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摘要: 研究复杂网络能控性鲁棒性对包括社会网络、生物和技术网络等在内的复杂系统的控制和应用具有重要价值. 复杂网络的能控性是指: 可通过若干控制节点和适当的输入, 在有限时间内将系统状态驱动至任意目标状态. 能控性鲁棒性则是指在受到攻击的情况下, 复杂网络依然维持能控性的能力. 设计具有优异能控性鲁棒性的复杂网络模型和优化实际网络的能控性鲁棒性一直是复杂网络领域的重要研究内容. 本文首先比较了常用的能控性鲁棒性定义及度量, 接着从攻击策略的角度分析了3类攻击的特点及效果, 包括随机攻击、基于特征的蓄意攻击和启发式攻击. 然后比较了常见模型网络的能控性鲁棒性. 介绍了常用优化策略, 包括模型设计和重新连边等. 目前的研究在攻击策略和拓扑结构优化方面都取得了进展, 也为进一步理论分析提供条件. 最后总结全文并提出潜在研究方向.Abstract: The study of controllability robustness is valuable to the control and application of various complex systems, including social, biological, and technological networks. The concept of controllability of complex networks refers to the ability of a network being steered by external inputs from any of its initial state to any desired target state under an admissible control input within a finite duration of time. The controllability robustness reflects how well the system can maintain the controllability against malicious attacks by means of node removals or edge removals. This survey gives a systematic investigation in the recent progress of the controllability robustness of complex networks. Firstly, the definitions and measures of controllability robustness are introduced. Then, the controllability robustness is considered from the perspective of attacks. Three types of attack strategies are discussed, including random attacks, feature-based targeted attacks, and heuristic-based attacks. Optimization methods toward stronger controllability robustness are investigated, including network modeling, edge rewiring, etc. Recent progresses have been achieved in both effective attack strategies and efficient topological optimizations, which provide a basis for further theoretical analysis. Finally, some potential future works are suggested.
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Key words:
- Complex network /
- controllability robustness /
- attack /
- optimization
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图 4 能控性鲁棒性度量方式比较举例
Fig. 4 Comparison of two different controllability robustness measurements
图 7 常见的网络模型在攻击下的能控性曲线变化
Fig. 7 The controllability curves of 9 network topologies under 4 different attack strategies
图 8 所有
$ N $ 节点和$ M $ 连边网络, 满足ENC的网络、全齐网络, 以及最优网络之间的关系图Fig. 8 The relationship diagram of the N-node M-edge networks, ENC networks, totally homogeneous networks, and the optimal networks
表 1 常用能控性鲁棒性优化策略的优点与不足
Table 1 Pros and cons of the strategies for controllability robustness optimization
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表 2 能控性鲁棒性优化的网络结构
Table 2 Comparison of network topologies with optimized controllability robustness
网络结构 优化设计 重新连边 全齐网络 模体 链 环 MCN √ √ QSN √ √ √ QSNR √ √ √ √ RTN √ √ √ RRN √ √ √ EH √ √ √ √ √
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$ A $ 邻接矩阵 $ {A}_{ij} $ 邻接矩阵中节点$ i $和$ j $之间的连边 $ {b}_{i} $ 节点$ i $的介数 $ B $ 输入矩阵 $ C $ 能控性矩阵 $ {\cal{E}} $ 网络连边集合 $ G $ 复杂网络 $ {k}_{i} $ 节点$ i $的度数 $ {k}_{i}^{{\rm{out}}} $ 节点$ i $的出度 $ {k}_{i}^{{\rm{in}}} $ 节点$ i $的入度 $ M $ 网络连边数 $ {M}_{c} $ 关键连边数 $ {M}_{r} $ 当前已攻击的连边数 $ N $ 网络节点数 $ {N}_{D} $ 网络所需控制节点数 $ \Delta {N}_{D} $ 网络所需控制节点数增量 $ {N}_{r} $ 当前已攻击的节点数 $ {n}_{D} $ 网络所需控制节点密度 $ p\left({M}_{c}\right) $ 关键连边比例 $ {R}_{c} $ 平均能控性鲁棒性 $ {R}_{{\rm{LCC}}} $ 平均连通鲁棒性 $ {\boldsymbol{u}} $ 控制向量 $ {\cal{V}} $ 网络节点集合 $ {\boldsymbol{x}} $ 状态向量 $ {\sigma }_{ij} $ 从节点$ i $到$ j $的最短路径 $ \Omega $ 适用解空间
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[1] Barabási A L. Network Science. Cambridge: Cambridge University Press, 2016. [2] Newman M E J. Networks: An Introduction. Oxford: Oxford University Press, 2010. [3] Chen G R, Wang X F, Li X. Fundamentals of Complex Networks: Models, Structures and Dynamics. John Wiley & Sons, 2014. [4] 汪小帆, 李翔, 陈关荣. 复杂网络理论及其应用. 清华大学出版社, 2006.Wang Xiao-Fan, Li Xiang, Chen Guan-Rong. Complex Network Theory and Its Applications. Beijing: Tsinghua University Press, 2006. [5] Chen D X, Shao Q, Liu Z Y, Yu W W, Chen C L P. Ridesourcing behavior analysis and prediction: A network perspective. IEEE Transactions on Intelligent Transportation Systems, 2022, 23(2): 1274-1283 doi: 10.1109/TITS.2020.3023951 [6] Chen G R, Lou Y. Naming Game: Models, Simulations and Analysis. Cham: Springer, 2019. [7] Ding Y. Scientific collaboration and endorsement: Network analysis of coauthorship and citation networks. Journal of Informetrics, 2011, 5(1): 187-203 doi: 10.1016/j.joi.2010.10.008 [8] Davis K F, D’’’Odorico P, Laio F, Ridolfi L. Global spatio-temporal patterns in human migration: A complex network perspective. PLoS One, 2013, 8(1): Article No. e53723 doi: 10.1371/journal.pone.0053723 [9] 熊熙, 乔少杰, 吴涛, 吴越, 韩楠, 张海清. 