所谓特征向量中心度

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看《蜘蛛：社会网络分析技术》，里面讲到特征向量中心度，觉得是个很有趣的东西，不过这本书里面语焉不详，baidu了下，中文的东西全是垃圾，弄了半天还是不明白，又谷歌了下，发现维基百科里面说的清晰明白靠谱，这里转帖一下，实在懒得翻译了。

Eigenvector centrality 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. Google's PageRank is a variant of the Eigenvector centrality measure.^[11] Another closely related centrality measure is Katz centrality.

Using the adjacency matrix to find eigenvector centrality

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For a given graph $G:=(V,E)$ with $|V|$ number of vertices let $A = (a_{v,t})$ be the adjacency matrix, i.e. $a_{v,t} = 1$ if vertex $v$ is linked to vertex $t$ , and $a_{v,t} = 0$ otherwise. The centrality score of vertex $v$ can be defined as:

$x_v = \frac{1}{\lambda} \sum_{t \in M(v)}x_t = \frac{1}{\lambda} \sum_{t \in G} a_{v,t}x_t$

where $M(v)$ is a set of the neighbors of $v$ and $\lambda$ is a constant. With a small rearrangement this can be rewritten in vector notation as the eigenvectorequation

$\mathbf{Ax} = {\lambda}\mathbf{x}$

In general, there will be many different eigenvalues $\lambda$ for which an eigenvector solution exists. However, the additional requirement that all the entries in the eigenvector be positive implies (by the Perron–Frobenius theorem) that only the greatest eigenvalue results in the desired centrality measure.^[12] The $v^{th}$ component of the related eigenvector then gives the centrality score of the vertex $v$ in the network. Power iteration is one of many eigenvalue algorithms that may be used to find this dominant eigenvector.^[11] Furthermore, this can be generalized so that the entries in A can be real numbers representing connection strengths, as in a stochastic matrix.

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