillator networks: Are smaller worlds easier to synchronize. Phys. Rev. Lett., 2003, 91(1): 014101 https://doi.org/10.1103/PhysRevLett.91.014101
20
Arenas A. , Díaz-Guilera A. , J. Pérez-Vicente C. . Synchronization reveals topological scales in complex networks. Phys. Rev. Lett., 2006, 96(11): 114102 https://doi.org/10.1103/PhysRevLett.96.114102
21
Hansel D. , Mato G. , Meunier C. . Clustering and slow switching in globally coupled phase oscillators. Phys. Rev. E, 1993, 48(5): 3470 https://doi.org/10.1103/PhysRevE.48.3470
22
Hasler M. , Maistrenko Yu. , Popovych O. . Simple example of partial synchronization of chaotic systems. Phys. Rev. E, 1998, 58(5): 6843 https://doi.org/10.1103/PhysRevE.58.6843
23
Zhang Y. , Hu G. , A. Cerdeira H. , Chen S. , Braun T. , Yao Y. . Partial synchronization and spontaneous spatial ordering in coupled chaotic systems. Phys. Rev. E, 2001, 63(2): 026211 https://doi.org/10.1103/PhysRevE.63.026211
24
Pikovsky A. , Popovych O. , Maistrenko Yu. . Resolving clusters in chaotic ensembles of globally coupled identical oscillators. Phys. Rev. Lett., 2001, 87(4): 044102 https://doi.org/10.1103/PhysRevLett.87.044102
25
A. Heisler I. , Braun T. , Zhang Y. , Hu G. , A. Cerdeira H. . Experimental investigation of partial synchronization in coupled chaotic oscillators. Chaos, 2003, 13(1): 185 https://doi.org/10.1063/1.1505811
26
R. S. Williams C. , E. Murphy T. , Roy R. , Sorrentino F. , Dahms T. , Schöll E. . Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators. Phys. Rev. Lett., 2013, 110(6): 064104 https://doi.org/10.1103/PhysRevLett.110.064104
M. Norton M. , Tompkins N. , Blanc B. , C. Cambria M. , Held J. , Fraden S. . Dynamics of reaction-diffusion oscillators in star and other networks with cyclic symmetries exhibiting multiple clusters. Phys. Rev. Lett., 2019, 123(14): 148301 https://doi.org/10.1103/PhysRevLett.123.148301
29
Fan H. , W. Kong L. , G. Wang X. , Hastings A. , C. Lai Y. . Synchronization within synchronization: Transients and intermittency in ecological networks. Natl. Sci. Rev., 2021, 8(10): nwaa269 https://doi.org/10.1093/nsr/nwaa269
30
Rodriguez E. , George N. , P. Lachaux J. , Martinerie J. , Renault B. , J. Varela F. . Perception’s shadow: Long-distance synchronization of human brain activity. Nature, 1999, 397(6718): 430 https://doi.org/10.1038/17120
31
Kitsunai S. , Cho W. , Sano C. , Saetia S. , Qin Z. , Koike Y. , Frasca M. , Yoshimura N. , Minati L. . Generation of diverse insect-like gait patterns using networks of coupled Rössler systems. Chaos, 2020, 30(12): 123132 https://doi.org/10.1063/5.0021694
32
F. Heagy J. , M. Pecora L. , L. Carroll T. . Short wavelength bifurcations and size instabilities in coupled oscillator systems. Phys. Rev. Lett., 1995, 774(21): 4185 https://doi.org/10.1103/PhysRevLett.74.4185
33
M. Pecora L. . Synchronization conditions and desynchronizing patterns in coupled limit-cycle and chaotic systems. Phys. Rev. E, 1998, 58(1): 347 https://doi.org/10.1103/PhysRevE.58.347
Fu C. , Deng Z. , Huang L. , G. Wang X. . Topological control of synchronous patterns in systems of networked chaotic oscillators. Phys. Rev. E, 2013, 87(3): 032909 https://doi.org/10.1103/PhysRevE.87.032909
37
Fu C. , Lin W. , Huang L. , G. Wang X. . Synchronization transition in networked chaotic oscillators: The viewpoint from partial synchronization. Phys. Rev. E, 2014, 89(5): 052908 https://doi.org/10.1103/PhysRevE.89.052908
38
M. Pecora L. , Sorrentino F. , M. Hagerstrom A. , E. Murphy T. , Roy R. . Cluster synchronization and isolated desynchronization in complex networks with symmetries. Nat. Commun., 2014, 5(1): 4079 https://doi.org/10.1038/ncomms5079
39
