更改

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。<br>

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。

 

 

[[Image:YYL1.png|right|thumb|Controlling a simple network.]]

[[Image:YYL1.png|right|thumb|Controlling a simple network.]]

其中，向量 <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> 表示在 <math>N</math> 个节点的系统时间 <math>t</math>时的状态。

其中，向量 <math>\mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T}</math> 表示在 <math>N</math> 个节点的系统时间 <math>t</math>时的状态。

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>

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>

They showed 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. 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.

They showed 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. 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.

<math>N \times N</math> 的矩阵 <math>\mathbf{A}</math> 描述了系统的连接图以及各组分之间的交互强度。<math>N \times M</math> 矩阵 <math>\mathbf{B}</math> 列出了由外部控制器控制的节点。控制器通过施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> 来实现对系统的控制。为了确定驱动节点的最小数目，<math>N_\mathrm{D}</math>，亦即确定对最少几个节点施加控制便足以完全控制系统的动力学进程，在这方面，Liu等人尝试了'''<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。<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> 他们发现，完全控制一个网络所需要的最少输入或驱动节点，取决于网络的'''<font color="#FF8000">最大匹配 maximum matching</font>'''，也就是不共享起始顶点和终止顶点的最大边集。从这个结论出发，一个基于''入-出度分布''的分析框架被开发出来，用于预测'''<font color="#FF8000">无标度网络 scale-free network</font>'''和'''<font color="#FF8000">ER随即图 Erdős–Rényi Graph</font>'''的<math>n_\mathrm{D} =N_\mathrm{D}/N </math>值。然而，更新的研究发现，网络的控制性，

<math>N \times N</math> 的矩阵 <math>\mathbf{A}</math> 描述了系统的连接图以及各组分之间的交互强度。<math>N \times M</math> 矩阵 <math>\mathbf{B}</math> 列出了由外部控制器控制的节点。控制器通过施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> 来实现对系统的控制。为了确定驱动节点的最小数目，<math>N_\mathrm{D}</math>，亦即确定对最少几个节点施加控制便足以完全控制系统的动力学进程，在这方面，Liu等人尝试了'''<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。<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> 他们发现，完全控制一个网络所需要的最少输入或驱动节点，取决于网络的'''<font color="#FF8000">最大匹配 maximum matching</font>'''，也就是不共享起始顶点和终止顶点的最大边集。从这个结论出发，一个基于''入-出度分布''的分析框架被开发出来，用于预测'''<font color="#FF8000">无标度网络 scale-free network</font>'''和'''<font color="#FF8000">ER随即图 Erdős–Rényi Graph</font>'''的<math>n_\mathrm{D} =N_\mathrm{D}/N </math>值。

 

==结构可控性 Structural Controllability==

===结构可控性 Structural Controllability===

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

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

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> 提出的，然后由 Glover 和 Silverman (1976)扩展。 <ref name="Shields-76">R. W. Shields and J. B. Pearson, ''IEEE Trans. Auto. Contr.'' '''21'''(1976).</ref>主要的问题是，对于可变系统参数来说，缺乏可控性或可观测性是否是普遍存在的现象。在结构控制的框架里，系统参数要么是独立自由变量，要么是固定的0。这对于物理系统的模型是一致的，因为参数值永远不会准确地获得，零值除外，零值代表没有交互和连接。

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> 提出的，然后由 Glover 和 Silverman (1976)扩展。 <ref name="Shields-76">R. W. Shields and J. B. Pearson, ''IEEE Trans. Auto. Contr.'' '''21'''(1976).</ref>主要的问题是，对于可变系统参数来说，缺乏可控性或可观测性是否是普遍存在的现象。在结构控制的框架里，系统参数要么是独立自由变量，要么是固定的0。这对于物理系统的模型是一致的，因为参数值永远不会准确地获得，零值除外，零值代表没有交互和连接。

==最大匹配 Maximum Matching==

===最大匹配 Maximum Matching===

'''<font color="#FF8000">最大匹配 Maximum Matching </font>'''<br>

'''<font color="#FF8000">最大匹配 Maximum Matching </font>'''<br>