更改

===Characterization by network flows==

===Characterization by network flows==

网络流特征

==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/>

Likewise, the type of path can be constrained to geodesics (shortest paths), paths  (no vertex is visited more than once), trails (vertices can be visited multiple times, no edge is traversed more than once), or walks (vertices and edges can be visited/traversed multiple times).

Likewise, the type of path can be constrained to geodesics (shortest paths), paths  (no vertex is visited more than once), trails (vertices can be visited multiple times, no edge is traversed more than once), or walks (vertices and edges can be visited/traversed multiple times).

同样，路径类型可以被限定为测地线(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===

===Characterization by walk structure===

行走结构特征

==行走结构特征==

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/>

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 degree and 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 betweenness centrality, the number of shortest paths which pass through the given vertex.

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 degree and 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 betweenness centrality, the number of shortest paths which pass through the given vertex.

可以从中心性的构造方式推导出另一种分类方法。这又分成了两个类。中心点可以是放射状的，也可以是内侧的。从给定顶点开始/结束的径向中心点数。度和特征值中心是径向中心的例子，计算长度为一或无穷大的行走的个数。内侧中心点计算通过给定顶点的步数。典型的例子是 "自由人"的中心性，即穿过给定顶点的最短路径的数量。

可以从中心性的构造方式推导出另一种分类方法。这又分成了两个类。中心点可以是放射状的，也可以是内侧的。从给定顶点开始/结束的径向中心点数。度和特征值中心是'''<font color="#ff8000"> 径向中心Radial centralities</font>'''的例子，计算长度为一或无穷大的行走的个数。'''<font color="#ff8000"> 中间中心性Medial centralities</font>'''计算通过给定顶点的步数。典型的例子是 弗里曼的'''<font color="#ff8000"> 中介中心性Betweenness centrality,</font>'''，即穿过给定顶点的最短路径的数量。

Likewise, the counting can capture either the volume or the length of walks. Volume is the total number of walks of the given type. The three examples from the previous paragraph fall into this category. Length captures the distance from the given vertex to the remaining vertices in the graph. Freeman's closeness centrality, the total geodesic distance from a given vertex to all other vertices, is the best known example. Note that this classification is independent of the type of walk counted (i.e. walk, trail, path, geodesic).

Likewise, the counting can capture either the volume or the length of walks. Volume is the total number of walks of the given type. The three examples from the previous paragraph fall into this category. Length captures the distance from the given vertex to the remaining vertices in the graph. Freeman's closeness centrality, the total geodesic distance from a given vertex to all other vertices, is the best known example. Note that this classification is independent of the type of walk counted (i.e. walk, trail, path, geodesic).

同样地，计数可以抓住行走的体积或长度。体积是给定类型的行走的总数。上一段的三个例子就属于这一类。长度则给出从给定顶点到图中其余顶点的距离。自由人 的贴近中心性，即从一个给定顶点到所有其他顶点的测地距离，是最著名的例子。请注意，这种分类独立于步行计数的类型(即。步行，小道，路径，测地线)。

同样地，计数可以抓住行走的体积或长度。体积是给定类型的行走的总数。上一段的三个例子就属于这一类。长度则给出从给定顶点到图中其余顶点的距离。弗里曼 的'''<font color="#ff8000"> 紧密中心性Closeness centrality</font>'''，即从一个给定顶点到所有其他顶点的测地距离，是最著名的例子。请注意，这种分类独立于步行计数的类型(即：步行，小道，路径，测地线)。

Borgatti and Everett propose that this typology provides insight into how best to compare centrality measures. Centralities placed in the same box in this 2×2  classification are similar enough to make plausible alternatives; one can reasonably compare which is better for a given application. Measures from different boxes, however, are categorically distinct. Any evaluation of relative fitness can only occur within the context of predetermining which category is more applicable, rendering the comparison moot.

Borgatti and Everett propose that this typology provides insight into how best to compare centrality measures. Centralities placed in the same box in this 2×2  classification are similar enough to make plausible alternatives; one can reasonably compare which is better for a given application. Measures from different boxes, however, are categorically distinct. Any evaluation of relative fitness can only occur within the context of predetermining which category is more applicable, rendering the comparison moot.

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程序更好。然而，不同盒子中的度量方法是截然不同的。任何对相对适应度的评估只能在预先确定哪个类别更适用的情况下发生，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===

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

光谱上存在的径向体积中心

==光谱上存在的径向体积中心==

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.

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.

Bonacich showed that if association is defined in terms of walks, then a family of centralities can be defined based on the length of walk considered. Degree centrality counts walks of length one, while 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 walks, then a family of centralities can be defined based on the length of walk considered. Degree centrality counts walks of length one, while 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集中性家族。度集中度计算长度为1的行走，特征值集中度计算长度为无穷大的行走。关联的其他定义也是合理的。阿尔法中心性允许顶点有一个外部影响源。埃斯特拉达的子图中心性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集中性家族。'''<font color="#ff8000"> 度集中度Degree centrality</font>'''计算长度为1的行走，'''<font color="#ff8000">特征值集中度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

for matrix exponentials, where

for matrix exponentials, where