基于时空特征的社交网络情绪传播分析与预测模型. 自动化学报, 2018, 44(12): 2290-2299).Xiong Xi, Qiao Shao-Jie, Wu Tao, Wu Yue, Han Nan, Zhang Hai-Qing. Spatio-temporal feature based emotional contagion analysis and prediction model for online social networks. Acta Automatica Sinica, 2018, 44(12): 2290-2299 [10] Lou Y, Chen G R. Analysis of the "naming game" with learning errors in communications. Scientific Reports, 2015, 5: Article No. 12191 doi: 10.1038/srep12191 [11] Lou Y, Chen G R, Hu J W. Communicating with sentences: A multi-word naming game model. Physica A: Statistical Mechanics and its Applications, 2018, 490: 857-868 doi: 10.1016/j.physa.2017.08.066 [12] Lou Y, Chen G R, Fan Z P, Xiang L N. Local communities obstruct global consensus: Naming game on multi-local-world networks. Physica A: Statistical Mechanics and its Applications, 2018, 492: 1741-1752 doi: 10.1016/j.physa.2017.11.094 [13] 黄春林, 刘兴武, 邓明华, 周杨, 卜东波. 复杂网络上疾病传播溯源算法综述. 计算机学报, 2018, 41(6): 1376-1399 doi: 10.11897/SP.J.1016.2018.01376Huang Chun-Lin, Liu Xing-Wu, Deng Ming-Hua, Zhou Yang, Bu Dong-Bo. A survey on algorithms for epidemic source identification on complex networks. Chinese Journal of Computers, 2018, 41(6): 1376-1399 doi: 10.11897/SP.J.1016.2018.01376 [14] Kabir K M A, Kuga K, Tanimoto J. Analysis of SIR epidemic model with information spreading of awareness. Chaos, Solitons & Fractals, 2019, 119: 118-125 [15] Firth J A, Hellewell J, Klepac P, Kissler S, CMMID COVID-19 Working Group, Kucharski A J, Spurgin L G. Using a real-world network to model localized COVID-19 control strategies. Nature Medicine 2020, 26(10): 1616-1622 doi: 10.1038/s41591-020-1036-8 [16] Zhang Y F, Wu G, Liu X L, Yu W W, Chen D X. Maximum Markovian order detection for collective behavior. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2020, 30(8): Article No. 083121 doi: 10.1063/5.0008397 [17] Chen D X, Liu X L, Xu B W, Zhang H T. Intermittence and connectivity of interactions in pigeon flock flights. Scientific Reports, 2017, 7(1): Article No. 10452 doi: 10.1038/s41598-017-09986-5 [18] Liu Y Y, Slotine J J, Barabási A L. Controllability of complex networks. Nature, 2011, 473(7346): 167-173 doi: 10.1038/nature10011 [19] Wang W X, Ni X, Lai Y C, Grebogi C. Optimizing controllability of complex networks by minimum structural perturbations. Physical Review E, 2012, 85(2): Article No. 026115 doi: 10.1103/PhysRevE.85.026115 [20] Yuan Z Z, Zhao C, Di Z R, Wang W X, Lai Y C. Exact controllability of complex networks. Nature Communications, 2013, 4: Article No. 2447 doi: 10.1038/ncomms3447 [21] Pósfai M, Liu Y Y, Slotine J J, Barabási A L. Effect of correlations on network controllability. Scientific Reports, 2013, 3: Article No. 1067 doi: 10.1038/srep01067 [22] Menichetti G, Dall’’’Asta L, Bianconi G. Network controllability is determined by the density of low in-degree and out-degree nodes. Physical Review Letters, 2014, 113(7): Article No. 078701 doi: 10.1103/PhysRevLett.113.078701 [23] 周涛, 张子柯, 陈关荣, 汪小帆, 史定华, 狄增如, 樊瑛, 方锦清, 韩筱璞, 刘建国, 刘润然, 刘宗华, 陆君安, 吕金虎, 吕琳媛, 荣智海, 汪秉宏, 许小可, 章忠志. 复杂网络研究的机遇与挑战. 电子科技大学学报, 2014, 43(1): 1-5 doi: 10.3969/j.issn.1001-0548.2014.01.001Zhou Tao, Zhang Zi-Ke, Chen Guan-Rong, Wang Xiao-Fan, Shi Ding-Hua, Di Zeng-Ru, Fan Ying, Fang Jin-Qing, Han Xiao-Pu, Liu Jian-Guo, Liu Run-Ran, Liu Zong-Hua, Lu Jun-An, Lü Jin-Hu, Lü Lin-Yuan, Rong Zhi-Hai, Wang Bing-Hong, Xu Xiao-Ke, Zhang Zhong-Zhi. The opportunities and challenges of complex networks research. Journal of University of Electronic Science and Technology of China, 2014, 43(1): 1-5 doi: 10.3969/j.issn.1001-0548.2014.01.001 [24] 侯绿林, 老松杨, 肖延东, 白亮. 复杂网络可控性研究现状综述. 物理学报, 2015, 64(18): Article No. 188901). doi: 10.7498/aps.64.188901Hou Lü-Lin, Lao Song-Yang, Xiao Yan-Dong, Bai Liang. Recent progress in controllability of complex network. Acta Physica Sinica, 2015, 64(18): Article No. 188901 doi: 10.7498/aps.64.188901 [25] 聂森. 