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Synchronizability and eigenvalues of multilayer star networks through unidirectionally coupling

Sun Juan Li Xiao-Xia Zhang Jin-Hao Shen Yu-Zhuo Li Yan-Yu

Synchronizability and eigenvalues of multilayer star networks through unidirectionally coupling

Sun Juan, Li Xiao-Xia, Zhang Jin-Hao, Shen Yu-Zhuo, Li Yan-Yu
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  • Previous studies on multilayer networks have found that properties of multilayer networks show great differences from those of the traditional complex networks. In this paper, we derive strictly the spectra of the Supra-Laplace matrix of three-layer star networks and multilayer star networks through unidirectionally coupling by using the master stability method to analyze the synchronizability of these two networks. Through mathematical analyses of the eigenvalues of the Supra-Laplace matrix, we explore how the node number, the intra-layer coupling strength the inter-layer coupling strength, and the layer number influence the synchronizability of multilayer star networks through unidirectionally coupling in two different ways. In particular, we focus on the layer number and the inter-layer coupling strength between the hub nodes, and then we conclude that the synchronizability of networks is greatly affected by the layer number. We find that when the synchronous region is unbounded, the synchronizability of the two different coupling multilayer star networks is related to not only the intra-layer coupling strength or the inter-layer coupling strength between the leaf nodes of the entire network, but also the layer number. If the synchronous region of two different coupling multilayer star networks is bounded, and the intra-layer coupling strength is weak, the synchronizability of the two different coupling multilayer star networks is different with the changing of the intra-layer coupling strength and the inter-layer coupling strength between the leaf nodes and the layer number. If the synchronous region of two different coupling multilayer star networks is bounded, and the inter-layer coupling strength between the hub nodes is weak, the two different coupling multilayer star networks are consistent with the changing of the intra-layer coupling strength and the layer number while different from the inter-layer coupling strength between the leaf nodes and the inter-layer coupling strength between the hub nodes. We find that the node number has no effect on the synchronizability of multilayer star networks through coupling from the hub node to the leaf node. The synchronizability of the network is directly proportional to the layer number, while inversely proportional to the inter-layer coupling strength between the hub nodes. Finally, the effects of the coupling strength, the layer number and the node number on the synchronizability of the two different coupling star networks can be extended from three-layer network to multilayer networks.
      Corresponding author: Li Xiao-Xia, lixiaoxia@hebut.edu.cn
    • Funds: Project supported by the Natural Science Foundation of Hebei Province, China (Grant No. E2011202051).
    [1]

    Tang L, Lu J A, Wu X, L J H 2013 Nonlinear Dyn. 73 1081

    [2]

    Wang X F 2002 Int. J. Bifur. Chaos 12 885

    [3]

    Barrat A, Barthélemy M, Vespignani A 2004 Phys. Rev. Lett. 92 228701

    [4]

    Wang W X, Wang B H, Hu B, Yan G, Ou Q 2005 Phys. Rev. Lett. 94 188702

    [5]

    Ide Y, Izuhara H, Machida T 2016 Physics 457 331

    [6]

    Dahan M, Levi S, Luccardini C, Rostaing P, Riveau B, Triller A 2003 Science 302 442

    [7]

    Raquel A B, Borgeholthoefer J, Wang N, Moreno Y, Gonzálezbailón S 2013 Entropy 15 4553

    [8]

    Boukobza E, Chuchem M, Cohen D, Vardi A 2009 Phys. Rev. Lett. 102 180403

    [9]

    Weber S, Htt M T, Porto M 2008 Europhys. Lett. 82 28003

    [10]

    Luan Z 2008 Phys. Rep. 469 93

    [11]

    Arenas A, Díaz-Guilera A, Kurths J, Moreno Y, Zhou C 2008 Phys. Reports 469 93

    [12]

    Timme M, Wolf F, Geisel T 2004 Phys. Rev. Lett. 92 074101

    [13]

    Motter A E, Zhou C, Kurths J 2005 Phys. Rev. E 71 016116

    [14]

    Xue M, Yeung E, Rai A, Roy S, Wan Y, Warnick S 2011 Complex Syst. 21 297

    [15]

    Cai S, Zhou P, Liu Z 2014 Nonlinear Dyn. 76 1677

    [16]

    Chen Y, Yu W, Tan S, Zhu H 2016 Automatica 70 189

    [17]

    Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang D U 2006 Phys. Rep. 424 175

    [18]

    Massah S, Hollebakken R, Labrecque M P, Kolybaba A M, Beischlag T V, Prefontaine G G 2004 Phys. Rev. Lett. 93 218701