T. Schaub M. , O’Clery N. , N. Billeh Y. , C. Delvenne J. , Lambiotte R. , Barahona M. . Graph partitions and cluster synchronization in networks of oscillators. Chaos, 2016, 26(9): 094821 https://doi.org/10.1063/1.4961065
40
Sorrentino F. , M. Pecora L. , M. Hagerstrom A. , E. Murphy T. , Roy R. . Complete characterization of the stability of cluster synchronization in complex dynamical networks. Sci. Adv., 2016, 2(4): e1501737 https://doi.org/10.1126/sciadv.1501737
41
D. Hart J. , Zhang Y. , Roy R. , E. Motter A. . Topological control of synchronization pattern: Trading symmetry for stability. Phys. Rev. Lett., 2019, 122(5): 058301 https://doi.org/10.1103/PhysRevLett.122.058301
42
M. Abrams D. , M. Pecora L. , E. Motter A. . Introduction to focus issue: Patterns of network synchronization. Chaos, 2016, 26(9): 094601 https://doi.org/10.1063/1.4962970
43
Golubitsky M. , Stewart I. . Recent advances in symmetric and network dynamics. Chaos, 2015, 25(9): 097612 https://doi.org/10.1063/1.4918595
44
Lin W. , Fan H. , Wang Y. , Ying H. , G. Wang X. . Controlling synchronous patterns in complex networks. Phys. Rev. E, 2016, 93(4): 042209 https://doi.org/10.1103/PhysRevE.93.042209
45
Lin W. , Li H. , Ying H. , G. Wang X. . Inducing isolated-desynchronization states in complex network of coupled chaotic oscillators. Phys. Rev. E, 2016, 94(6): 062303 https://doi.org/10.1103/PhysRevE.94.062303
46
Nishikawa T. , E. Motter A. . Network-complement transitions, symmetries, and cluster synchronization. Chaos, 2016, 26(9): 094818 https://doi.org/10.1063/1.4960617
Cao B. , F. Wang Y. , Wang L. , Z. Yu Y. , G. Wang X. . Cluster synchronization in complex network of coupled chaotic circuits: An experimental study. Front. Phys., 2018, 13(5): 130505 https://doi.org/10.1007/s11467-018-0775-1
49
F. Wang Y. , Wang L. , Fan H. , G. Wang X. . Cluster synchronization in networked nonidentical chaotic oscillators. Chaos, 2019, 29(9): 093118 https://doi.org/10.1063/1.5097242
50
Wang L. , Guo Y. , Wang Y. , Fan H. , G. Wang X. . Pinning control of cluster synchronization in regular networks. Phys. Rev. Res., 2020, 2(2): 023084 https://doi.org/10.1103/PhysRevResearch.2.023084
51
Long Y. , Zhai Z. , Tang M. , Liu Y. , C. Lai Y. . Structural position vectors and symmetries in complex networks. Chaos, 2022, 32(9): 093132 https://doi.org/10.1063/5.0107583
52
M. Cardoso D. , Delorme C. , Rama P. . Laplacian eigenvectors and eigenvalues and almost equitable partitions. Eur. J. Combin., 2007, 28(3): 665 https://doi.org/10.1016/j.ejc.2005.03.006
53
A. D. Aguiar M. , P. S. Dias A. , Golubitsky M. , C. A. Leite M. . Bifurcations from regular quotient networks: A first insight. Physica D, 2009, 238(2): 137 https://doi.org/10.1016/j.physd.2008.10.006
54
O’Clery N. , Yuan Y. , B. Stan G. , Barahona M. . Observability and coarse graining of consensus dynamics through the external equitable partition. Phys. Rev. E, 2013, 88(4): 042805 https://doi.org/10.1103/PhysRevE.88.042805
55
Irving D. , Sorrentino F. . Synchronization of dynamical hypernetworks: Dimensionality reduction through simultaneous block-diagonalization of matrices. Phys. Rev. E, 2012, 86(5): 056102 https://doi.org/10.1103/PhysRevE.86.056102
56
Zhang Y. , E. Motter A. . Symmetry-independent stability analysis of synchronization patterns. SIAM Rev., 2020, 86: 056102
57
Zhang Y. , E. Motter A. . Unified treatment of synchronization patterns in generalized networks with higher-order, multilayer, and temporal interactions. Commun. Phys., 2021, 4(1): 195 https://doi.org/10.1038/s42005-021-00695-0
58
