复杂网络可控性研究现状综述

侯绿林; 老松杨; 肖延东; 白亮

doi:10.7498/aps.64.188901

摘要

控制复杂系统是人们对复杂系统模型结构及相关动力学进行研究的最终目标, 反映人们对复杂系统的认识能力. 近年来, 通过控制理论和复杂性科学相结合,复杂网络可控性的研究引起了人们的广泛关注. 在过去的几年内, 来自国内外不同领域的研究人员从不同的角度对复杂网络可控性进行了深入的分析研究, 取得了丰硕的成果. 本文重点讨论了复杂网络的结构可控性研究进展, 详细介绍了基于最大匹配方法的复杂网络结构可控性分析框架, 综述了自2011年以来复杂网络可控性的相关研究成果, 具体论述了不同类型的可控性、可控性与网络拓扑结构统计特征的关联、基于可控性的网络及节点度量、控制的鲁棒性和可控性的相关优化方法. 最后, 对网络可控性未来的研究动态进行了展望, 有助于国内同行开展网络可控性的相关研究.

关键词:

Abstract

The model, structure and dynamics of complex systems and networks are studied to control complex systems, which reflects the ability to understanding complex systems. Recently, the research on controllability of complex networks by using control theory and complexity science has attracted much attention. It has been investigated extensively by many scientists from various fields, and many meaningful achievements have been obtained in the past few years. In this paper, the process of controllability of complex networks is discussed, the framework of structural controllability based on maximum matching is introduced in detail, and the relevant research status since 2011 is summarized. Controllabilities of complex networks are introduced in the following aspects: different types of controllabilities, relationship between controllability and network statistical characteristics, classification and measures based on controllability, robustness of controllability, and optimization methods of controllability. Finally, the questions urgent to solve in controllability are discussed, so as to give a help to the the study in this respect.#br#There are five sections in this paper, which involve with different aspects of controllability. In the introduction section, the research work of controllability since 2011 is briefly mentioned, and the difference between controllability and previous pinning controllability is clarified. In the second section, the concept of controllability and different types of controllabilities are discussed in detail, including structural controllability, exact controllability, controllability with edge dynamics and controllability with nodal dynamics. In the third section, the relationship between controllability and network structure is investigated, especially the effects of common statistical characteristics and low-degree nodes on controllability. In the fourth section, the measures based on controllability are introduced, which includes control profiles, control range, control centrality, control capacity and control modality. In the fifth section, the research work about control robustness is discussed from robustness measures to optimization methods. In the fifth section, the optimization methods of controllability are introduced, which are classified into two different strategies: topology and edge direction.

Keywords:

作者及机构信息

1.
国防科技大学, 信息系统工程国家重点实验室, 长沙 410073;

2.
School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia

通信作者: 侯绿林, houlvlin@gmail.com

Authors and contacts

1.
Science and Technology on Information Systems Enginering Laboratory, National University of Defense Technology, Changsha 410073, China;

