两层双向加权星型网络的同步能力分析
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摘要
对多层网络的同步性研究大多集中在无向无权的规则网络上,但是无权网络仅仅反映了不同节点之间的连通关系及网络的拓扑特征,并不能具体地描述节点间相互作用的强弱程度。针对这一问题,在两层无权星型网络的基础上,对网络中节点间的连接边进行加权,构建网络的超拉普拉斯矩阵,利用主稳定性函数法,求取该矩阵的特征值,讨论了两层双向加权星型网络的同步性在不同同步域情况下,网络各参数对同步能力的影响。同步域无界时,网络的同步能力与层间耦合强度成正比,与层内耦合强度、权重和等参数无关,与节点接收信号能力β成反比;同步域有界时,网络的同步能力与层间耦合强度、节点接收信号能力β成正比,与层内耦合强度、权重和成反比。对于双向加权网络而言,无论同步域是否有界,增强层间耦合强度,都能增强网络的同步能力。
The researches on synchronization of multilayer networks are mostly concentrating on the undirected and unweighted networks. However,the unweighted network cannot describe the interaction intensity level between nodes,only to show the connectivity and topological network characteristics. Based on the two-layer unweighted star network,this paper weighted connecting edges of nodes,constructed a super-Laplace matrix network and abstracted matrix eigenvalues by the master stability method. It analyzed the impact of network parameters on the synchronizability under the two-layer bidirectional weighted star network in different synchronization regions,and then it simulated the real network preferably. In particular,when the synchronization region was unbounded,the synchronizability of the network is not only directly proportional to the inter-layer coupling strength,but also inversely proportional to the signal reception capability of the node. Whereas,it was irrelevant with the intralayer coupling strength and the weighted sum. On the other hand,when the synchronization region was bounded,the synchronizability of the network was directly proportional to the inter-layer coupling strength and the signal reception capability of the nodes,while inversely proportional to the intra-layer coupling strength and the weighted sum. For undirected weighted networks,the increasing of the inter-layer coupling strength can enhance the synchronizability of the network both in bounded and unbounded synchronous regions.
The researches on synchronization of multilayer networks are mostly concentrating on the undirected and unweighted networks. However,the unweighted network cannot describe the interaction intensity level between nodes,only to show the connectivity and topological network characteristics. Based on the two-layer unweighted star network,this paper weighted connecting edges of nodes,constructed a super-Laplace matrix network and abstracted matrix eigenvalues by the master stability method. It analyzed the impact of network parameters on the synchronizability under the two-layer bidirectional weighted star network in different synchronization regions,and then it simulated the real network preferably. In particular,when the synchronization region was unbounded,the synchronizability of the network is not only directly proportional to the inter-layer coupling strength,but also inversely proportional to the signal reception capability of the node. Whereas,it was irrelevant with the intralayer coupling strength and the weighted sum. On the other hand,when the synchronization region was bounded,the synchronizability of the network was directly proportional to the inter-layer coupling strength and the signal reception capability of the nodes,while inversely proportional to the intra-layer coupling strength and the weighted sum. For undirected weighted networks,the increasing of the inter-layer coupling strength can enhance the synchronizability of the network both in bounded and unbounded synchronous regions.
引文
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[2] Duan Zhisheng,Liu Chao,Chen Guanrong,et al. Network synchronizability analysis:the theory of subgraphs and complementary graphs[J]. Physica D:Nonlinear Phenomena,2008,237(7):1006-1012.
[3] Zhou Jin,Lu Jun'an,Lyu Jinhu. Pinning adaptive synchronization of a general complex dynamical network[J]. Automatica,2008,44(4):996-1003.
[4] Chen Guanrong,Duan Zhisheng. Network synchronizability analysis:a graph-theoretic approach[J]. Chaos,2008,18(3):037102.
[5]陆君安.从单层网络到多层网络——结构、动力学和功能[J].现代物理知识,2015,27(4):3-8.
