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第65行： − 示意图显示了一个有向网络的控制。对于给定的有向网络(图)。A) ，计算它的最大匹配: 没有共同的正面或反面的最大边集。最大匹配将由一组顶点不相交的有向路径和有向循环组成(见图 b 中的红边)。如果一个节点是匹配边的头部，则匹配该节点(图 b 中的绿色节点)。否则，它是不匹配的(图 b 中的白色节点)。那些不匹配的节点是需要控制的节点，即。驱动节点。通过向这些驱动节点注入信号，可以获得一组有向路径，起始点作为输入(参见图 c)。这些路径被称为“茎”。由此产生的有向图称为根 u 阶乘连接。通过“嫁接”那些“茎”的定向循环，一个人得到“芽”。得到的有向图称为 cacti (见图 d)。根据结构能控性定理，由于存在跨越受控网络的 cacti 结构(见图 e) ，系统是可控的。控制网络(Fig.e)底层的 cacti 结构(Fig.d)是维持可控性的“骨架”。+ +
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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，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。
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>'''。
A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem, since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.
A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem, since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability.
示意图显示了定向网络的控制。对于给定的定向网络（图a），可以计算其最大匹配：没有共同头部或尾部的最大边缘集。最大匹配将由一组顶点不相交的有向路径和有向循环组成（请参见图b中的红色边缘）。如果一个节点是匹配边的头部，则该节点是匹配的（图b中的绿色节点）。否则，它是不匹配的（图b中的白色节点）。那些不匹配的节点是需要控制的节点，即驱动节点。通过向这些驱动节点注入信号，可以得到一组以起点为输入的有向路径（见图c）。这些路径被称为“茎”。得到的有向图称为U根因子连接。通过将定向周期“嫁接”到那些“茎”，人们就会得到“芽”。得到的有向图称为cacti（见图d）。根据结构可控性定理，由于存在跨越受控网络的cacti结构（参见图e），因此该系统是可控的。受控网络（图e）下的cacti结构（图d）是维持可控性的“骨架”。
===Structural Controllability===
===Structural Controllability===
结构可控性<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'''