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The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the minimum number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al. attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed

The <math>N \times N</math>matrix <math>\mathbf{A}</math> describes the system's wiring diagram and the interaction strength between the components. The <math>N \times M</math> matrix <math>\mathbf{B}</math> identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> that the controller imposes on the system. To identify the minimum number of driver nodes, denoted by <math>N_\mathrm{D}</math>, whose control is sufficient to fully control the system's dynamics, Liu et al. attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed

<math>N \times N</math> 矩阵 <math>\mathbf{A}</math> 描述了系统的接线图和元件之间的交互强度。<math>N \times M</math> 矩阵 <math>\mathbf{B}</math> 识别由外部控制器控制的节点。系统通过控制器强加给系统的时间相关向量 <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> 来控制。为了确定驱动节点的最小数目，用<math>N_\mathrm{D}</math>来表示，其控制足以完全控制系统的动力学进程，在这方面，Liu等人成功做到了将结构控制理论、图论和统计物理的工具的结合。

<math>N \times N</math> 矩阵 <math>\mathbf{A}</math> 描述了系统的接线图和元件之间的交互强度。<math>N \times M</math> 矩阵 <math>\mathbf{B}</math> 识别由外部控制器控制的节点。系统通过控制器强加给系统的时间相关向量 <math>\mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T}</math> 来控制。为了确定驱动节点的最小数目，用<math>N_\mathrm{D}</math>来表示，其控制足以完全控制系统的动力学进程，在这方面，Liu等人成功做到了'''<font color="#FF8000">将结构控制理论 Structural Control Theory </font>'''、'''<font color="#FF8000">图论 Graph Theory </font>'''和'''<font color="#FF8000">统计物理 Statistical Physics </font>'''的工具的结合。

It is also notable, that Liu's et al. formulation  questions whether degree, which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely,  food webs, neuronal and metabolic  networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued  (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown that network controllability can be altered only by using betweenness centrality and closeness centrality, without using degree (graph theory) or degree correlations at all.

It is also notable, that Liu's et al. formulation  questions whether degree, which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely,  food webs, neuronal and metabolic  networks, the mismatch in values of <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> calculated by Liu et al. is notable. If controllability is decided mainly by degree, why are <math>{n_\mathrm{D}}^{real}</math> and <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> so different for many real world networks? They argued  (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown that network controllability can be altered only by using betweenness centrality and closeness centrality, without using degree (graph theory) or degree correlations at all.

同样值得关注的是，刘等人的发现。他们提出'''<font color="#FF8000">度 Degree </font>'''是网络中一种纯粹的局部度量，能够完全描述网络的可控性，即便是稍微远一点的节点也能确定它对网络的可控性是否有影响。事实上，对于许多'''<font color="#FF8000">实词网络 Real-Word Networks </font>'''，像'''<font color="#FF8000">食物网络 Food Webs </font>'''、'''<font color="#FF8000">神经元网络 Neuronal Network </font>''' 和'''<font color="#FF8000">代谢网络 Metabolic Network </font>'''，Liu等人计算的<math>{n_\mathrm{D}}^{real}</math><math> 和 {n_\mathrm{D}}^\mathrm{rand\_degree}</math> 的值并不匹配。值得注意的是。如果可控性主要是由度决定，那么为什么对于许多现实世界的网络来说，<math>{n_\mathrm{D}}^{real}</math> 和 <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> 如此不同？他们认为，这可能是由于度相关性的影响。然而，已有的研究表明，网络的可控性只能通过中间性和封闭性来改变，而完全不需要使用度（图论）或'''<font color="#FF8000">度关联 Degree Correlations </font>'''。

同样值得关注的是，刘等人的发现。他们提出'''<font color="#FF8000">度 Degree </font>'''是网络中一种纯粹的局部度量，能够完全描述网络的可控性，即便是稍微远一点的节点也能确定它对网络的可控性是否有影响。事实上，对于许多'''<font color="#FF8000">实词网络 Real-Word Networks </font>'''，像'''<font color="#FF8000">食物网络 Food Webs </font>'''、'''<font color="#FF8000">神经元网络 Neuronal Network </font>''' 和'''<font color="#FF8000">代谢网络 Metabolic Network </font>'''，Liu等人计算的<math>{n_\mathrm{D}}^{real}</math><math> 和 {n_\mathrm{D}}^\mathrm{rand\_degree}</math> 的值并不匹配。值得注意的是。如果可控性主要是由度决定，那么为什么对于许多现实世界的网络来说，<math>{n_\mathrm{D}}^{real}</math> 和 <math>{n_\mathrm{D}}^\mathrm{rand\_degree}</math> 如此不同？他们认为，这可能是由于度相关性的影响。然而，已有的研究表明，网络的可控性只能通过中间性和封闭性来改变，而完全不需要使用度（图论）或'''<font color="#FF8000">度关联 Degree Correlations </font>'''。(arXiv:1203.5161v1)

===Structural Controllability===

===Structural Controllability===

结构可控性<br>

'''<font color="#FF8000">结构可控性 Structural Controllability </font><br>

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, IEEE Trans. Auto. Contr. 19

The concept of the structural properties was first introduced by Lin (1974)<ref name="Lin-74">C.-T. Lin, IEEE Trans. Auto. Contr. 19

结构性质的概念最早是由 Lin (1974) ref name"Lin-74"c. t. Lin，IEEE Trans。自动。女名女子名。19

结构性质的概念最早是由 Lin (1974) <ref name="Lin-74">C.-T. Lin, ''IEEE Trans. Auto. Contr.'' '''19'''(1974).</ref>

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, ''IEEE Trans. Auto. Contr.'' '''21'''

(1974).</ref> and then extended by Shields and Pearson (1976)<ref name="Shields-76">R. W. Shields and J. B. Pearson, ''IEEE Trans. Auto. Contr.'' '''21'''

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, IEEE Trans. Auto. Contr. 21

(1976).</ref> and alternatively derived by Glover and Silverman (1976).<ref name="Glover-76">K. Glover and L. M. Silverman, IEEE Trans. Auto. Contr. 21

(1976) . / ref 和另外由 Glover 和 Silverman (1976)派生。 ref name"Glover-76"k。格洛弗和 l. m. Silverman，IEEE Trans。自动。女名女子名。21

和另外由 Glover 和 Silverman (1976)派生。 <ref name="Glover-76">K. Glover and L. M. Silverman, ''IEEE Trans. Auto. Contr.'' '''21'''(1976).</ref>

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

(1976).</ref> The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.