[6] Aguirre J,Sevillaescoboza R,Gutiérrez R,et al. Synchronization of interconnected networks:the role of connector nodes[EB/OL].(2014-06-16). https://doi. org/10. 1103/PhysR evL ett. 112. 248701.
[7] Boccaletti S,Bianconi G,Criado R,et al. The structure and dynamics of multilayer networks[J]. Physics Reports,2014,544(1):1-122.
[8] Lee K M,Kim J Y,Cho W K,et al. Correlated multiplexity induces unusual connectivity in multiplex random networks[J]. New Journal of Physics,2011,14(3):33027-33038.
[9] D'Agostino G,Scala A. Networks of networks:the last frontier of complexity[M]. Berlin:Springer,2014.
[10]Gao Jianxi,Buldyrev S V,Havlin S,et al. Robustness of a network of networks[J]. Physical Review Letters,2010,107(19):3096-3100.
[11]Salehi M,Sharma R,Marzolla M,et al. Spreading processes in multilayer networks[J]. IEEE Trans on Network Science and Engineering,2015,2(2):65-83.
[12]郭世泽,陆哲明.复杂网络基础理论[M].北京:科学出版社,2012.
[13]Liu Jianguo,Zhou Tao,Guo Qiang,et al. Structural effects on synchronizability of scale-free networks[J]. International Journal of Mo-dern Physics C,2008,19(9):1359-1366.
[14]傅晨波.复杂网络同步若干问题研究[D].杭州:浙江大学,2013.
[15] Li Yang,Wu Xiaoqun,Lu Jun'an,et al. Synchronizability of duplex networks[J]. IEEE Trans on Circuits and Systems II:Express Briefs,2016,63(2):206-210.
[16]Kenett D Y,Perc M,Boccaletti S,et al. Networks of networks-an introduction[J]. Chaos,Solitons&Fractals,2015,80(11):1-6.
[17] Li Changpin,Sun Weigang,Kurths J,et al. Synchronization between two coupled complex networks[J]. Physical Review E,2007,76(4):046204.
[18] Wu Xiaoqun,Zheng Weixing,Zhou Jin,et al. Generalized outer synchronization between complex dynamical networks[J]. Chaos:an Interdiscipcinary Journal of Nonlinear Science,2009,19(1):013109.
[19]Song Qiang,Cao Jinde,Liu Fang,et al. Synchronization of complex dynamical networks with nonidentical nodes[J]. Physics Letters A,2010,374(4):544-551.
[20]严洲.复杂网络随机稳定性和同步性研究[D].杭州:浙江大学,2012.
[21]徐明明,陆君安,周进.两层星型网络的特征值谱及同步能力[J].物理学报,2016,65(2):387-399.
[22]Parshani R,Buldyrev S V,Havlin S. Interdependent networks:reducing the coupling strength leads to a change from a first to second order percolation transition[J]. Physical Review Letters,2010,105(4):048701.
[23]Gao Jianxi,Buldyrev S V,Stanley H E,et al. Networks formed from interdependent networks[J]. Nature physics,2011,8(12):40-48.
[24]马腾.图拉普拉斯矩阵特征值优化及其应用[D].天津:天津大学,2013
[25]张晓转.基于超网络的超拉普拉斯矩阵研究[J].山东工业技术,2016(2):264.
[26]Cardillo A,Zanin M,Gómez-Gardenes J,et al. Modeling the multi-layer nature of the european air transport network:resilience and passengers re-scheduling under random failures[J]. European Physical Journal Special Topics,2012,215(1):23-33.
[27] Parshani R,Rozenblat C,Ietri D,et al. Inter-similarity between coupled networks[J]. Europhysics Letters,2010,92(6):68002.
[28]Dickison M,Havlin S,Stanley H E,et al. Epidemics on interconnected networks[J]. Physical Review E,2012,85(6):066109.
[29] Buono C,Braunstein L A. Immunization strategy for epidemic spreading on multilayer networks[J]. Europhysics Letters,2014,109(2):26001.
[30]Chen Hanshuang,He Gang,Huang Feng,et al. Explosive synchronization transitions in complex neural network[J]. Chaos,2012,23(3):033124.