多层单向耦合星形网络的特征值谱及同步能力分析
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摘要
随着复杂网络同步的进一步发展,对复杂网络的研究重点由单层网络转向更加接近实际网络的多层有向网络.本文分别严格推导出三层、多层的单向耦合星形网络的特征值谱,并分析了耦合强度、节点数、层数对网络同步能力的影响,重点分析了层数和层间中心节点之间的耦合强度对多层单向耦合星形网络同步能力的影响,得出了层数对多层网络同步能力的影响至关重要.当同步域无界时,网络的同步能力与耦合强度、层数有关,同步能力随其增大而增强;当同步域有界时,对于叶子节点向中心节点耦合的多层星形网络,当层内耦合强度较弱时,层内耦合强度的增大会使同步能力增强,而层间叶子节点之间的耦合强度、层数的增大反而会使同步能力减弱;当层间中心节点之间的耦合强度较弱时,层间中心节点之间的耦合强度、层数的增大会使同步能力增强,层内耦合强度、层间叶子节点之间的耦合强度的增大反而会使同步能力减弱.对于中心节点向叶子节点耦合的多层星形网络,层间叶子节点之间的耦合强度、层数的增大会使同步能力增强,层内耦合强度、节点数、层间中心节点之间的耦合强度的增大反而会使同步能力减弱.
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.
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.
引文
[1]Tang L,Lu J A,Wu X,LüJ H 2013 Nonlinear Dyn.731081
[2]Wang X F 2002 Int.J.Bifur.Chaos 12 885
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[7]Raquel A B,Borgeholthoefer J,Wang N,Moreno Y,Gonzálezbailón S 2013 Entropy 15 4553
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[10]Luan Z 2008 Phys.Rep.469 93
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[14]Xue M,Yeung E,Rai A,Roy S,Wan Y,Warnick S 2011Complex 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 DU 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 1910
[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 2014Nonlinear 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-Garde?es J,Moreno Y,Arenas A 2007 Phys.Rev.Lett.98 034101
[28]Murase Y,T?r?k J,Jo H H,Kaski K,Kertész J 2014Phys.Rev.E 90 052810
[29]Cardillo A,Zanin M,Gómez-Garde?es 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.USA108 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]LüJ 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 2008Eur.Phys.J.Special Topics 165 119
[36]Lee T H,Ju H P,Wu Z G,Lee S C,Dong H L 2012Nonlinear 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,Gómezgarde?es J,Pérezvicente C J,Moreno Y,Arenas A 2013 Phys.Rev.Lett.110028701
[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65028902]
[46]Aguirre J,Sevillaescoboza R,Gutiérrez R,Papo D,BuldúJ 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 1998Phys.Rev.Lett.80 3956
[49]Sun J,Bollt E M,Nishikawa T 2008 Europhys.Lett.8560011
[50]Xu M,Zhou J,Lu J A,Wu X 2015 Euro.Phys.J.B 88240
[51]Xu W G,Wang L G 2016 Acta Math.Appl.Sin.39 801
[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,Hütt M T,Porto M 2008 Europhys.Lett.8228003
[10]Luan Z 2008 Phys.Rep.469 93
[11]Arenas A,Díaz-Guilera A,Kurths J,Moreno Y,Zhou C2008 Phys.Reports 469 93
[12]Timme M,Wolf F,Geisel T 2004 Phys.Rev.Lett.92074101
[13]Motter A E,Zhou C,Kurths J 2005 Phys.Rev.E 71016116
[14]Xue M,Yeung E,Rai A,Roy S,Wan Y,Warnick S 2011Complex 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 DU 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 1910
[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 2014Nonlinear 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-Garde?es J,Moreno Y,Arenas A 2007 Phys.Rev.Lett.98 034101
[28]Murase Y,T?r?k J,Jo H H,Kaski K,Kertész J 2014Phys.Rev.E 90 052810
[29]Cardillo A,Zanin M,Gómez-Garde?es 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.USA108 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]LüJ 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 2008Eur.Phys.J.Special Topics 165 119
[36]Lee T H,Ju H P,Wu Z G,Lee S C,Dong H L 2012Nonlinear 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,Gómezgarde?es J,Pérezvicente C J,Moreno Y,Arenas A 2013 Phys.Rev.Lett.110028701
[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65028902]
[46]Aguirre J,Sevillaescoboza R,Gutiérrez R,Papo D,BuldúJ 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 1998Phys.Rev.Lett.80 3956
[49]Sun J,Bollt E M,Nishikawa T 2008 Europhys.Lett.8560011
[50]Xu M,Zhou J,Lu J A,Wu X 2015 Euro.Phys.J.B 88240
[51]Xu W G,Wang L G 2016 Acta Math.Appl.Sin.39 801