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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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