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当根据内聚力方法对中心性进行分类时，很明显大多数中心性都将被划分于同一类别。起始于给定顶点的步数总和仅取决于步数的定义以及计数方式。这种分类方式的不足表现为它仅能较弱的描绘中心性特征，即按照一步步度（'''<font color="#ff8000">度中心性 degree centrality</font>'''）到无穷步步长（'''<font color="#ff8000">特征向量中心性 eigenvalue centrality</font>'''）的方式将中心性置于一种光谱状的分类中。<ref name=Bonacich1987/><ref name="Benzi2013">{{cite journal | last1=Benzi | first1=Michele | last2=Klymko| first2=Christine | year=2013 |title= A matrix analysis of different centrality measures |arxiv=1312.6722 | doi=10.1137/130950550 | volume=36 | issue=2 | journal=SIAM Journal on Matrix Analysis and Applications | pages=686–706}}</ref> 对于采用类似方式定义的各种中心性的观察也表明了这些指数之间的高阶相关性。

当根据内聚力方法对中心性进行分类时，很明显大多数中心性都将被划分于同一类别。起始于给定顶点的步数总和仅取决于步数的定义以及计数方式。这种分类方式的不足表现为它仅能较弱的描绘中心性特征，即按照一步步度（'''<font color="#ff8000">度中心性 degree centrality</font>'''）到无穷步步长（'''<font color="#ff8000">特征向量中心性 eigenvalue centrality</font>'''）的方式将中心性置于一种光谱状的分类中。<ref name=Bonacich1987/><ref name="Benzi2013">{{cite journal | last1=Benzi | first1=Michele | last2=Klymko| first2=Christine | year=2013 |title= A matrix analysis of different centrality measures |arxiv=1312.6722 | doi=10.1137/130950550 | volume=36 | issue=2 | journal=SIAM Journal on Matrix Analysis and Applications | pages=686–706}}</ref> 对于采用类似方式定义的各种中心性的观察也表明了这些指数之间的高阶相关性。

===Characterization by network flows==

===网络流特征Characterization by network flows==

==网络流特征==

A network can be considered a description of the paths along which something flows. This allows a characterization based on the type of flow and the type of path encoded by the centrality. A flow can be based on transfers, where each indivisible item goes from one node to another, like a package delivery going from the delivery site to the client's house. A second case is serial duplication, in which an item is replicated so that both the source and the target have it. An example is the propagation of information through gossip, with the information being propagated in a private way and with both the source and the target nodes being informed at the end of the process. The last case is parallel duplication, with the item being duplicated to several links at the same time, like a radio broadcast which provides the same information to many listeners at once.<ref name=Borgatti2005/>

A network can be considered a description of the paths along which something flows. This allows a characterization based on the type of flow and the type of path encoded by the centrality. A flow can be based on transfers, where each indivisible item goes from one node to another, like a package delivery going from the delivery site to the client's house. A second case is serial duplication, in which an item is replicated so that both the source and the target have it. An example is the propagation of information through gossip, with the information being propagated in a private way and with both the source and the target nodes being informed at the end of the process. The last case is parallel duplication, with the item being duplicated to several links at the same time, like a radio broadcast which provides the same information to many listeners at once.<ref name=Borgatti2005/>

同样，路径类型可以被限定为'''<font color="#ff8000"> 测地线Distance </font>'''(最短路径)、路径(对顶点的访问不超过一次)、小径(可以访问多次顶点，没有边被访问超过一次)或者步子(可以访问/遍历多次顶点和边)。

同样，路径类型可以被限定为'''<font color="#ff8000"> 测地线Distance </font>'''(最短路径)、路径(对顶点的访问不超过一次)、小径(可以访问多次顶点，没有边被访问超过一次)或者步子(可以访问/遍历多次顶点和边)。

===Characterization by walk structure===

 

==行走结构特征==

===行走结构特征Characterization by walk structure===

An alternative classification can be derived from how the centrality is constructed. This again splits into two classes. Centralities are either ''radial'' or ''medial.'' Radial centralities count walks which start/end from the given vertex. The [[Centrality#Degree centrality|degree]] and [[Centrality#Eigenvector centrality|eigenvalue]] centralities are examples of radial centralities, counting the number of walks of length one or length infinity. Medial centralities count walks which pass through the given vertex. The canonical example is Freeman's [[Centrality#Betweenness centrality|betweenness]] centrality, the number of shortest paths which pass through the given vertex.<ref name=Borgatti2006/>