复杂网络可控性研究 [博士学位论文], 中国科学技术大学, 中国, 2015.Nie Sen. Research on Controllability of Complex Networks [Ph.D. dissertation], University of Science and Technology of China, China, 2015. [26] Motter A E. Networkcontrology. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2015, 25(9): Article No. 097621 doi: 10.1063/1.4931570 [27] Wang L, Chen G R, Wang X F, Tang W K S. Controllability of networked MIMO systems. Automatica, 2016, 69: 405-409 doi: 10.1016/j.automatica.2016.03.013 [28] Liu Y Y, Barabási A L. Control principles of complex systems. Reviews of Modern Physics, 2016, 88(3): Article No. 035006 doi: 10.1103/RevModPhys.88.035006 [29] Wang L, Wang X F, Chen G R. Controllability of networked higher-dimensional systems with one-dimensional communication. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2017, 375(2088): Article No. 20160215 doi: 10.1098/rsta.2016.0215 [30] Wang L Z, Chen Y Z, Wang W X, Lai Y C. Physical controllability of complex networks. Scientific Reports, 2017, 7: Article No. 40198 doi: 10.1038/srep40198 [31] Zhang Y, Zhou T. Controllability analysis for a networked dynamic system with autonomous subsystems. IEEE Transactions on Automatic Control, 2017, 62(7): 3408-3415 doi: 10.1109/TAC.2016.2612831 [32] Xiang L Y, Chen F, Ren W, Chen G R. Advances in network controllability. IEEE Circuits and Systems Magazine, 2019, 19(2): 8-32 doi: 10.1109/MCAS.2019.2909446 [33] 段广仁. 高阶系统方法-Ⅱ. 能控性与全驱性. 自动化学报, 2020, 46(8): 1571-1581Duan Guang-Ren. High-order system approaches: Ⅱ. Controllability and full-actuation. Acta Automatica Sinica, 2020, 46(8): 1571-1581 [34] Chen G R. Pinning control and synchronization on complex dynamical networks. International Journal of Control, Automation and Systems, 2014, 12(2): 221-230 doi: 10.1007/s12555-014-9001-2 [35] Holme P, Kim B J, Yoon C N, Han S K. Attack vulnerability of complex networks. Physical Review E, 2002, 65(5): Article No. 056109 doi: 10.1103/PhysRevE.65.056109 [36] Shargel B, Sayama H, Epstein I R, Bar-Yam Y. Optimization of robustness and connectivity in complex networks. Physical Review Letters, 2003, 90(6): Article No. 068701 doi: 10.1103/PhysRevLett.90.068701 [37] Schneider C M, Moreira A A, Andrade J S Jr, Havlin S, Herrmann H J. Mitigation of malicious attacks on networks. Proceedings of the National Academy of Sciences of the United States of America, 2011, 108(10): 3838-3841 doi: 10.1073/pnas.1009440108 [38] Liu Y Y, Slotine J J, Barabási A L. Control centrality and hierarchical structure in complex networks. PLoS One, 2012, 7(9): Article No. e44459 doi: 10.1371/journal.pone.0044459 [39] Bashan A, Berezin Y, Buldyrev S V, Havlin S. The extreme vulnerability of interdependent spatially embedded networks. Nature Physics, 2013, 9(10): 667-672 doi: 10.1038/nphys2727 [40] Xiao Y D, Lao S Y, Hou L L, Bai L. Optimization of robustness of network controllability against malicious attacks. Chinese Physics B, 2014, 23(11): Article No. 118902 doi: 10.1088/1674-1056/23/11/118902 [41] 李文锋, 符修文. 无线传感器网络抗毁性. 计算机学报, 2015, 38(3): 625-647Li Wen-Feng, Fu Xiu-Wen. Survey on invulnerability of wireless sensor networks. Chinese Journal of Computers, 2015, 38(3): 625-647 [42] 董政呈, 方彦军, 田猛. 相互依存网络抗毁性研究综述. 复杂系统与复杂性科学, 2017, 14(3): 30-44Dong Zheng-Cheng, Fang Yan-Jun, Tian Meng. Review on invulnerability of interdependent networks. Complex Systems and Complexity Science, 2017, 14(3): 30-44 [43] 王尔申, 王玉伟, 庞涛, 曲萍萍, 姜毅. 基于边攻击成本的复杂网络鲁棒性研究. 电子学报, 2018, 46(5): 1166-1172 doi: 10.3969/j.issn.0372-2112.2018.05.022Wang Er-Shen, Wang Yu-Wei, Pang Tao, Qu Ping-Ping, Jiang Yi. Research on robustness of complex networks with edge’’’s attack cost. Acta Electronica Sinica, 2018, 46(5): 1166-1172 doi: 10.3969/j.issn.0372-2112.2018.05.022 [44] 赵志刚, 周根贵, 杜辉. 复杂加权供应链网络级联抗毁性研究. 小型微型计算机系统, 2019, 40(12): 2591-2596 doi: 10.3969/j.issn.1000-1220.2019.12.022Zhao Zhi-Gang, Zhou Gen-Gui, Du Hui. Research on cascading invulnerability of complex weighted supply chain networks. Journal of Chinese Computer Systems, 2019, 40(12): 2591-2596 doi: 10.3969/j.issn.1000-1220.2019.12.022 [45] 王哲, 李建华, 康东, 冉淏丹. 复杂网络鲁棒性增强策略研究综述. 