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复杂网络控制反映了人类对复杂系统的认识深度和改造能力. 最新研究成果基于线性系统控制理论建立了复杂网络可控性的理论架构, 能够发现任意拓扑结构的线性时不变复杂网络中控制全部节点状态的最小驱动节点集, 但是该模型未考虑免疫节点或失效节点对控制信号传播的阻断.在继承该模型优点的前提下, 重新构建了基于传播免疫的复杂网络控制模型.在采用分属于随机免疫和目标免疫两种策略的 4个方法确定免疫节点的情况下,分析14个真实网络的可控性.结果表明:如果将网络中度数、 介数和紧密度指标较高的节点作为免疫节点,将极大地提高控制复杂网络的难度. 从而在一定程度上丰富了以往模型的结论.Control of complex network reflects humans' comprehension of complex system and the ability to reform it. Up-to-date research establishes the controllability theory of the complex networks based on linear system control theory. The theory could find a minimal set of driver nodes which controls all nodes' state in a linear time invariant complex network with any topology. However, this theory does not take into account the immune node or failure node which blocks the control signal. While inheriting the advantages of the theory, in the paper we first refine the complex network control model based on propagation immunization. Second we adopt four methods which belong to random immunization strategy and targeted immunization strategy to determine the immune nodes, and analyze the controllability of 14 real networks. The experimental results show that when the nodes which have higher degrees, betweeness or closeness are treated as immune nodes, the control of complex networks will become more difficult.
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Keywords:
- control of complex network /
- propagation immunization /
- immune nodes
[1] Wang X F, Su H S 2008 Advances in Mechanics 751 (in Chinese) [汪小帆, 苏厚胜 2008 力学进展 751]
[2] Liu Y Y, Slotine J J, Baraba/si A L 2011 Nature 473 167
[3] Albert R, Jeong H, Barab'asi AL 2000 Nature 406 378
[4] Kitsak M, Gallos L K, Havlin S, Liljeros F, Muchnik L, Stanley H E, Makse H A 2010 Nature Physics 888
[5] Romualdo P S, Alessandro V 2011 Phys. Rev. Lett. 86 3200
[6] Centola D 2010 Science 329 1194
[7] Wang X F, Li X, Chen G R 2006 Complex network theory and its application (1st Ed.) (Beijing: Tsinghua University Press) p78 (in Chinese) [汪小帆, 李翔, 陈关荣 2006 复杂网络理论及其应用 (第一版) (北京:清华大学出版社) 第78页]
[8] Lin C T 1974 IEEE Transactions on Automatic Control 19 201
[9] Hopcroft J E, Karp R M 1973 SWAT 225
[10] Cho A 2011 Science 332 777
[11] Magnus E 2011 Nature 473 158
[12] Xu D, Li X, Wang X F 2007 Acta Phys. Sin. 56 3 (in Chinese) [许丹, 李翔, 汪小帆 2007 物理学报 56 3]
[13] Wang Y Q, Jiang G P 2011 Acta Phys. Sin. 60 060202 (in Chinese) [王亚奇, 蒋国平 2011 物理学报 60 060202]
[14] Wasserman S, Faust K 1994 Social Network Analysis:Methods and Application (New York: Cambridge University Press) 65-94
[15] Freeman L C 1977 Sociometry 40 35
[16] Costenbader E, Valente T W 2003 Social Network 25 238
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[1] Wang X F, Su H S 2008 Advances in Mechanics 751 (in Chinese) [汪小帆, 苏厚胜 2008 力学进展 751]
[2] Liu Y Y, Slotine J J, Baraba/si A L 2011 Nature 473 167
[3] Albert R, Jeong H, Barab'asi AL 2000 Nature 406 378
[4] Kitsak M, Gallos L K, Havlin S, Liljeros F, Muchnik L, Stanley H E, Makse H A 2010 Nature Physics 888
[5] Romualdo P S, Alessandro V 2011 Phys. Rev. Lett. 86 3200
[6] Centola D 2010 Science 329 1194
[7] Wang X F, Li X, Chen G R 2006 Complex network theory and its application (1st Ed.) (Beijing: Tsinghua University Press) p78 (in Chinese) [汪小帆, 李翔, 陈关荣 2006 复杂网络理论及其应用 (第一版) (北京:清华大学出版社) 第78页]
[8] Lin C T 1974 IEEE Transactions on Automatic Control 19 201
[9] Hopcroft J E, Karp R M 1973 SWAT 225
[10] Cho A 2011 Science 332 777
[11] Magnus E 2011 Nature 473 158
[12] Xu D, Li X, Wang X F 2007 Acta Phys. Sin. 56 3 (in Chinese) [许丹, 李翔, 汪小帆 2007 物理学报 56 3]
[13] Wang Y Q, Jiang G P 2011 Acta Phys. Sin. 60 060202 (in Chinese) [王亚奇, 蒋国平 2011 物理学报 60 060202]
[14] Wasserman S, Faust K 1994 Social Network Analysis:Methods and Application (New York: Cambridge University Press) 65-94
[15] Freeman L C 1977 Sociometry 40 35
[16] Costenbader E, Valente T W 2003 Social Network 25 238
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