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From the study of multilayer networks, scientists have found that the properties of the multilayer networks show great difference from those of the traditional complex networks. In this paper, we derive strictly the spectrum of the super-Laplacian matrix and the synchronizability of two-layer star networks by applying the master stabi- lity method. Through mathematical analysis of the eigenvalues of the super-Laplacian matrix, we study how the node number, the inter-layer and the intra-layer coupling strengths influence the synchronizability of a two-layer star net-work. We find that when the synchronous region is unbounded, the synchronizability of a two-layer star network is only related to the intra-layer coupling strength between the leaf nodes or the inter-layer coupling strength of the entire network. If the synchronous region of a two-layer star network is bounded, not only the inter-layer coupling strength of the network and the intra-layer coupling strength between the leaf nodes, but also the intra-layer coupling strength between the hub nodes and the network size have influence on the synchronizability of the networks. Provided that the same inter-layer and intra-layer coupling strengths are concerned, we would further discuss the opti-mal ways of strengthening the synchronizability of a two-layer star network. If the inter-layer and intra-layer coupling strengths are far less than unity, changing the intra-layer coupling strength is the best way to enhance the synchronizability no matter what the synchronous region is. While if the coupling strengths are the same as, less than or more than unity, there will be different scenarios for the network with bounded and unbounded synchronous regions. Besides, we also discuss the synchronizability of the multilayer network with more than two layers. And then, we carry out numerical simulations and theoretical analysis of the two-layer BA scale-free networks coupled with 200 nodes and obtain very similar conclusions to that of the two-layer star networks. Finally, conclusion and discussion are given to summarize the main results and our future research interests.
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Keywords:
- multilayer networks /
- star networks /
- synchronization /
- eigenvalue spectrum
[1] Mucha P J, Richardson T, Macon K, PorterM A, Onnela J P 2010 Science 328 876
[2] D'Agostino G, Scala A 2014 Networks of Networks: The Last Frontier of Complexity (Berlin: Springer International Publishing) pp53-73
[3] Kivel M, Arenas A, Barthelemy M, Gleeson J P, Moreno Y, Porter M A 2014 J. Com. Net. 2 203
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[5] Um J, Minnhagen P, Kim B J 2011 Chaos 21 5712
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[7] Zhang X Y, Boccaletti S, Guan S G, Liu Z H 2015 Phys. Rev. Lett. 114 038701
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[12] Bauch C T, Galvani A P 2013 Science 342 47
[13] Wang W, Tang M, Yang H, Do Y, Lai Y C, Lee G W 2014 Sci. Rep. 4 2154
[14] Granell C, Gmez S, Arenas A 2013 Phys. Rev. Lett. 111 128701
[15] Wang H J, Li Q, D'Agostino G, Havlin S, Stanley H E, Van Mieghem P 2013 Phys. Rev. E 88 022801
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[19] Blekhman I I 1988 Synchronization in Science and Technology(American Society of Mechanical Engineers Press) pp1-255
[20] Buck J 1988 Q. Revs. Biol. 63 265
[21] Walker T J 1969 Science 166 891
[22] Hansel D, Sompolinsky H 1992 Phys. Rev. Lett. 68 718
[23] Peskin C S 1975 Mathematical Aspects of Heart Physiology (New York: New York University) pp1-278
[24] Uhlhaas P J, Singer W 2006 Neuron 52 155
[25] Liu H, Chen J, Lu J A, Cao M 2010 Phys. A 389 1759
[26] Lu W L, Liu B, Chen T P 2010 Chaos 20 013120
[27] Chen L, Lu J A, Tse C K 2009 IEEE Tran. Circuits Syst.-II 56 310
[28] Suykens J A K, Osipov G V 2008 Chaos 18 037101
[29] Arenas A, Daz-Guilera A, Kurths J, Moreno Y, Zhou C 2008 Phys. Rep. 469 93
[30] Wu W, Chen T P 2008 IEEE T. Neur. Net. 19 319
[31] Han X P, Lu J A, Wu X Q 2008 Int. J. Bifurcat. Chaos 18 1539
[32] Zhou J, Lu J A, L J H 2008 Automatica 44 996
[33] Liu Q, Fang J Q, Li Y 2007 Commun. Theor. Phys. 47 752
[34] Zhou J, Lu J A, L J H 2006 IEEE T. Autmat. Contr. 51 652
[35] Lu X B, Wang X F, Fang J Q 2006 Phys. A 371 841