复杂系统与复杂性科学, 2020, 17(3): 1-26, 46Wang Zhe, Li Jian-Hua, Kang Dong, Ran Hao-Dan. Review on strategies enhancing the robustness of complex network. Complex Systems and Complexity Science, 2020, 17(3): 1-26, 46 [46] Yan G, Vértes P E, Towlson E K, Chew Y L, Walker D S, Schafer W R, Barabási A L. Network control principles predict neuron function in the Caenorhabditis elegans connectome. Nature, 2017, 550(7677): 519-523 doi: 10.1038/nature24056 [47] Liu J, Zhou M X, Wang S, Liu P H. A comparative study of network robustness measures. Frontiers of Computer Science, 2017, 11(4): 568-584 doi: 10.1007/s11704-016-6108-z [48] 谭跃进, 吴俊, 邓宏钟. 复杂网络抗毁性研究进展. 上海理工大学学报, 2011, 33(6): 653-668 doi: 10.3969/j.issn.1007-6735.2011.06.022Tan Yue-Jin, Wu Jun, Deng Hong-Zhong. Progress in invulnerability of complex networks. Journal of University of Shanghai for Science and Technology, 2011, 33(6): 653-668 doi: 10.3969/j.issn.1007-6735.2011.06.022 [49] Yamashita K, Yasuda Y, Nakamura R, Ohsaki H. On the predictability of network robustness from spectral measures. In: Proceedings of the 43rd IEEE Annual Computer Software and Applications Conference (COMPSAC). Milwaukee, USA: IEEE, 2019. 24−29 [50] Chen G R, Lou Y, Wang L. A comparative study on controllability robustness of complex networks. IEEE Transactions on Circuits and Systems II: Express Briefs, 2019, 66(5): 828-832 doi: 10.1109/TCSII.2019.2908435 [51] Li X Y, Zhang Z J, Liu J M, Gai K K. A new complex network robustness attack algorithm. In: Proceedings of the 2019 ACM International Symposium on Blockchain and Secure Critical Infrastructure. Auckland, New Zealand: Association for Computing Machinery, 2019. 13−17 [52] Lou Y, He Y D, Wang L, Chen G R. Predicting network controllability robustness: A convolutional neural network approach. IEEE Transactions on Cybernetics, 2022, 52(5): 4052−4063 [53] Fan C J, Zeng L, Sun Y Z, Liu Y Y. Finding key players in complex networks through deep reinforcement learning. Nature Machine Intelligence, 2020, 2(6): 317-324 doi: 10.1038/s42256-020-0177-2 [54] Kalman R E. On the general theory of control systems. In: Proceedings of the 1st International Conference on Automatic Control. Moscow, 1960. 481−492 [55] Ding J, Lu Y Z. Control backbone: An index for quantifying a node’’’s importance for the network controllability. Neurocomputing, 2015, 153: 309-318 doi: 10.1016/j.neucom.2014.11.024 [56] Hosoe S. Determination of generic dimensions of controllable subspaces and its application. IEEE Transactions on Automatic Control, 1980, 25(6): 1192-1196 doi: 10.1109/TAC.1980.1102506 [57] Lou Y, Wang L, Chen G R. A framework of hierarchical attacks to network controllability. Communications in Nonlinear Science and Numerical Simulation, 2021, 98: Article No. 105780 doi: 10.1016/j.cnsns.2021.105780 [58] Usman U, Mahmood A, Wang L. Robust control centrality. In: Proceedings of the 2019 Chinese Control Conference (CCC). Guangzhou, China: IEEE, 2019. 5486−5491 [59] Annamalai C. Finding perfect matchings in bipartite hypergraphs. In: Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms. Arlington, USA: Society for Industrial and Applied Mathematics, 2016. 1814−1823 [60] Williams V V. Multiplying matrices faster than coppersmith-winograd. In: Proceedings of the 44th Annual ACM Symposium on Theory of Computing. New York, USA: Association for Computing Machinery, 2012. 887−898 [61] Sun P, Kooij R E, He Z D, Van Mieghem P. Quantifying the robustness of network controllability. In: Proceedings of the 4th International Conference on System Reliability and Safety (ICSRS). Rome, Italy: IEEE, 2019. 66−76 [62] 林景栋, 吴欣怡, 柴毅, 尹宏鹏. 卷积神经网络结构优化综述. 自动化学报, 2020, 46(1): 24-37Lin Jing-Dong, Wu Xin-Yi, Chai Yi, Yin Hong-Peng. Structure optimization of convolutional neural networks: A survey. Acta Automatica Sinica, 2020, 46(1): 24-37 [63] Lou Y, He Y D, Wang L, Tsang K F, Chen G R. Knowledge-based prediction of network controllability robustness. IEEE Transactions on Neural Networks and Learning Systems, 2021, DOI: 10.1109/TNNLS.2021.3071367"> 10.1109/TNNLS.2021.3071367 [64] Lou Y, He Y D, Wang L, Tsang K F, Chen G R. Predicting the robustness of undirected network controllability. In: Proceedings of the 39th Chinese Control Conference (CCC). Shenyang, China: IEEE, 2020. 