Panahi S. , Amaya N. , Klickstein I. , Novello G. , Sorrentino F. . Failure of the simultaneous block diagonalization technique applied to complete and cluster synchronization of random networks. Phys. Rev. E, 2022, 105(1): 014313 https://doi.org/10.1103/PhysRevE.105.014313
59
Golubitsky M.Stewart I.G. Schaeffer D., Singularities and Groups in Bifurcation Theory, Springer-Verlag, 1985
60
Zhou C. , Zemanová L. , Zamora G. , C. Hilgetag C. , Kurths J. . Hierarchical organization unveiled by functional connectivity in complex brain networks. Phys. Rev. Lett., 2006, 97(23): 238103 https://doi.org/10.1103/PhysRevLett.97.238103
61
Zhou C. , Zemanová L. , Zamora-López G. , C. Hilgetag C. , Kurths J. . Structure-function relationship in complex brain networks expressed by hierarchical synchronization. New J. Phys., 2007, 9(6): 178 https://doi.org/10.1088/1367-2630/9/6/178
62
Wang R. , Lin P. , Liu M. , Wu Y. , Zhou T. , Zhou C. . Hierarchical connectome modes and critical state jointly maximize human brain functional diversity. Phys. Rev. Lett., 2019, 123(3): 038301 https://doi.org/10.1103/PhysRevLett.123.038301
63
Huo S. , Tian C. , Zheng M. , Guan S. , Zhou C. , Liu Z. . Spatial multi-scaled chimera states of cerebral cortex network and its inherent structure-dynamics relationship in human brain. Natl. Sci. Rev., 2020, 8(1): nwaa125 https://doi.org/10.1093/nsr/nwaa125
Huang L. , Park K. , C. Lai Y. , Yang L. , Yang K. . Abnormal synchronization in complex clustered networks. Phys. Rev. Lett., 2006, 97(16): 164101 https://doi.org/10.1103/PhysRevLett.97.164101
66
G. Wang X. , Huang L. , C. Lai Y. , H. Lai C. . Optimization of synchronization in gradient clustered networks. Phys. Rev. E, 2007, 76(5): 056113 https://doi.org/10.1103/PhysRevE.76.056113
L. Hindmarsh J. , M. Rose R. . A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. B, 1984, 221(1222): 87 https://doi.org/10.1098/rspb.1984.0024
G. Wang X. , C. Lai Y. , H. Lai C. . Enhancing synchronization based on complex gradient networks. Phys. Rev. E, 2007, 75(5): 056205 https://doi.org/10.1103/PhysRevE.75.056205
71
W. Scannell J. , A. P. C. Burns G. , C. Hilgetag C. , A. O’Neil M. , P. Young M. . The connectional organization of the cortico-thalamic system of the cat. Cereb. Cortex, 1999, 9(3): 277 https://doi.org/10.1093/cercor/9.3.277
72
Hagmann P. , Cammoun L. , Gigandet X. , Meuli R. , J. Honey C. , J. Wedeen V. , Sporns O. . Mapping the structural core of human cerebral cortex. PLoS Biol., 2008, 6(7): e157 https://doi.org/10.1371/journal.pbio.0060159
73
J. Honey C. , Sporns O. , Cammoun L. , Gigandet X. , P. Thiran J. , Meuli R. , Hagmann P. . Predicting human resting-state functional connectivity from structural connectivity. Proc. Natl. Acad. Sci. USA, 2009, 106(6): 2035 https://doi.org/10.1073/pnas.0811168106
Poel W. , Zakharova A. , Schöll E. . Partial synchronization and partial amplitude death in mesoscale network motifs. Phys. Rev. E, 2015, 91(2): 022915 https://doi.org/10.1103/PhysRevE.91.022915
76
Khanra P. , Ghosh S. , Alfaro-Bittner K. , Kundu P. , Boccaletti S. , Hens C. , Pal P. . Identifying symmetries and predicting cluster synchronization in complex networks. Chaos Solitons Fractals, 2022, 155: 111703 https://doi.org/10.1016/j.chaos.2021.111703
77
B. Denton F. , J. Parke S. , Tao T. , Zhang X. . Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra. Bull. Am. Math. Soc., 2022, 59(1): 31 https://doi.org/10.1090/bull/1722
78
Wang Y. , Zhang D. , Wang L. , Li Q. , Cao H. , G. Wang X. . Cluster synchronization induced by manifold deformation. Chaos, 2022, 32(9): 093139 https://doi.org/10.1063/5.0107866
79
Ma J. . Biophysical neurons, energy, and synapse controllability: A review. J. Zhejiang Univ. – Sci. A, 2023, 24: 109 https://doi.org/10.1631/jzus.A2200469
