2009/12/01 by Sebastiano Vigna, Vigna, Sebastiano · 1 citation
Computer Science · #Advanced Algebra and Logic #FOS: Computer and information sciences #FOS: Physical sciences #Information Retrieval (cs.IR) #Physics and Society (physics.soc-ph) #Social and Information Networks (cs.SI)
paper · pdf · doi:10.48550/arxiv.0912.0238
openalex publication_date 2009/12/01 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
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.