4550−4553 [65] Chan H, Akoglu L. Optimizing network robustness by edge rewiring: A general framework. Data Mining and Knowledge Discovery, 2016, 30(5): 1395-1425 doi: 10.1007/s10618-015-0447-5 [66] Yan G, Martinez N D, Liu Y Y. Degree heterogeneity and stability of ecological networks. Journal of the Royal Society Interface, 2017, 14(131): Article No. 20170189 doi: 10.1098/rsif.2017.0189 [67] Wang S, Liu J, Jin Y C. Surrogate-assisted robust optimization of large-scale networks based on graph embedding. IEEE Transactions on Evolutionary Computation, 2020, 24(4): 735-749 doi: 10.1109/TEVC.2019.2950935 [68] 阮逸润, 老松杨, 王竣德, 白亮, 陈立栋. 基于领域相似度的复杂网络节点重要度评估算法. 物理学报, 2017, 66(3): Article No. 038902 doi: 10.7498/aps.66.038902Ruan Yi-Run, Lao Song-Yang, Wang Jun-De, Bai Liang, Chen Li-Dong. Node importance measurement based on neighborhood similarity in complex network. Acta Physica Sinica, 2017, 66(3): Article No. 038902 doi: 10.7498/aps.66.038902 [69] Šimon M, Luptáková D I, Huraj L, Hos?ovecký M, Pospíchal J. Combined heuristic attack strategy on complex networks. Mathematical Problems in Engineering, 2017, 2017: Article No. 6108563 [70] Yang H H, An S. Critical nodes identification in complex networks. Symmetry, 2020, 12(1): Article No. 123 doi: 10.3390/sym12010123 [71] Wu-Yan E, Betzel R F, Tang E, Gu S, Pasqualetti F, Bassett D S. Benchmarking measures of network controllability on canonical graph models. Journal of Nonlinear Science, 2020, 30(5): 2195-2233 doi: 10.1007/s00332-018-9448-z [72] Zhang R, Wang X M, Cheng M, Jia T. The evolution of network controllability in growing networks. Physica A: Statistical Mechanics and its Applications, 2019, 520: 257-266 doi: 10.1016/j.physa.2019.01.042 [73] Hao Y Q, Duan Z S, Chen G R. Further on the controllability of networked MIMO LTI systems. International Journal of Robust and Nonlinear Control, 2018, 28(5): 1778-1788 doi: 10.1002/rnc.3986 [74] Bröhl T, Lehnertz K. Centrality-based identification of important edges in complex networks. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2019, 29(3): Article No. 033115 doi: 10.1063/1.5081098 [75] Thomas J, Ghosh S, Parek D, Ruths D, Ruths J. Robustness of network controllability to degree-based edge attacks. In: Proceedings of the 5th International Workshop on Complex Networks and their Applications. Milan, Italy: Springer, 2017. 525−537 [76] Nie T Y, Guo Z, Zhao K, Lu Z M. New attack strategies for complex networks. Physica A: Statistical Mechanics and its Applications, 2015, 424: 248-253 doi: 10.1016/j.physa.2015.01.004 [77] Nguyen Q, Pham H D, Cassi D, Bellingeri M. Conditional attack strategy for real-world complex networks. Physica A: Statistical Mechanics and its Applications, 2019, 530: Article No. 121561 doi: 10.1016/j.physa.2019.121561 [78] Gao Y L, Chen S M, Nie S, Ma F, Guan J J. Robustness analysis of interdependent networks under multiple-attacking strategies. Physica A: Statistical Mechanics and its Applications, 2018, 496: 495-504 doi: 10.1016/j.physa.2017.12.085 [79] Hao Y C, Jia L M, Wang Y H. Edge attack strategies in interdependent scale-free networks. Physica A: Statistical Mechanics and its Applications, 2020, 540: Article No. 122759 doi: 10.1016/j.physa.2019.122759 [80] Huang X Q, Gao J X, Buldyrev S V, Havlin S, Stanley H E. Robustness of interdependent networks under targeted attack. Physical Review E, 2011, 83(6): Article No. 065101 doi: 10.1103/PhysRevE.83.065101 [81] Dong G G, Gao J X, Tian L X, Du R J, He Y H. Percolation of partially interdependent networks under targeted attack. Physical Review E, 2012, 85(1): Article No. 016112 doi: 10.1103/PhysRevE.85.016112 [82] Cui P S, Zhu P D, Wang K, Xun P, Xia Z Q. Enhancing robustness of interdependent network by adding connectivity and dependence links. Physica A: Statistical Mechanics and its Applications, 2018, 497: 185-197 doi: 10.1016/j.physa.2017.12.142 [83] Dong G G, Gao J X, Du R J, Tian L X, Stanley H E, Havlin S. Robustness of network of networks under targeted attack. Physical Review E, 2013, 87(5): Article No. 052804 doi: 10.1103/PhysRevE.87.052804 [84] Liu X M, Peng H, Gao J X. Vulnerability and controllability of networks of networks. Chaos, Solitons & Fractals, 2015, 80: 125-138 [85] Bellingeri M, Cassi D. Robustness of weighted networks. Physica A: Statistical Mechanics and its Applications, 2018, 489: 47-55 doi: 10.1016/j.physa.2017.07.020 [86] Pu C L, Pei W J, Michaelson A. Robustness analysis of network controllability. Physica A: Statistical Mechanics and its Applications, 2012, 391(18): 4420-4425 doi: 