    [19]

    Song Q, Cao J, Liu F 2010 Phys. Lett. A 374 544

    [20]

    He P, Jing C G, Fan T, Chen C Z 2014 Complexity 19 10

    [21]

    Arenas A, Díazguilera A, Pérezvicente C J 2006 Phys. Rev. Lett. 96 114102

    [22]

    Zhang Q, Zhao J 2012 Nonlinear Dyn. 67 2519

    [23]

    Zhang Q, Luo J, Wan L 2013 Nonlinear Dyn. 71 353

    [24]

    Ling L, Li C, Wang W, Sun Y, Wang Y, Sun A 2014 Nonlinear Dyn. 77 1

    [25]

    Pacheco J M, Traulsen A, Nowak M A 2006 Phys. Rev. Lett. 97 258103

    [26]

    Zhang J, Small M 2006 Phys. Rev. Lett. 96 238701

    [27]

    Gómez-Gardeñes J, Moreno Y, Arenas A 2007 Phys. Rev. Lett. 98 034101

    [28]

    Murase Y, Török J, Jo H H, Kaski K, Kertész J 2014 Phys. Rev. E 90 052810

    [29]

    Cardillo A, Zanin M, Gómez-Gardeóes J, Romance M, Amo A J G D, Boccaletti S 2013 Eur. Phys. J. Special Topics 215 23

    [30]

    Bassett D S, Wymbs N F, Porter M A, Mucha P J, Carlson J M, Grafton S T 2011 Proc. Nat. Acad. Sci. USA 108 7641

    [31]

    Li Y, Wu X, Lu J, Lu J 2015 IEEE Trans. Circ. Syst. Ⅱ Express Briefs 63 206

    [32]

    Kivel M, Arenas A, Barthelemy M, Gleeson J P, Moreno Y, Porter M A 2014 J. Complex Netw. 2 203

    [33]

    L J H 2008 Adv. Mech. 38 713(in Chinese)[吕金虎2008力学进展 38 713]

    [34]

    Lu J A 2010 Complex Syst. Complex Sci. 7 19(in Chinese)[陆君安2010复杂系统与复杂性科学 7 19]

    [35]

    Baptista M S, Garcia S P, Dana S K, Kurths J 2008 Eur. Phys. J. Special Topics 165 119

    [36]

    Lee T H, Ju H P, Wu Z G, Lee S C, Dong H L 2012 Nonlinear Dyn. 70 559

    [37]

    Qin J, Yu H J 2007 Acta Phys. Sin. 56 6828(in Chinese)[秦洁, 于洪洁2007物理学报 56 6828]

    [38]

    Wang J, Zhang Y 2010 Phys. Lett. A 374 1464

    [39]

    Zhao M, Wang B H, Jiang P Q, Zhou T 2005 Prog. Phys. 25 273(in Chinese)[赵明, 汪秉宏, 蒋品群, 周涛2005物理学进展 25 273]

    [40]

    Gómez S, Díazguilera A, Gómezgardeñes J, Pérezvicente C J, Moreno Y, Arenas A 2013 Phys. Rev. Lett. 110 028701

    [41]

    Almendral J A, Díazguilera A 2007 New J. Phys. 9 187

    [42]

    Granell C, Gómez S, Arenas A 2013 Phys. Rev. Lett. 111 128701

    [43]

    Pecora L M, Carroll T L 1998 Phys. Rev. Lett. 80 2109

    [44]

    Liang Y, Wang X Y 2012 Acta Phys. Sin. 61 038901(in Chinese)[梁义, 王兴元2012物理学报 61 038901]

    [45]

    Xu M M, Lu J A, Zhou J 2016 Acta Phys. Sin. 65 028902(in Chinese)[徐明明, 陆君安, 周进2016物理学报 65 028902]

    [46]

    Aguirre J, Sevillaescoboza R, Gutiérrez R, Papo D, Buldú J M 2014 Phys. Rev. Lett. 112 248701

    [47]

    Dabrowski A 2012 Nonlinear Dyn. 69 1225

    [48]

    Johnson G A, Mar D J, Carroll T L, Pecora L M 1998 Phys. Rev. Lett. 80 3956

    [49]

    Sun J, Bollt E M, Nishikawa T 2008 Europhys. Lett. 85 60011

    [50]

    Xu M, Zhou J, Lu J A, Wu X 2015 Euro. Phys. J. B 88 240

    [51]

    Xu W G, Wang L G 2016 Acta Math. Appl. Sin. 39 801

  • [1]

