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==Eigenvector centrality==

==Eigenvector centrality==

 

{{main|Eigenvector centrality}}

{{main|Eigenvector centrality}}

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，所以有一个唯一的最大特征值，它是实的和正的，由 Perron-弗罗贝尼乌斯定理提供。这个最大的特征值导致期望的中心性度量。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规范化，例如，使所有顶点的和为1或顶点的总数。此外，这可以通用化，使得 a 中的条目可以是真实的数字，表示连接强度，就像转移矩阵一样。

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，就像随机矩阵中一样。

==Katz centrality==

==Katz centrality==