An alternative classification can be derived from how the centrality is constructed. This again splits into two classes. Centralities are either ''radial'' or ''medial.'' Radial centralities count walks which start/end from the given vertex. The [[Centrality#Degree centrality|degree]] and [[Centrality#Eigenvector centrality|eigenvalue]] centralities are examples of radial centralities, counting the number of walks of length one or length infinity. Medial centralities count walks which pass through the given vertex. The canonical example is Freeman's [[Centrality#Betweenness centrality|betweenness]] centrality, the number of shortest paths which pass through the given vertex.<ref name=Borgatti2006/>

博尔加蒂和埃弗雷特提出，这种类型学为如何最好地比较中心性度量提出了建议。在这个22分类中，放在同一盒子中的中心点足够相似，可以做出合理的选择; 人们可以合理地比较哪个对于给定的应用更好。然而，不同盒子中的度量方法是截然不同的。只有在预先确定哪个类别更适用的情况下，对相对适应性的评估才会发生，这使得比较变得毫无意义。

博尔加蒂和埃弗雷特提出，这种类型学为如何最好地比较中心性度量提出了建议。在这个22分类中，放在同一盒子中的中心点足够相似，可以做出合理的选择; 人们可以合理地比较哪个对于给定的应用更好。然而，不同盒子中的度量方法是截然不同的。只有在预先确定哪个类别更适用的情况下，对相对适应性的评估才会发生，这使得比较变得毫无意义。

===Radial-volume centralities exist on a spectrum===

===光谱上存在的径向体积中心Radial-volume centralities exist on a spectrum===

 

The characterization by walk structure shows that almost all centralities in wide use are radial-volume measures. These encode the belief that a vertex's centrality is a function of the centrality of the vertices it is associated with. Centralities distinguish themselves on how association is defined.

The characterization by walk structure shows that almost all centralities in wide use are radial-volume measures. These encode the belief that a vertex's centrality is a function of the centrality of the vertices it is associated with. Centralities distinguish themselves on how association is defined.

步行结构的特征表明，几乎所有广泛使用的中心性都是径向体积的衡量。这得出结论顶点的中心性是与之相关联的顶点中心性的函数。中心性根据如何定义关联而不同。

步行结构的特征表明，几乎所有广泛使用的中心性都是径向体积的衡量。这得出结论顶点的中心性是与之相关联的顶点中心性的函数。中心性根据如何定义关联而不同。

Bonacich showed that if association is defined in terms of [[Glossary of graph theory terms#walk|walks]], then a family of centralities can be defined based on the length of walk considered.<ref name="Bonacich1987"/> [[Centrality#Degree centrality|Degree centrality]] counts walks of length one, while [[Centrality#Eigenvector centrality|eigenvalue centrality]] counts walks of length infinity. Alternative definitions of association are also reasonable. [[Alpha centrality]] allows vertices to have an external source of influence. Estrada's subgraph centrality proposes only counting closed paths (triangles, squares, etc.).

Bonacich showed that if association is defined in terms of [[Glossary of graph theory terms#walk|walks]], then a family of centralities can be defined based on the length of walk considered.<ref name="Bonacich1987"/> [[Centrality#Degree centrality|Degree centrality]] counts walks of length one, while [[Centrality#Eigenvector centrality|eigenvalue centrality]] counts walks of length infinity. Alternative definitions of association are also reasonable. [[Alpha centrality]] allows vertices to have an external source of influence. Estrada's subgraph centrality proposes only counting closed paths (triangles, squares, etc.).

Bonacich 指出，如果联想是根据行走来定义的，那么可以根据考虑的行走长度来定义一个中心性家族。'''<font color="#ff8000"> 度中心性Degree centrality</font>'''计算长度为1的行走，'''<font color="#ff8000">特征值中心性Eigenvalue centrality  </font>'''计算长度为无穷大的行走。关联的其他定义也是合理的。'''<font color="#ff8000"> 阿尔法中心性Alpha centrality</font>'''允许顶点有一个外部影响源。埃斯特拉达的'''<font color="#ff8000"> 子图中心性Subgraph centrality </font>'''提出只计算封闭路径(三角形、正方形等)。).

Bonacich 指出，如果联想是根据行走来定义的，那么可以根据考虑的行走长度来定义一个中心性家族。'''<font color="#ff8000"> 度中心性Degree centrality</font>'''计算长度为1的行走，'''<font color="#ff8000">特征值中心性Eigenvalue centrality  </font>'''计算长度为无穷大的行走。关联的其他定义也是合理的。'''<font color="#ff8000"> 阿尔法中心性Alpha centrality</font>'''允许顶点有一个外部影响源。埃斯特拉达的'''<font color="#ff8000"> 子图中心性Subgraph centrality </font>'''提出只计算封闭路径(三角形、正方形等)。).

