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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变输入向量<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>表示，其控制足以完全控制系统的动力学，Liu 等人尝试将结构控制理论、图论和统计物理的工具相结合，他们做到了。

<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，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。

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.

同样值得注意的是，刘氏等人的发现，他们提出度是网络中一种纯粹的局部度量，它是否能完全描述网络的可控性，即使是稍微远一点的节点在决定网络的可控性方面是否没有作用。事实上，对于许多实词网络，即食物网络、神经元网络和代谢网络，Liu 等人计算的数学数学和数学数学的值不匹配。值得注意的是。如果可控性主要是由程度决定的，那么为什么对于许多现实世界的网络来说，数学、数学和数学如此不同？他们认为(arXiv: 1203.5161 v1) ，这可能是由于度相关性的影响。然而，研究表明，网络的可控性只能通过介于中心性和接近中心性之间来改变，完全不需要度(图论)或度相关性。

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，对于许多实词网络，像食物网络、神经元网络和代谢网络，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(arXiv: 1203.5161 v1) ，这可能是由于度相关性的影响。然而，研究表明，网络的可控性只能通过介于中心性和接近中心性之间来改变，完全不需要度(图论)或度相关性。