    Tang L, Lu J A, Wu X, L J H 2013 Nonlinear Dyn. 73 1081

    [2]

    Wang X F 2002 Int. J. Bifur. Chaos 12 885

    [3]

    Barrat A, Barthélemy M, Vespignani A 2004 Phys. Rev. Lett. 92 228701

    [4]

    Wang W X, Wang B H, Hu B, Yan G, Ou Q 2005 Phys. Rev. Lett. 94 188702

    [5]

    Ide Y, Izuhara H, Machida T 2016 Physics 457 331

    [6]

    Dahan M, Levi S, Luccardini C, Rostaing P, Riveau B, Triller A 2003 Science 302 442

    [7]

    Raquel A B, Borgeholthoefer J, Wang N, Moreno Y, Gonzálezbailón S 2013 Entropy 15 4553

    [8]

    Boukobza E, Chuchem M, Cohen D, Vardi A 2009 Phys. Rev. Lett. 102 180403

    [9]

    Weber S, Htt M T, Porto M 2008 Europhys. Lett. 82 28003

    [10]

    Luan Z 2008 Phys. Rep. 469 93

    [11]

    Arenas A, Díaz-Guilera A, Kurths J, Moreno Y, Zhou C 2008 Phys. Reports 469 93

    [12]

    Timme M, Wolf F, Geisel T 2004 Phys. Rev. Lett. 92 074101

    [13]

    Motter A E, Zhou C, Kurths J 2005 Phys. Rev. E 71 016116

    [14]

    Xue M, Yeung E, Rai A, Roy S, Wan Y, Warnick S 2011 Complex Syst. 21 297

    [15]

    Cai S, Zhou P, Liu Z 2014 Nonlinear Dyn. 76 1677

    [16]

    Chen Y, Yu W, Tan S, Zhu H 2016 Automatica 70 189

    [17]

    Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang D U 2006 Phys. Rep. 424 175

    [18]

    Massah S, Hollebakken R, Labrecque M P, Kolybaba A M, Beischlag T V, Prefontaine G G 2004 Phys. Rev. Lett. 93 218701

    [19]

    Song Q, Cao J, Liu F 2010 Phys. Lett. A 374 544

    [20]

    He P, Jing C G, Fan T, Chen C Z 2014 Complexity 19 10

    [21]

    Arenas A, Díazguilera A, Pérezvicente C J 2006 Phys. Rev. Lett. 96 114102

    [22]

    Zhang Q, Zhao J 2012 Nonlinear Dyn. 67 2519

    [23]

    Zhang Q, Luo J, Wan L 2013 Nonlinear Dyn. 71 353

    [24]

    Ling L, Li C, Wang W, Sun Y, Wang Y, Sun A 2014 Nonlinear Dyn. 77 1

    [25]

    Pacheco J M, Traulsen A, Nowak M A 2006 Phys. Rev. Lett. 97 258103

    [26]

    Zhang J, Small M 2006 Phys. Rev. Lett. 96 238701

    [27]

    Gómez-Gardeñes J, Moreno Y, Arenas A 2007 Phys. Rev. Lett. 98 034101

    [28]

    Murase Y, Török J, Jo H H, Kaski K, Kertész J 2014 Phys. Rev. E 90 052810

    [29]

    Cardillo A, Zanin M, Gómez-Gardeóes J, Romance M, Amo A J G D, Boccaletti S 2013 Eur. Phys. J. Special Topics 215 23

    [30]

    Bassett D S, Wymbs N F, Porter M A, Mucha P J, Carlson J M, Grafton S T 2011 Proc. Nat. Acad. Sci. USA 108 7641

    [31]

    Li Y, Wu X, Lu J, Lu J 2015 IEEE Trans. Circ. Syst. Ⅱ Express Briefs 63 206

    [32]

    Kivel M, Arenas A, Barthelemy M, Gleeson J P, Moreno Y, Porter M A 2014 J. Complex Netw. 2 203

    [33]

    L J H 2008 Adv. Mech. 38 713(in Chinese)[吕金虎2008力学进展 38 713]

    [34]

    Lu J A 2010 Complex Syst. Complex Sci. 7 19(in Chinese)[陆君安2010复杂系统与复杂性科学 7 19]

    [35]

    Baptista M S, Garcia S P, Dana S K, Kurths J 2008 Eur. Phys. J. Special Topics 165 119

    [36]

    Lee T H, Ju H P, Wu Z G, Lee S C, Dong H L 2012 Nonlinear Dyn. 70 559

    [37]

    Qin J, Yu H J 2007 Acta Phys. Sin. 56 6828(in Chinese)[秦洁, 于洪洁2007物理学报 56 6828]