2.
School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia

Corresponding author: Hou Lü-Lin, houlvlin@gmail.com

参考文献

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[2]	Barabási A L, Albert R 1999 Science 286 509
[3]	Lombardi A, Hörnquist M 2007 Phys. Rev. E 75 056110
[4]	Liu Y Y, Slotine J J, Barabrási A L 2011 Nature 473 167
[5]	Wang X F, Chen G R 2002 Physica A 310 521
[6]	Li X, Wang X F, Chen G R 2004 IEEE Trans. Circ. I 51 2074
[7]	Chen G R 2013 Acta Autom. Sin. 39 4
[8]	Chen G R 2014 Int. J. Control Autom. 12 221
[9]	Wang X F, Su H S 2005 Adv. Mech. 38 6 (in Chinese) [汪小帆, 苏厚胜 2005 力学进展 38 6]
[10]	Chen T P, Liu X W, Lu W L 2007 IEEE Trans. Circ. I 54 1317
[11]	Zhou J, Lu J A, L J H 2008 Automatica 44 996
[12]	Chen G R, Duan Z S 2008 Chaos 18 037102
[13]	Guo W L, Austin F, Chen S H, Sun W 2009 Phys. Lett. A 373 1565
[14]	Lin C T 1974 IEEE Trans. Automat. Control 19 201
[15]	Shields R W, Pearson J B 1976 IEEE Trans. Automat. Control. 21 203
[16]	Reinschke K J, Wiedemann G 1997 Linear Algebra Appl. 266 199
[17]	Sontag E D 1998 Mathematical Control Theory: Deterministic Finite Dimensional Systems (New York: Springer) p12
[18]	Lovász L, Plummer M D 1986 Matching Theory (North Holland: Elsevier Science Publishing Company) p113
[19]	Mulmuley K, Vazirani U V, Vazirani V V 1987 Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing p345
[20]	Umeyama S 1988 IEEE Trans. Pattern Anal. 10 695
[21]	Hopcroft J E, Karp R M 1973 Siam J. Comput. 2 225
[22]	Kalman R E 1963 J. Soc. Indus. Appl. Math. Ser. A 1 152
[23]	Poljak S 1990 IEEE Trans. Automat. Control 35 367
[24]	Yuan Z Z, Zhao C, Di Z R, Wang W X, Lai Y C 2013 Nat. Comm. 4 3447
[25]	Nepusz T, Vicsek T 2012 Nat. Phys. 8 568
[26]	Cowan N J, Chastain E J, Vilhena D A, Freudenberg J S, Bergstrom C T 2012 PloS One 7 e38398
[27]	Pósfai M, Liu Y Y, Slotine J J, Barabási A L 2013 Sci. Rep. 3 1065
[28]	Barrat A, Weigt M 2000 Eur. Phys. J. B 13 547
[29]	Leicht E A, Newman M E J 2008 Phys. Rev. Lett. 100 118703
[30]	Foster J G, Foster D V, Grassberger P, Paczuski M 2010 Proc. Natl. Acad. Sci. USA 107 10815
[31]	Menichetti G, Dall’Asta L, Bianconi G 2014 Phys. Rev. Lett. 113 078701
[32]	Hou L L, Small M, Lao S Y 2014 Phys. Lett. A 378 3426
[33]	Ruths J, Ruths D 2014 Science 343 1373
[34]	Onnela J P 2014 Science 21 1325
[35]	Wang B B, Gao L, Gao Y 2012 J. Stat. Mech. 2012 P04011
[36]	Liu Y Y, Slotine J J, Barabási A L 2012 PloS One 7 e44459
[37]	Jia T, Barabási A L 2013 Sci. Rep. 3 2354
[38]	Jia T, Liu Y Y, Csoka E, Posfai M, Slotine J J, Barabási A L 2013 Nat. Comm. 4 2002
[39]	Liu Y Y, Csoka E, Zhou H J, Posfai M 2012 Phys. Rev. Lett. 109 205703
[40]	Jia T, Posfai M 2014 Sci. Rep. 4 5379
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[47]	Pu C L, Pei W J, Michaelson A 2012 Physica A 391 4420
[48]	Nie S, Wang X, Zhang H, Li Q, Wang B 2014 PLoS One 9 e89066
[49]	Wang B, Gao L, Gao Y, Deng Y 2013 EPL 101 58003
[50]	Xiao Y D, Lao S Y, Hou L L, Bai L 2014 Chin. Phys. B 23 118902
[51]	Xiao Y D, Lao S Y, Hou L L, Bai L 2013 Acta Phys. Sin. 62 180201(in Chinese) [肖延东, 老松杨, 侯绿林, 白亮 2013 62 180201]
[52]	L T Y, Piao X F, Xie W Y, Huang S B 2012 Acta Phys. Sin. 61 170512(in Chinese) [吕天阳, 朴秀峰, 谢文艳, 黄少滨 2012 61 170512]
[53]	Ruths J, Ruths D 2013 Complex Networks IV (New York: Springer) 476 185
[54]	Wang W X, Ni X, Lai Y C, Grebogi C 2012 Phys. Rev. E 85 026115
[55]	Hou L L, Lao S Y, Bu J, Bai L 2013 International Conference on Intelligent System Design and Engineering Applications 709
[56]	Hou L L, Lao S Y, Liu G, Bai L 2012 Chin. Phys. Lett. 29 108901
[57]	Xiao Y D, Lao SY, Hou L L, Bai L 2014 Phys. Rev. E 90 042804
[58]	Ding J, Lu Y Z, Chu J 2013 Physica A 392 6603
[59]	Yan G, Ren J, Lai Y C, Lai C H, Li B W 2012 Phys. Rev. Lett. 108 218703
[60]	Delpini D, Battiston S, Riccaboni M, Gabbi G, Pammolli F, Caldarelli G 2013 Sci. Rep. 3 1626
[61]	Wuchty S 2014 Proc. Natl. Acad. Sci. USA 111 7156

