光谱等级
Spectral Ranking
摘要：
我们简述了谱排序的历史，谱排序是将线性映射理论（特别是特征值和特征向量）应用于矩阵的技术的总称，这些矩阵不代表几何变换，而是实体之间的某种关系。虽然最近因Google的PageRank算法的大量新闻报道而闻名，但光谱排名是在一个多世纪前设计的，并且已经在比赛排名，心理学，社会科学，文献计量学，经济学和选择理论中进行了研究。我们描述的贡献，以前的学者在精确和现代的数学术语：一路上，我们展示了如何表达在一般的方式阻尼排名，如卡茨指数，作为主导特征向量的扰动矩阵，然后使用结果的Drazin逆回到主导特征向量的限制过程。结果表明，一个正规化的定义的频谱排名，产生一个唯一的向量取决于一个边界条件的一般矩阵。
英文摘要：
We sketch the history of spectral ranking, a general umbrella name for techniques that apply the theory of linear maps (in particular, eigenvalues and eigenvectors) to matrices that do not represent geometric transformations, but rather some kind of relationship between entities. Albeit recently made famous by the ample press coverage of Google's PageRank algorithm, spectral ranking was devised more than a century ago, and has been studied in tournament ranking, psychology, social sciences, bibliometrics, economy and choice theory. We describe the contribution given by previous scholars in precise and modern mathematical terms: along the way, we show how to express in a general way damped rankings, such as Katz's index, as dominant eigenvectors of perturbed matrices, and then use results on the Drazin inverse to go back to the dominant eigenvectors by a limit process. The result suggests a regularized definition of spectral ranking that yields for a general matrix a unique vector depending on a boundary condition.
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