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Consider the canonical linear time-invariant dynamics on a complex network

Consider the canonical linear time-invariant dynamics on a complex network

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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>.   

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>.   

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>

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。

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>

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.<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

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

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。