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 智能系统学报  2018, Vol. 13 Issue (4): 602-609  DOI: 10.11992/tis.201703020 0

### 引用本文

CHEN Shiming, CHENG Yunhong, DENG Bing. Research on the controllability of directed interdependent networks[J]. CAAI Transactions on Intelligent Systems, 2018, 13(4), 602-609. DOI: 10.11992/tis.201703020.

### 文章历史

Research on the controllability of directed interdependent networks
CHEN Shiming, CHENG Yunhong, DENG Bing
School of Electrical and Automation Engineering, East China Jiaotong University, Nanchang 330013, China
Abstract: In this paper, we consider the influence of interdependency on the controllability of interdependent directed networks and investigate the controllability of interdependent directed networks with different types of interdependency. We build a basic interdependent directed network model and generate a controllability index by introducing the theory of exact controllability. We propose three kinds of interdependent directed network models for classical directed random networks and directed scale-free networks. In addition, we investigate the controllability of the interdependent directed networks with random interdependencies. Based on the results, we propose three kinds of interdependencies and compare and analyze the controllability of interdependent directed networks with different types of interdependency. The results show that, with the same proportion of interdependence, the best controllability of an interdependent directed network is that with an interdependency of lowest in-degree and lowest out-degree nodes, whereas the poorest controllability of an interdependent directed network is that with an interdependency of highest in-degree and highest out-degree nodes. The research results provide a useful reference and guidance for the construction of actual interdependent directed networks.
Key words: directed network    interdependent network    interdependency    exact controllability

1 有向相依网络可控性 1.1 有向相依网络

 $\begin{array}{*{20}{l}}{{F_A} = N_A^I/{N_A}}\\{{F_B} = N_B^I/{N_B}}\end{array}$ (1)

1.2 严格可控性

 ${\dot x} = {{Ax}}\left( {{t}} \right) + {{Bu}}\left( t \right)\; = \left[ \begin{gathered} \mathop A\nolimits_1 \;\;\mathop A\nolimits_{12} \\ \mathop A\nolimits_{21} \;\; \mathop A\nolimits_2 \\ \end{gathered} \right]{x}\left( {t} \right) + {B u}\left( {t} \right)$ (2)

 $\mathop n\nolimits_D = \frac{{\mathop N\nolimits_D }}{{2N}}$ (3)

1.3 理论分析

 ${N_0} = N_1+N_2$ (4)

 ${N_0} = \min (N_1)+\max(N_2)$ (5)

 $\frac{\min (N_1)}{N_0}=1-\frac{\max(N_2)}{N_0}$ (6)

 $\mathop n\nolimits_D = \frac{{\min \left( {\mathop N\nolimits_1 } \right)}}{{\mathop N\nolimits_0 }}$ (7)

 $\mathop N\nolimits_D = \min \left\{ {{\rm{rank}}({B})} \right\}$ (8)

 $2N = {\rm{rank}}\left[ {\mathop \lambda \nolimits_i {I} - {{A}},{{B}}} \right]$ (9)

 $\begin{gathered} 2N ={\rm{rank}}\left[ {\mathop \lambda \nolimits_i {I} - {A},{B}} \right] \leqslant \\ {\rm{rank}}\left[ {\mathop \lambda \nolimits_i {I} - {A}} \right] + {\rm{rank}}\left[ {B} \right] \\ \end{gathered}$ (10)

 ${\rm{rank}}\left[ {B} \right] \geqslant 2N - {\rm{rank}}\left[ {\mathop \lambda \nolimits_i {I} - {A}} \right]$ (11)

 $\begin{gathered} {\rm{rank}}\left[ {B} \right] \geqslant \max \left\{ {2N - {\rm{rank}}\left[ {\mathop \lambda \nolimits_i {I} - {A}} \right]} \right\} = \\ \mathop {\max }\limits_i \left\{ {\mu \left( {\mathop \lambda \nolimits_i } \right)} \right\} = \mu \left( {\mathop \lambda \nolimits^M } \right) \\ \end{gathered}$ (12)

 $\begin{gathered} \min \left( { {\rm{rank}}\left[ {B} \right]} \right) \geqslant \max \left\{ {N - {\rm{rank}}\left[ {\mathop \lambda \nolimits_i {I} - {A}} \right]} \right\} = \\ \mathop {\max }\limits_i \left\{ {\mu \left( {\mathop \lambda \nolimits_i } \right)} \right\} = \mu \left( {\mathop \lambda \nolimits^M } \right) \\ \end{gathered}$ (13)

 $\mathop N\nolimits_D = \mu \left( {\mathop \lambda \nolimits^M } \right)$ (14)

$\mathop N\nolimits_D$ 的定义可得：

 $\mathop N\nolimits_D = \min \left( {{N_1}} \right)$ (15)

 $\mathop n\nolimits_D = \frac{{\mathop N\nolimits_D }}{{2N}}$ (16)

 Download: 图 2 有向相依网络可控性 Fig. 2 Controllability of the interdependent directed network

 ${A} = \left[ {\begin{array}{*{20}{c}} 0&1&1&0&1&0&0&0&1&0 \\ 0&0&0&0&1&0&0&0&0&0 \\ 0&0&0&1&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&1&0&0&1&0&0&0 \\ 0&0&0&0&0&0&1&1&0&1 \\ 0&0&0&0&1&0&0&1&0&1 \\ 0&0&0&0&0&0&0&0&1&0 \\ 1&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&1 \end{array}} \right]$ (17)
 ${{\lambda}} = \mathop {\left[ {\begin{array}{*{20}{c}} 1&{ - 1}&0&0&0&1&{ - 1}&0&0&0 \end{array}} \right]}\nolimits^{\rm T}$ (18)

2 随机相依方式下网络可控性的仿真与分析

 Download: 图 3 单一相依比例下DER-DER网络的可控性 Fig. 3 Controllability of DER-DER network with a single dependency ratio

 Download: 图 4 连续相依比例下网络的可控性 Fig. 4 Controllability of the network under continuous dependency ratio
3 有向相依网络的相依方式及其可控性对比分析

 Download: 图 5 有向相依网络的相依方式 Fig. 5 Dependent way of the interdependent directed network

 Download: 图 6 不同相依方式下有向相依网络可控性 Fig. 6 Controllability of the interdependent directed network under different dependent way
4 结束语

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