施引文献

[1]	Watts D J, Strogatz S H 1998 Nature 393 440
[2]	Barabási A L, Albert R 1999 Science 286 509
[3]	Lombardi A, Hörnquist M 2007 Phys. Rev. E 75 056110
[4]	Liu Y Y, Slotine J J, Barabrási A L 2011 Nature 473 167
[5]	Wang X F, Chen G R 2002 Physica A 310 521
[6]	Li X, Wang X F, Chen G R 2004 IEEE Trans. Circ. I 51 2074
[7]	Chen G R 2013 Acta Autom. Sin. 39 4
[8]	Chen G R 2014 Int. J. Control Autom. 12 221
[9]	Wang X F, Su H S 2005 Adv. Mech. 38 6 (in Chinese) [汪小帆, 苏厚胜 2005 力学进展 38 6]
[10]	Chen T P, Liu X W, Lu W L 2007 IEEE Trans. Circ. I 54 1317
[11]	Zhou J, Lu J A, L J H 2008 Automatica 44 996
[12]	Chen G R, Duan Z S 2008 Chaos 18 037102
[13]	Guo W L, Austin F, Chen S H, Sun W 2009 Phys. Lett. A 373 1565
[14]	Lin C T 1974 IEEE Trans. Automat. Control 19 201
[15]	Shields R W, Pearson J B 1976 IEEE Trans. Automat. Control. 21 203
[16]	Reinschke K J, Wiedemann G 1997 Linear Algebra Appl. 266 199
[17]	Sontag E D 1998 Mathematical Control Theory: Deterministic Finite Dimensional Systems (New York: Springer) p12
[18]	Lovász L, Plummer M D 1986 Matching Theory (North Holland: Elsevier Science Publishing Company) p113
[19]	Mulmuley K, Vazirani U V, Vazirani V V 1987 Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing p345
[20]	Umeyama S 1988 IEEE Trans. Pattern Anal. 10 695
[21]	Hopcroft J E, Karp R M 1973 Siam J. Comput. 2 225
[22]	Kalman R E 1963 J. Soc. Indus. Appl. Math. Ser. A 1 152
[23]	Poljak S 1990 IEEE Trans. Automat. Control 35 367
[24]	Yuan Z Z, Zhao C, Di Z R, Wang W X, Lai Y C 2013 Nat. Comm. 4 3447
[25]	Nepusz T, Vicsek T 2012 Nat. Phys. 8 568
[26]	Cowan N J, Chastain E J, Vilhena D A, Freudenberg J S, Bergstrom C T 2012 PloS One 7 e38398
[27]	Pósfai M, Liu Y Y, Slotine J J, Barabási A L 2013 Sci. Rep. 3 1065
[28]	Barrat A, Weigt M 2000 Eur. Phys. J. B 13 547
[29]	Leicht E A, Newman M E J 2008 Phys. Rev. Lett. 100 118703
[30]	Foster J G, Foster D V, Grassberger P, Paczuski M 2010 Proc. Natl. Acad. Sci. USA 107 10815
[31]	Menichetti G, Dall’Asta L, Bianconi G 2014 Phys. Rev. Lett. 113 078701
[32]	Hou L L, Small M, Lao S Y 2014 Phys. Lett. A 378 3426
