561
个编辑
更改
→Eigenvector centrality
==Eigenvector centrality==
==Eigenvector centrality==
特征向量中心性
=='''<font color="#ff8000">特征向量中心性 Eigenvector centrality</font>'''==
{{main|Eigenvector centrality}}
{{main|Eigenvector centrality}}
Eigenvector centrality (also called eigencentrality) is a measure of the influence of a node in a network. It assigns relative scores to all nodes in the network based on the concept that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes.
Eigenvector centrality (also called eigencentrality) is a measure of the influence of a node in a network. It assigns relative scores to all nodes in the network based on the concept that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes.
特征向量中心性(也称为特征中心性)是衡量网络中节点影响的一个指标。它将相对得分分配给网络中的所有节点，这是基于这样一个概念: 与得分较高的节点相比，与得分较低的节点相同的连接对所涉及的节点的得分贡献更大。
'''<font color="#ff8000">特征向量中心性 Eigenvector centrality</font>'''(也称为特征中心性)是衡量网络中节点影响的一个指标。它将相对得分分配给网络中的所有节点，这是基于这样一个概念: 与得分较高的节点相比，与得分较低的节点相同的连接对所涉及的节点的得分贡献更大。
=== Using the adjacency matrix to find eigenvector centrality ===
=== Using the adjacency matrix to find eigenvector centrality ==
==使用'''<font color="#ff8000"> 邻接矩阵The adjacency matrix</font>'''发现'''<font color="#ff8000">特征中心性 Eigenvector centrality</font>'''==
For a given graph <math>G:=(V,E)</math> with <math>|V|</math> number of vertices let <math>A = (a_{v,t})</math> be the [[adjacency matrix]], i.e. <math>a_{v,t} = 1</math> if vertex <math>v</math> is linked to vertex <math>t</math>, and <math>a_{v,t} = 0</math> otherwise. The relative centrality score of vertex <math>v</math> can be defined as:
For a given graph <math>G:=(V,E)</math> with <math>|V|</math> number of vertices let <math>A = (a_{v,t})</math> be the [[adjacency matrix]], i.e. <math>a_{v,t} = 1</math> if vertex <math>v</math> is linked to vertex <math>t</math>, and <math>a_{v,t} = 0</math> otherwise. The relative centrality score of vertex <math>v</math> can be defined as:
For a given graph <math>G:=(V,E)</math> with <math>|V|</math> number of vertices let <math>A = (a_{v,t})</math> be the adjacency matrix, i.e. <math>a_{v,t} = 1</math> if vertex <math>v</math> is linked to vertex <math>t</math>, and <math>a_{v,t} = 0</math> otherwise. The relative centrality score of vertex <math>v</math> can be defined as:
For a given graph <math>G:=(V,E)</math> with <math>|V|</math> number of vertices let <math>A = (a_{v,t})</math> be the adjacency matrix, i.e. <math>a_{v,t} = 1</math> if vertex <math>v</math> is linked to vertex <math>t</math>, and <math>a_{v,t} = 0</math> otherwise. The relative centrality score of vertex <math>v</math> can be defined as:
对于一个给定的图 g: = (v，e) </math > 与 < math > | v | </math > 让 < math > a = (a { v，t }) </math > 成为邻接矩阵，即。如果顶点 < math > > v </math > 与 math > t </math > 相连，而 < math > a { v，t } = 0 </math > 否则。顶点 < math > v </math > 的相对中心性评分可以定义为:
对于一个给定的图 g: = (v，e) </math > 与 < math > | v | </math > 让 < math > a = (a { v，t }) </math > 成为'''<font color="#ff8000"> 邻接矩阵The adjacency matrix</font>'''，即。如果顶点 < math > > v </math > 与 math > t </math > 相连，而 < math > a { v，t } = 0 </math > 否则。顶点 < math > v </math > 的相对中心性评分可以定义为:
In general, there will be many different eigenvalues <math>\lambda</math> for which a non-zero eigenvector solution exists. Since the entries in the adjacency matrix are non-negative, there is a unique largest eigenvalue, which is real and positive, by the Perron–Frobenius theorem. This greatest eigenvalue results in the desired centrality measure. The <math>v^{th}</math> component of the related eigenvector then gives the relative centrality score of the vertex <math>v</math> in the network. The eigenvector is only defined up to a common factor, so only the ratios of the centralities of the vertices are well defined. To define an absolute score one must normalise the eigenvector, e.g., such that the sum over all vertices is 1 or the total number of vertices n. Power iteration is one of many eigenvalue algorithms that may be used to find this dominant eigenvector. Furthermore, this can be generalized so that the entries in A can be real numbers representing connection strengths, as in a stochastic matrix.
In general, there will be many different eigenvalues <math>\lambda</math> for which a non-zero eigenvector solution exists. Since the entries in the adjacency matrix are non-negative, there is a unique largest eigenvalue, which is real and positive, by the Perron–Frobenius theorem. This greatest eigenvalue results in the desired centrality measure. The <math>v^{th}</math> component of the related eigenvector then gives the relative centrality score of the vertex <math>v</math> in the network. The eigenvector is only defined up to a common factor, so only the ratios of the centralities of the vertices are well defined. To define an absolute score one must normalise the eigenvector, e.g., such that the sum over all vertices is 1 or the total number of vertices n. Power iteration is one of many eigenvalue algorithms that may be used to find this dominant eigenvector. Furthermore, this can be generalized so that the entries in A can be real numbers representing connection strengths, as in a stochastic matrix.
一般情况下，存在许多不同的特征值，对于这些特征值存在一个非零特征向量解。由于邻接矩阵中的条目是非负的，所以由 Perron-弗罗贝尼乌斯定理得出，它有一个实的和正的唯一最大特征值。由这个最大的特征值得出期望的中心性度量。相关特征向量的 < math > v ^ { th } </math > 分量给出了网络中顶点 < math > v </math > 的相对中心性评分。特征向量只定义了一个公共因子，所以只有顶点中心的比例是明确定义的。要定义一个绝对分数，必须对特征向量进行标准化，例如，使所有顶点的和为1或顶点的总数n。此外，这可以通用，使得 a 中的条目可以是表示连接强度的真实数字，就像随机矩阵中一样。
一般情况下，存在许多不同的特征值，对于这些特征值存在一个非零特征向量解。由于邻接矩阵中的条目是非负的，所以由 Perron-弗罗贝尼乌斯定理得出，它有一个实的和正的唯一最大特征值。由这个最大的特征值得出期望的中心性度量。相关特征向量的 < math > v ^ { th } </math > 分量给出了网络中顶点 < math > v </math > 的相对中心性评分。特征向量只定义了一个公共因子，所以只有顶点中心的比例是明确定义的。要定义一个绝对分数，必须对特征向量进行标准化，例如，使所有顶点的和为1或顶点的总数n。此外，这可以通用，使得 a 中的条目可以是表示连接强度的真实数字，就像'''<font color="#ff8000"> 随机矩阵Stochastic matrix</font>'''中一样。
==Katz centrality==
==Katz centrality==