Spectral Ranking

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.