[36] Lu W L, Chen T P, Chen G R 2006 Phys. D 221 118
[37] Nishikawa T, Motter A E, Lai Y C, Hoppensteadt F C 2003 Phys. Rev. Lett. 91 014201
[38] Barahona M, Pecora L M 2002 Phys. Rev. Lett. 89 716
[39] Kocarev L, Parlitz U 1996 Phys. Rev. Lett. 76 1816
[40] The Algebra Group of Teaching and Research Section of Algebra and Gemotry, Mathematics Department, Beijing University, 2003 Advanced Algebra(Third Edition) (Beijing: Higher Education Press) pp43-82 (in Chinese) [ 北京大学数学系几何与代数教研室代数小组 2003 高等代数(第三版) (北京, 高等教育出版社)第43-82页]
[41] Pecora L M, Carroll T L 1998 Phys. Rev. Lett. 80 3956
[42] Pecora L M, Carroll T L, Johnson G, Mar D, Fink K S 2000 Int. J. Bifurcat. Chaos 10 273
[43] Tang L K, Lu J A, L J H, Yu X H 2012 Int. J. Bifurcat. Chaos 22 1250282
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[1] Mucha P J, Richardson T, Macon K, PorterM A, Onnela J P 2010 Science 328 876
[2] D'Agostino G, Scala A 2014 Networks of Networks: The Last Frontier of Complexity (Berlin: Springer International Publishing) pp53-73
[3] Kivel M, Arenas A, Barthelemy M, Gleeson J P, Moreno Y, Porter M A 2014 J. Com. Net. 2 203
[4] Aguirre J, Sevilla-Escoboza R, Gutirrez R, Papo D, Buld J M 2014 Phys. Rev. Lett. 112 248701
[5] Um J, Minnhagen P, Kim B J 2011 Chaos 21 5712
[6] Lu R Q, Yu W W, L J H, Xue A K 2014 IEEE T. Neur. Net. Lear. 25 2110
[7] Zhang X Y, Boccaletti S, Guan S G, Liu Z H 2015 Phys. Rev. Lett. 114 038701
[8] Xu M M, Zhou J, Lu J A, Wu X Q 2015 Eur. Phys. J. B 88 1
[9] Boccaletti S, Bianconi G, Criado R, Del Genio C I, Gmez-Gardees J, Romance M, Sendia-Nadal I, Wang Z, Zanin M 2014 Phys. Rep. 544 1
[10] Gmez S, Daz-Guilera A, Gmez-Gardees J, Prez-Vicente C J, Moreno Y, Arenas A 2013 Phys. Rev. Lett. 110 028701
[11] Sol-Ribalta A, De Domenico M, Kouvaris N E, Daz-Guilera A, Gmez S, Arenas A 2013 Phys. Rev. E 88 032807
[12] Bauch C T, Galvani A P 2013 Science 342 47
[13] Wang W, Tang M, Yang H, Do Y, Lai Y C, Lee G W 2014 Sci. Rep. 4 2154
[14] Granell C, Gmez S, Arenas A 2013 Phys. Rev. Lett. 111 128701
[15] Wang H J, Li Q, D'Agostino G, Havlin S, Stanley H E, Van Mieghem P 2013 Phys. Rev. E 88 022801
[16] Ouyang B, Jin X Y, Xia Y X, Jiang L R, Wu T P 2014 Acta Phys. Sin. 63 218902 (in Chinese) [欧阳博, 金心宇, 夏永祥, 蒋路茸, 吴端坡 2014 物理学报 63 218902]
[17] Peng X Z, Yao H, Du J, Wang Z, Ding C 2015 Acta Phys. Sin. 64 048901 (in Chinese) [彭兴钊, 姚宏, 杜军, 王哲, 丁超 2015 物理学报 64 048901]
[18] Chen S M, L H, Xu Q G, Xu Y F, Lai Q 2015 Acta Phys. Sin. 64 048902 (in Chinese) [陈世明, 吕辉, 徐青刚, 许云飞, 赖强 2015 物理学报 64 048902]
[19] Blekhman I I 1988 Synchronization in Science and Technology(American Society of Mechanical Engineers Press) pp1-255
[20] Buck J 1988 Q. Revs. Biol. 63 265
[21] Walker T J 1969 Science 166 891
[22] Hansel D, Sompolinsky H 1992 Phys. Rev. Lett. 68 718
[23] Peskin C S 1975 Mathematical Aspects of Heart Physiology (New York: New York University) pp1-278
[24] Uhlhaas P J, Singer W 2006 Neuron 52 155
[25] Liu H, Chen J, Lu J A, Cao M 2010 Phys. A 389 1759
[26] Lu W L, Liu B, Chen T P 2010 Chaos 20 013120
[27] Chen L, Lu J A, Tse C K 2009 IEEE Tran. Circuits Syst.-II 56 310
[28] Suykens J A K, Osipov G V 2008 Chaos 18 037101
[29] Arenas A, Daz-Guilera A, Kurths J, Moreno Y, Zhou C 2008 Phys. Rep. 469 93
[30] Wu W, Chen T P 2008 IEEE T. Neur. Net. 19 319
[31] Han X P, Lu J A, Wu X Q 2008 Int. J. Bifurcat. Chaos 18 1539
[32] Zhou J, Lu J A, L J H 2008 Automatica 44 996
[33] Liu Q, Fang J Q, Li Y 2007 Commun. Theor. Phys. 47 752
[34] Zhou J, Lu J A, L J H 2006 IEEE T. Autmat. Contr. 51 652
[35] Lu X B, Wang X F, Fang J Q 2006 Phys. A 371 841
[36] Lu W L, Chen T P, Chen G R 2006 Phys. D 221 118
[37] Nishikawa T, Motter A E, Lai Y C, Hoppensteadt F C 2003 Phys. Rev. Lett. 91 014201
[38] Barahona M, Pecora L M 2002 Phys. Rev. Lett. 89 716
[39] Kocarev L, Parlitz U 1996 Phys. Rev. Lett. 76 1816
[40] The Algebra Group of Teaching and Research Section of Algebra and Gemotry, Mathematics Department, Beijing University, 2003 Advanced Algebra(Third Edition) (Beijing: Higher Education Press) pp43-82 (in Chinese) [ 北京大学数学系几何与代数教研室代数小组 2003 高等代数(第三版) (北京, 高等教育出版社)第43-82页]
[41] Pecora L M, Carroll T L 1998 Phys. Rev. Lett. 80 3956
[42] Pecora L M, Carroll T L, Johnson G, Mar D, Fink K S 2000 Int. J. Bifurcat. Chaos 10 273
[43] Tang L K, Lu J A, L J H, Yu X H 2012 Int. J. Bifurcat. Chaos 22 1250282
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