The heart of such measures is the observation that powers of the graph's adjacency matrix gives the number of walks of length given by that power.  Similarly, the matrix exponential is also closely related to the number of walks of a given length. An initial transformation of the adjacency matrix allows a different definition of the type of walk counted. Under either approach, the centrality of a vertex can be expressed as an infinite sum, either

The heart of such measures is the observation that powers of the graph's adjacency matrix gives the number of walks of length given by that power.  Similarly, the matrix exponential is also closely related to the number of walks of a given length. An initial transformation of the adjacency matrix allows a different definition of the type of walk counted. Under either approach, the centrality of a vertex can be expressed as an infinite sum, either

< math > sum _ { k = 0} ^ infty a _ { r } ^ { k } beta ^ k </math >  

< math > sum _ { k = 0} ^ infty a _ { r } ^ { k } beta ^ k </math >  

博纳奇的一系列措施并没有改变邻接矩阵。阿尔法集中性用它的解决方案替代了邻接矩阵。子图集中性用它的踪迹取代了邻接矩阵。一个令人吃惊的结论是，不管邻接矩阵最初的转变是什么，所有这些方法都有共同的限制行为。随着贝塔系数趋近于零，指数收敛到度中心性。随着贝塔系数接近其最大值，指数收敛到特征值的中心性。

博纳奇的一系列措施并没有改变邻接矩阵。阿尔法集中性用它的解决方案替代了邻接矩阵。子图集中性用它的踪迹取代了邻接矩阵。一个令人吃惊的结论是，不管邻接矩阵最初的转变是什么，所有这些方法都有共同的限制行为。随着贝塔系数趋近于零，指数收敛到度中心性。随着贝塔系数接近其最大值，指数收敛到特征值的中心性。

===Game-theoretic centrality===

===博弈论中心性Game-theoretic centrality===

The common feature of most of the aforementioned standard measures is that they assess the importance of a node by focusing only on the role that a node plays by itself. However, in many applications such an approach is inadequate because of synergies that may occur

The common feature of most of the aforementioned standard measures is that they assess the importance of a node by focusing only on the role that a node plays by itself. However, in many applications such an approach is inadequate because of synergies that may occur

同样，'''<font color="#CD32CD"> 加权分布概念的解()The solution concept authority distribution () </font>'''采用 Shapley-Shubik 幂指数，而不是 Shapley 值来衡量双方之间的直接影响。这种分布确实是一种产生'''<font color="#ff8000"> 特征向量中心性Engenvector centrality</font>'''的类型。它用于对 Hu (2020)中的大数据对象进行排序，比如美国大学排名。

同样，'''<font color="#CD32CD"> 加权分布概念的解()The solution concept authority distribution () </font>'''采用 Shapley-Shubik 幂指数，而不是 Shapley 值来衡量双方之间的直接影响。这种分布确实是一种产生'''<font color="#ff8000"> 特征向量中心性Engenvector centrality</font>'''的类型。它用于对 Hu (2020)中的大数据对象进行排序，比如美国大学排名。

== Important limitations ==

== 重要限制Important limitations ==

 

Centrality indices have two important limitations, one obvious and the other subtle. The obvious limitation is that a centrality which is optimal for one application is often sub-optimal for a different application. Indeed, if this were not so, we would not need so many different centralities. An illustration of this phenomenon is provided by the [[Krackhardt kite graph]], for which three different notions of centrality give three different choices of the most central vertex.<ref>{{cite journal|title=Assessing the Political Landscape: Structure, Cognition, and Power in Organizations|first=David|last=Krackhardt|authorlink=David Krackhardt|journal=Administrative Science Quarterly|volume=35|issue=2|date=June 1990|pages=342–369|doi=10.2307/2393394|jstor=2393394}}</ref>

Centrality indices have two important limitations, one obvious and the other subtle. The obvious limitation is that a centrality which is optimal for one application is often sub-optimal for a different application. Indeed, if this were not so, we would not need so many different centralities. An illustration of this phenomenon is provided by the [[Krackhardt kite graph]], for which three different notions of centrality give three different choices of the most central vertex.<ref>{{cite journal|title=Assessing the Political Landscape: Structure, Cognition, and Power in Organizations|first=David|last=Krackhardt|authorlink=David Krackhardt|journal=Administrative Science Quarterly|volume=35|issue=2|date=June 1990|pages=342–369|doi=10.2307/2393394|jstor=2393394}}</ref>