10.1016/j.physa.2012.04.019 [87] Lu Z M, Li X F. Attack vulnerability of network controllability. PLoS One, 2016, 11(9): Article No. e0162289 doi: 10.1371/journal.pone.0162289 [88] Li X, Chen G R. A local-world evolving network model. Physica A: Statistical Mechanics and its Applications, 2003, 328(1-2): 274-286 doi: 10.1016/S0378-4371(03)00604-6 [89] Fan Z P, Chen G R, Zhang Y N. A comprehensive multi-local-world model for complex networks. Physics Letters A, 2009, 373(18-19): 1601-1605 doi: 10.1016/j.physleta.2009.02.072 [90] Sun S W, Ma Y L, Wu Y F, Wang L, Xia C Y. Towards structural controllability of local-world networks. Physics Letters A, 2016, 380(22-23): 1912-1917 doi: 10.1016/j.physleta.2016.03.048 [91] Nie S, Wang X W, Zhang H F, Li Q L, Wang B H. Robustness of controllability for networks based on edge-attack. PLoS One, 2014, 9(2): Article No. e89066 doi: 10.1371/journal.pone.0089066 [92] Wang H, Huang J Y, Xu X M, Xiao Y H. Damage attack on complex networks. Physica A: Statistical Mechanics and its Applications, 2014, 408: 134-148 doi: 10.1016/j.physa.2014.04.001 [93] da Cunha B R, González-Avella J C, Gonçalves S. Fast fragmentation of networks using module-based attacks. PLoS One, 2015, 10(11): Article No. e0142824 doi: 10.1371/journal.pone.0142824 [94] Shai S, Kenett D Y, Kenett Y N, Faust M, Dobson S, Havlin S. Critical tipping point distinguishing two types of transitions in modular network structures. Physical Review E, 2015, 92(6): Article No. 062805 doi: 10.1103/PhysRevE.92.062805 [95] Ma L L, Liu J, Duan B P. Evolution of network robustness under continuous topological changes. Physica A: Statistical Mechanics and its Applications, 2016, 451: 623-631 doi: 10.1016/j.physa.2016.01.088 [96] Zhang X K, Wu J, Wang H, Xiong J, Yang K W. Optimization of disintegration strategy for multi-edges complex networks. In: Proceedings of the 2016 IEEE Congress on Evolutionary Computation (CEC). Vancouver, Canada: IEEE, 2016. 522−528 [97] 孙昱, 姚佩阳, 张杰勇, 付凯. 基于优化理论的复杂网络节点攻击策略. 电子与信息学报, 2017, 39(3): 518-524Sun Yu, Yao Pei-Yang, Zhang Jie-Yong, Fu Kai. Node attack strategy of complex networks based on optimization theory. Journal of Electronics & Information Technology, 2017, 39(3): 518-524 [98] Ventresca M. Global search algorithms using a combinatorial unranking-based problem representation for the critical node detection problem. Computers & Operations Research, 2012, 39(11): 2763-2775 [99] Deng Y, Wu J, Tan Y J. Optimal attack strategy of complex networks based on tabu search. Physica A: Statistical Mechanics and its Applications, 2016, 442: 74-81 doi: 10.1016/j.physa.2015.08.043 [100] Qi M Z, Deng Y, Deng H Z, Wu J. Optimal disintegration strategy in multiplex networks. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2018, 28(12): Article No. 121104 doi: 10.1063/1.5078449 [101] Lozano M, García-Martínez C, Rodríguez F J, Trujillo H M. Optimizing network attacks by artificial bee colony. Information Sciences, 2017, 377: 30-50 doi: 10.1016/j.ins.2016.10.014 [102] Li Q, Liu S Y, Yang X S. Neighborhood information-based probabilistic algorithm for network disintegration. Expert Systems With Applications, 2020, 139: Article No. 112853 doi: 10.1016/j.eswa.2019.112853 [103] Erdös P, Rényi A. On the evolution of random graphs. Mathematical Institute of the Hungarian Academy of Sciences, 1960, 5: 17-61 [104] Newman M E J, Watts D J. Renormalization group analysis of the small-world network model. Physics Letters A, 1999, 263(4-6): 341-346 doi: 10.1016/S0375-9601(99)00757-4 [105] Watts D J, Strogatz S H. Collective dynamics of `small-world' networks. Nature, 1998, 393(6684): 440-442 doi: 10.1038/30918 [106] Yang D, Liu M Y, Zhang Y C, Lin D, Fan Z P, Chen G R. Henneberg growth of social networks: Modeling the Facebook. IEEE Transactions on Network Science and Engineering, 2020, 7(2): 701-712 doi: 10.1109/TNSE.2018.2856280 [107] Goh K I, Kahng B, Kim D. Universal behavior of load distribution in scale-free networks. Physical Review Letters, 2001, 87(27): Article No. 278701 doi: 10.1103/PhysRevLett.87.278701 [108] Sorrentino F. Effects of the network structural properties on its controllability. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2007, 17(3): Article No. 033101 doi: 10.1063/1.2743098 [109] Herrmann H J, Schneider C M, Moreira A A, Andrade J S Jr, Havlin S. Onion-like network topology enhances robustness against malicious attacks. Journal of Statistical Mechanics: Theory and Experiment, 2011, 2011: Article No. P01027 [110] Wu Z X, Holme P. Onion structure and network robustness. Physical Review E, 2011, 84(2): Article No. 026106 doi: 10.1103/PhysRevE.84.026106 [111] Tanizawa T, Havlin S, Stanley H E. Robustness of onionlike correlated networks against targeted attacks. Physical Review E, 2012, 85(4): Article No. 046109 doi: 10.1103/PhysRevE.85.046109 [112] Hayashi Y, Uchiyama N. Onion-like networks are both robust and resilient. Scientific Reports, 2018, 8(1): Article No. 11241 doi: 10.1038/s41598-018-29626-w [113] Lou Y, Wang L, Chen G R. Toward stronger robustness of network controllability: A snapback network model. IEEE Transactions on Circuits and Systems I: Regular Papers, 2018, 65(9): 2983-2991 doi: 10.1109/TCSI.2018.2821124 [114] Lou Y, Wang L, Chen G R. Local diversity-stability of the q-snapback network model. Physica A: Statistical Mechanics and its Applications, 2019, 536: Article No. 121020 doi: 10.1016/j.physa.2019.04.256 [115] Lou Y, Wang L, Chen G R. Enhancing controllability robustness of q-snapback networks through redirecting edges. Research, 2019, 2019: Article No. 7857534 [116] Lou Y, Wang L, Tsang K F, Chen G R. Towards optimal robustness of network controllability: An empirical necessary condition. IEEE Transactions on Circuits and Systems I: Regular Papers, 2020, 67(9): 3163-3174 doi: 10.1109/TCSI.2020.2986215 [117] Eiben A E, Smith J. From evolutionary computation to the evolution of things. Nature, 2015, 521(7553): 476-482 doi: 10.1038/nature14544 [118] 丁青锋, 尹晓宇. 差分进化算法综述. 智能系统学报, 2017, 12(4): 431-442Ding Qing-Feng, Yin Xiao-Yu. Research survey of differential evolution algorithms. CAAI Transactions on Intelligent Systems, 2017, 12(4): 431-442 [119] Lou Y, Xie S L, Chen G R. Searching better rewiring strategies and objective functions for stronger controllability robustness. IEEE Transactions on Circuits and Systems II: Express Briefs, 2021, 68(6): 2112-2116 doi: 10.1109/TCSII.2020.3047538 [120] 林诗洁, 董晨, 陈明志, 张凡, 陈景辉. 新型群智能优化算法综述. 计算机工程与应用, 2018, 54(12): 1-9 doi: 10.3778/j.issn.1002-8331.1803-0260Lin Shi-Jie, Dong Chen, Chen Ming-Zhi, Zhang Fan, Chen Jing-Hui. Summary of new group intelligent optimization algorithms. Computer Engineering and Applications, 2018, 54(12): 1-9 doi: 10.3778/j.issn.1002-8331.1803-0260 [121] Lou Y, Yuen S Y, Chen G R. Non-revisiting stochastic search revisited: Results, perspectives, and future directions. Swarm and Evolutionary Computation, 2021, 61: Article No. 100828. doi: 10.1016/j.swevo.2020.100828 [122] Yan X Y, Wang W X, Chen G R, Shi D H. Multiplex congruence network of natural numbers. Scientific Reports, 2016, 6: Article No. 23714 doi: 10.1038/srep23714 [123] Bai L, Xiao Y D, Hou L L, Lao S Y. Smart rewiring: Improving network robustness faster. Chinese Physics Letters, 2015, 32(7): Article No. 078901 doi: 10.1088/0256-307X/32/7/078901 [124] Wu J, Barahona M, Tan Y J, Deng H Z. Natural connectivity of complex networks. Chinese Physics Letters, 2010, 27(7): Article No. 078902 doi: 10.1088/0256-307X/27/7/078902 [125] Chakrabarti D, Wang Y, Wang C X, Leskovec J, Faloutsos C. Epidemic thresholds in real networks. ACM Transactions on Information and System Security, 2008, 10(4): Article No. 1 [126] Estrada E, Hatano N, Benzi M. The physics of communicability in complex networks. Physics Reports, 2012, 514(3): 89-119 doi: 10.1016/j.physrep.2012.01.006 [127] Fiedler M. Algebraic connectivity of graphs. Czechoslovak Mathematical Journal, 1973, 23(2): 298-305 doi: 10.21136/CMJ.1973.101168 [128] Ghosh A, Boyd S, Saberi A. Minimizing effective resistance of a graph. SIAM Review, 2008, 50(1): 37-66 doi: 10.1137/050645452 [129] Wu J, Barahona M, Tan Y J, Deng H Z. Spectral measure of structural robustness in complex networks. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 2011, 41(6): 1244-1252 doi: 10.1109/TSMCA.2011.2116117 [130] Xiao S, Xiao G, Cheng T H, Ma S, Fu X, Soh H. Robustness of scale-free networks under rewiring operations. Europhysics Letters, 2010, 89(3): Article No. 38002 doi: 10.1209/0295-5075/89/38002 [131] Newman M E J. Mixing patterns in networks. Physical Review E, 2003, 67(2): Article No. 026126 doi: 10.1103/PhysRevE.67.026126 [132] Louzada V H P, Daolio F, Herrmann H J, Tomassini M. Smart rewiring for network robustness. Journal of Complex Networks, 2013, 1(2): 150-159 doi: 10.1093/comnet/cnt010 [133] Buesser P, Daolio F, Tomassini M. Optimizing the robustness of scale-free networks with simulated annealing. In: Proceedings of the 10th International Conference on Adaptive and Natural Computing Algorithms. Ljubljana, Slovenia: Springer, 2011. 167−176 [134] Zhou M X, Liu J. A memetic algorithm for enhancing the robustness of scale-free networks against malicious attacks. Physica A: Statistical Mechanics and its Applications, 2014, 410: 131-143 doi: 10.1016/j.physa.2014.05.002 [135] Zeng A, Liu W P. Enhancing network robustness against malicious attacks. Physical Review E, 2012, 85(6): Article No. 066130 doi: 10.1103/PhysRevE.85.066130 [136] Liu J, Abbass H A, Tan K C. Evolving robust networks using evolutionary algorithms. Evolutionary Computation and Complex Networks. Cham: Springer, 2019. 117-140 [137] Wang S, Liu J. Designing comprehensively robust networks against intentional attacks and cascading failures. Information Sciences, 2019, 478: 125-140 doi: 10.1016/j.ins.2018.11.005 [138] Gunasekara R C, Mohan C K, Mehrotra K. Multi-objective optimization to improve robustness in networks. Multi-Objective Optimization: Evolutionary to Hybrid Framework. Singapore: Springer, 2018. 115−139 [139] Hou L L, Lao S Y, Jiang B, Bai L. Enhancing complex network controllability by rewiring links. In: Proceedings of the 3rd International Conference on Intelligent System Design and Engineering Applications (ISDEA). Hong Kong, China: IEEE, 2013. 709−711 [140] Xu J Q, Wang J F, Zhao H, Jia S Y. Improving controllability of complex networks by rewiring links regularly. In: Proceedings of the 26th Chinese Control and Decision Conference (CCDC). Changsha, China: IEEE, 2014. 642−645 [141] Wang S, Liu J. A multi-objective evolutionary algorithm for promoting the emergence of cooperation and controllable robustness on directed networks. IEEE Transactions on Network Science and Engineering, 2018, 5(2): 92-100 doi: 10.1109/TNSE.2017.2742522 [142] Hou L L, Lao S Y, Small M, Xiao Y D. Enhancing complex network controllability by minimum link direction reversal. Physics Letters A, 2015, 379(20-21): 1321-1325 doi: 10.1016/j.physleta.2015.03.018 [143] Shi D H, Lü L Y, Chen G R. Totally homogeneous networks. National Science Review, 2019, 6(5): 962-969 doi: 10.1093/nsr/nwz050 [144] Shi D H, Chen G R, Thong W W K, Yan X Y. Searching for optimal network topology with best possible synchronizability. IEEE Circuits and Systems Magazine, 2013, 13(1): 66-75 doi: 10.1109/MCAS.2012.2237145 [145] Fan T L, Lv L Y, Shi D H, Zhou T. Characterizing cycle structure in complex networks. Communications Physics, 2021, 4, Article number: 272 [146] Milo R, Shen-Orr S, Itzkovitz S, Kashtan N, Chklovskii D, Alon U. Network motifs: Simple building blocks of complex networks. Science, 2002, 298(5594): 824-827 doi: 10.1126/science.298.5594.824 [147] Menck P J, Heitzig J, Kurths J, Schellnhuber H J. How dead ends undermine power grid stability. Nature Communications, 2014, 5: Article No. 3969 doi: 10.1038/ncomms4969 [148] Gorochowski T E, Grierson C S, di Bernardo M. Organization of feed-forward loop motifs reveals architectural principles in natural and engineered networks. Science Advances, 2018, 4(3): Article No. eaap9751 doi: 10.1126/sciadv.aap9751 [149] Badhwar R, Bagler G. Robust sigmoidal control response of C. elegans neuronal network. In: Proceedings of the 2017 International Joint Conference on Rough Sets. Olsztyn, Poland: Springer, 2017. 393−402 [150] Dey A K, Gel Y R, Poor H V. What network motifs tell us about resilience and reliability of complex networks. Proceedings of the National Academy of Sciences of the United States of America, 2019, 116(39): 19368-19373 doi: 10.1073/pnas.1819529116 [151] 贾承丰, 韩华, 完颜娟, 吕亚楠. 基于网络模体特征攻击的网络抗毁性研究. 复杂系统与复杂性科学, 2017, 14(4): 43-50Jia Cheng-Feng, Han Hua, Wan Yan-Juan, Lü Ya-Nan. Network destruction resistance based on network motif feature. Complex Systems and Complexity Science, 2017, 14(4): 43-50 -
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