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<math>
<math>
\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)
\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)
\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)
\dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t)
</math>
</math>
</math>
</math>
where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>. The <math>N \times N</math>
where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>. The <math>N \times N</math>
where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>. The <math>N \times N</math>
where the vector <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> captures the state of a system of <math>N</math> nodes at time <math>t</math>. The <math>N \times N</math>
其中，向量 math { x }(t)(x1(t) ，cdots，xn (t)) ^ mathrm { t } / math 捕获了时间 math t / math 时数学 n / math 节点系统的状态。数学 n 乘以 n / 数学
其中，向量<math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math>表示用<math>t</math><math>时间捕获<math>N</math>个节点系统。<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.<ref name="Liu-Nature-11">{{cite journal | last=Liu | first=Yang-Yu | last2=Slotine | first2=Jean-Jacques | last3=Barabási | first3=Albert-László | title=Controllability of complex networks | journal=Nature | publisher=Springer Science and Business Media LLC | volume=473 | issue=7346 | year=2011 | issn=0028-0836 | doi=10.1038/nature10011 | pages=167–173}}</ref> attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed<ref name="Liu-Nature-11"/> that the minimum number of inputs or driver nodes needed to maintain full control of the network is determined by the 'maximum matching’ in the network, that is, the maximum set of links that do not share start or end nodes. From this result, an analytical framework, based on the in-out degree distribution, was developed to predict <math>n_\mathrm{D} =N_\mathrm{D}/N </math> for scale-free and Erdős–Rényi Graphs.<ref name="Liu-Nature-11"/> However, more recently it has been demonstrated that network controllability (and other structure-only methods which use exclusively the connectivity of a graph, <math>\mathbf{A}</math>, to simplify the underlying dynamics), both undershoot and overshoot the number and which sets of driver nodes best control network dynamics, highlighting the importance of redundancy (e.g. canalization) and non-linear dynamics in determining control.<ref name="gates_rocha_scirep">{{cite journal | last=Gates | first=Alexander J. | last2=Rocha | first2=Luis M. | title=Control of complex networks requires both structure and dynamics | journal=Scientific Reports | publisher=Springer Science and Business Media LLC | volume=6 | issue=1 | date=2016-04-18 | issn=2045-2322 | doi=10.1038/srep24456 | page=24456|doi-access=free}}</ref>
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.<ref name="Liu-Nature-11">{{cite journal | last=Liu | first=Yang-Yu | last2=Slotine | first2=Jean-Jacques | last3=Barabási | first3=Albert-László | title=Controllability of complex networks | journal=Nature | publisher=Springer Science and Business Media LLC | volume=473 | issue=7346 | year=2011 | issn=0028-0836 | doi=10.1038/nature10011 | pages=167–173}}</ref> attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed<ref name="Liu-Nature-11"/> that the minimum number of inputs or driver nodes needed to maintain full control of the network is determined by the 'maximum matching’ in the network, that is, the maximum set of links that do not share start or end nodes. From this result, an analytical framework, based on the in-out degree distribution, was developed to predict <math>n_\mathrm{D} =N_\mathrm{D}/N </math> for scale-free and Erdős–Rényi Graphs.<ref name="Liu-Nature-11"/> However, more recently it has been demonstrated that network controllability (and other structure-only methods which use exclusively the connectivity of a graph, <math>\mathbf{A}</math>, to simplify the underlying dynamics), both undershoot and overshoot the number and which sets of driver nodes best control network dynamics, highlighting the importance of redundancy (e.g. canalization) and non-linear dynamics in determining control.<ref name="gates_rocha_scirep">{{cite journal | last=Gates | first=Alexander J. | last2=Rocha | first2=Luis M. | title=Control of complex networks requires both structure and dynamics | journal=Scientific Reports | publisher=Springer Science and Business Media LLC | volume=6 | issue=1 | date=2016-04-18 | issn=2045-2322 | doi=10.1038/srep24456 | page=24456|doi-access=free}}</ref>
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
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
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。
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。