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Spectral ranking

Published online by Cambridge University Press:  21 November 2016

SEBASTIANO VIGNA
SEBASTIANO VIGNA*
Affiliation:
Dipartimento di informatica, Università degli Studi di Milano, Milano, 20122, Italy (e-mail: [email protected])
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Abstract

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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 60 years ago, almost exactly in the same terms, and has been studied in 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.

Keywords

rankingcentralityeigenvectors
Type
Research Article
Information
Network Science , Volume 4 , Issue 4 , December 2016 , pp. 433 - 445
Copyright
Copyright © Cambridge University Press 2016 

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Supplementary material: Link

Vigna Supplementary Material Link

Author’s updates to this short history of spectral ranking are maintained on arXiv

https://arxiv.org/abs/0912.0238
Link