    [38]

    Wang J, Zhang Y 2010 Phys. Lett. A 374 1464

    [39]

    Zhao M, Wang B H, Jiang P Q, Zhou T 2005 Prog. Phys. 25 273(in Chinese)[赵明, 汪秉宏, 蒋品群, 周涛2005物理学进展 25 273]

    [40]

    Gómez S, Díazguilera A, Gómezgardeñes J, Pérezvicente C J, Moreno Y, Arenas A 2013 Phys. Rev. Lett. 110 028701

    [41]

    Almendral J A, Díazguilera A 2007 New J. Phys. 9 187

    [42]

    Granell C, Gómez S, Arenas A 2013 Phys. Rev. Lett. 111 128701

    [43]

    Pecora L M, Carroll T L 1998 Phys. Rev. Lett. 80 2109

    [44]

    Liang Y, Wang X Y 2012 Acta Phys. Sin. 61 038901(in Chinese)[梁义, 王兴元2012物理学报 61 038901]

    [45]

    Xu M M, Lu J A, Zhou J 2016 Acta Phys. Sin. 65 028902(in Chinese)[徐明明, 陆君安, 周进2016物理学报 65 028902]

    [46]

    Aguirre J, Sevillaescoboza R, Gutiérrez R, Papo D, Buldú J M 2014 Phys. Rev. Lett. 112 248701

    [47]

    Dabrowski A 2012 Nonlinear Dyn. 69 1225

    [48]

    Johnson G A, Mar D J, Carroll T L, Pecora L M 1998 Phys. Rev. Lett. 80 3956

    [49]

    Sun J, Bollt E M, Nishikawa T 2008 Europhys. Lett. 85 60011

    [50]

    Xu M, Zhou J, Lu J A, Wu X 2015 Euro. Phys. J. B 88 240

    [51]

    Xu W G, Wang L G 2016 Acta Math. Appl. Sin. 39 801

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  • Received Date:  01 April 2017
  • Accepted Date:  16 May 2017
  • Published Online:  05 September 2017

Synchronizability and eigenvalues of multilayer star networks through unidirectionally coupling

    Corresponding author: Li Xiao-Xia, lixiaoxia@hebut.edu.cn
  • 1. State Key Laboratory of Reliability and Intelligence of Electrical Equipment, Hebei University of Technology, Tianjin 300130, China;
  • 2. Key Laboratory of Electromagnetic Field and Electrical Apparatus Reliability of Hebei Province, Hebei University of Technology, Tianjin 300130, China
Fund Project:  Project supported by the Natural Science Foundation of Hebei Province, China (Grant No. E2011202051).

Abstract: Previous studies on multilayer networks have found that properties of multilayer networks show great differences from those of the traditional complex networks. In this paper, we derive strictly the spectra of the Supra-Laplace matrix of three-layer star networks and multilayer star networks through unidirectionally coupling by using the master stability method to analyze the synchronizability of these two networks. Through mathematical analyses of the eigenvalues of the Supra-Laplace matrix, we explore how the node number, the intra-layer coupling strength the inter-layer coupling strength, and the layer number influence the synchronizability of multilayer star networks through unidirectionally coupling in two different ways. In particular, we focus on the layer number and the inter-layer coupling strength between the hub nodes, and then we conclude that the synchronizability of networks is greatly affected by the layer number. We find that when the synchronous region is unbounded, the synchronizability of the two different coupling multilayer star networks is related to not only the intra-layer coupling strength or the inter-layer coupling strength between the leaf nodes of the entire network, but also the layer number. If the synchronous region of two different coupling multilayer star networks is bounded, and the intra-layer coupling strength is weak, the synchronizability of the two different coupling multilayer star networks is different with the changing of the intra-layer coupling strength and the inter-layer coupling strength between the leaf nodes and the layer number. If the synchronous region of two different coupling multilayer star networks is bounded, and the inter-layer coupling strength between the hub nodes is weak, the two different coupling multilayer star networks are consistent with the changing of the intra-layer coupling strength and the layer number while different from the inter-layer coupling strength between the leaf nodes and the inter-layer coupling strength between the hub nodes. We find that the node number has no effect on the synchronizability of multilayer star networks through coupling from the hub node to the leaf node. The synchronizability of the network is directly proportional to the layer number, while inversely proportional to the inter-layer coupling strength between the hub nodes. Finally, the effects of the coupling strength, the layer number and the node number on the synchronizability of the two different coupling star networks can be extended from three-layer network to multilayer networks.

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