[33]	Ruths J, Ruths D 2014 Science 343 1373
[34]	Onnela J P 2014 Science 21 1325
[35]	Wang B B, Gao L, Gao Y 2012 J. Stat. Mech. 2012 P04011
[36]	Liu Y Y, Slotine J J, Barabási A L 2012 PloS One 7 e44459
[37]	Jia T, Barabási A L 2013 Sci. Rep. 3 2354
[38]	Jia T, Liu Y Y, Csoka E, Posfai M, Slotine J J, Barabási A L 2013 Nat. Comm. 4 2002
[39]	Liu Y Y, Csoka E, Zhou H J, Posfai M 2012 Phys. Rev. Lett. 109 205703
[40]	Jia T, Posfai M 2014 Sci. Rep. 4 5379
[41]	Albert R, Jeong H, Barabási A L 2000 Nature 406 378
[42]	Buldyrev S V, Parshani R, Paul G, Stanley H E 2010 Nature 464 1025
[43]	Holme P, Kim B J, Yoon C N, Han S K 2002 Phys. Rev. E 65 056109
[44]	Callaway D S, Newman M E J, Strogatz S H, Watts D J 2000 Phys. Rev. Lett. 85 5468
[45]	Schwarte N, Cohen R, Ben-Avraham D, Barabási A L, Havlin S 2002 Phys. Rev. E 66 015104
[46]	Shargel B, Sayama H, Epstein I R, Bar-Yam Y 2003 Phys. Rev. Lett. 90 068701
[47]	Pu C L, Pei W J, Michaelson A 2012 Physica A 391 4420
[48]	Nie S, Wang X, Zhang H, Li Q, Wang B 2014 PLoS One 9 e89066
[49]	Wang B, Gao L, Gao Y, Deng Y 2013 EPL 101 58003
[50]	Xiao Y D, Lao S Y, Hou L L, Bai L 2014 Chin. Phys. B 23 118902
[51]	Xiao Y D, Lao S Y, Hou L L, Bai L 2013 Acta Phys. Sin. 62 180201(in Chinese) [肖延东, 老松杨, 侯绿林, 白亮 2013 62 180201]
[52]	L T Y, Piao X F, Xie W Y, Huang S B 2012 Acta Phys. Sin. 61 170512(in Chinese) [吕天阳, 朴秀峰, 谢文艳, 黄少滨 2012 61 170512]
[53]	Ruths J, Ruths D 2013 Complex Networks IV (New York: Springer) 476 185
[54]	Wang W X, Ni X, Lai Y C, Grebogi C 2012 Phys. Rev. E 85 026115
[55]	Hou L L, Lao S Y, Bu J, Bai L 2013 International Conference on Intelligent System Design and Engineering Applications 709
[56]	Hou L L, Lao S Y, Liu G, Bai L 2012 Chin. Phys. Lett. 29 108901
[57]	Xiao Y D, Lao SY, Hou L L, Bai L 2014 Phys. Rev. E 90 042804
[58]	Ding J, Lu Y Z, Chu J 2013 Physica A 392 6603
[59]	Yan G, Ren J, Lai Y C, Lai C H, Li B W 2012 Phys. Rev. Lett. 108 218703
[60]	Delpini D, Battiston S, Riccaboni M, Gabbi G, Pammolli F, Caldarelli G 2013 Sci. Rep. 3 1626
[61]	Wuchty S 2014 Proc. Natl. Acad. Sci. USA 111 7156

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