Abstract
Consider a set of elements which we want to rate using information about their bilateral relationships. For instance sports teams and the outcomes of their games, journals and their mutual citations, web sites and their link structure, or social alternatives and the tournament derived from the voters' preferences. A wide variety of scoring methods have been proposed to deal with this problem. In this paper we axiomatically characterize two of these scoring methods, variants of which are used to rank web pages by their relevance to a query, and academic journals according to their impact. These methods are based on the Perron–Frobenius theorem for non-negative matrices.
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Notes
An anonymous referee suggested Saaty's (1980) work on the analytic hierarchy process in the context of multicriteria decision making as another successful application of the Perron–Frobenius theory.
Readers interested in learning some of the subtle properties of this procedure can consult Merlin and Saari (1996).
Note that in this context, A is the transpose of the adjacency matrix of the WWW graph.
Note, however, that this restriction should not be applied for ranking journals, since self-references do matter. See Palacios-Huerta and Volij (2002).
That every irreducible stochastic matrix has a unique stationary distribution is a standard result in finite Markov Chain theory. See Kemeny and Snell (1976).
This interpretation is based on Lemma 3.1 in Freidlin and Wentzell (1998).
Note that since A is irreducible, this ratio is well-defined.
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We thank Herbert A. David for his insightful comments and an anonymous referee for several useful references.
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Slutzki, G., Volij, O. Scoring of web pages and tournaments—axiomatizations. Soc Choice Welfare 26, 75–92 (2006). https://doi.org/10.1007/s00355-005-0033-7
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DOI: https://doi.org/10.1007/s00355-005-0033-7