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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 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.

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Berge, C. (1958). Théorie des graphes et ses applications. Paris, France: Dunod.
Bergstrom, C. T., West, J. D., & Wiseman, M. A. (2008). The Eigenfactor™ metrics. Journal of Neuroscience, 28 (45), 1143311434.
Boldi, P., Santini, M., & Vigna, S. (2009). PageRank: Functional dependencies. ACM Transactions on Information Systems, 27 (4), 123.
Bonacich, P. (1972). Factoring and weighting approaches to status scores and clique identification. Journal of Mathematical Sociology, 2 (1), 113120.
Bonacich, P. (1987). Power and centrality: A family of measures. The American Journal of Sociology, 92 (5), 11701182.
Bonacich, P. (1991). Simultaneous group and individual centralities. Social Networks, 13 (2), 155168.
Bonacich, P., & Lloyd, P. (2001). Eigenvector-like measures of centrality for asymmetric relations. Social Networks, 23 (3), 191201.
Brauer, A. (1952). Limits for the characteristic roots of a matrix. IV: Applications to stochastic matrices. Duke Mathematical Journal, 19 (1), 7591.
Drazin, M. P. (1958). Pseudo-inverses in associative rings and semigroups. The American Mathematical Monthly, 65 (7), 506514.
Festinger, L. (1949). The analysis of sociograms using matrix algebra. Human Relations, 2 (2), 153–9.
Franceschet, M. (2011). PageRank: Standing on the shoulders of giants. Communications of the ACM, 54 (6), 92101.
French, J. R. P. Jr. (1956). A formal theory of social power. Psychological Review, 63 (3), 181194.
Friedkin, N. E. (1991). Theoretical foundations for centrality measures. The American Journal of Sociology, 96 (6), 14781504.
Frisse, M. E. (1988). Searching for information in a hypertext medical handbook. Communications of the ACM, 31 (7), 880886.
Geller, N. L. (1978). On the citation influence methodology of Pinski and Narin. Information Processing & Management, 14 (2), 9395.
Gleich, D. F. (2015). PageRank beyond the web. SIAM Review, 57 (3), 321363.
Gould, P. R. (1967). On the geographical interpretation of eigenvalues. Transactions of the Institute of British Geographers, 42, 5386.
Hoede, C. (1978). A new status score for actors in a social network. Memorandum 243. Twente University Department of Applied Mathematics.
Hubbell, C. H. (1965). An input-output approach to clique identification. Sociometry, 28 (4), 377399.
Huberman, B. A., Pirolli, P. L.T., Pitkow, J. E., & Lukose, R. M. (1998). Strong regularities in world wide web surfing. Science, 280 (5360), 95.
Jeh, G., & Widom, J. (2003). Scaling personalized web search. In Proceedings of the 12th International World Wide Web Conference. New York, NY: ACM Press.
Katz, L. (1953). A new status index derived from sociometric analysis. Psychometrika, 18 (1), 3943.
Keener, J. P. (1993). The Perron–Frobenius theorem and the ranking of football teams. SIAM Review, 35 (1), 8093.
Kendall, M. G. (1955). Further contributions to the theory of paired comparisons. Biometrics, 11 (1), 4362.
Kleinberg, J. M. (1999). Authoritative sources in a hyperlinked environment. Journal of the ACM, 46 (5), 604632.
Leontief, W. W. (1941). The structure of American economy, 1919-1929: An empirical application of equilibrium analysis. Cambridge, MA: Harvard University Press.
Luce, R. D., & Perry, A. D. (1949). A method of matrix analysis of group structure. Psychometrika, 14 (2), 95116.
Marchiori, M. (1997). The quest for correct information on the Web: Hyper search engines. Computer Networks and ISDN Systems, 29 (8), 12251235.
Markov, A. A. (1906). Rasprostranenie zakona bolshih chisel na velichiny, zavisyaschie drug ot druga. Izvestiya Fiziko-matematicheskogo obschestva pri Kazanskom universitete, 2 (15), 135156.
Meyer, C. D. Jr. (1974). Limits and the index of a square matrix. SIAM Journal on Applied Mathematics, 26 (3), 469478.
Page, L., Brin, S., Motwani, R., & Winograd, T. (1998). The PageRank citation ranking: Bringing order to the web. Tech. rept. SIDL-WP-1999-0120. Stanford Digital Library Technologies Project, Stanford University.
Pinski, G., & Narin, F. (1976). Citation influence for journal aggregates of scientific publications: Theory, with application to the literature of physics. Information Processing & Management, 12 (5), 297312.
Rothblum, U. G. (1981). Expansions of sums of matrix powers. SIAM Review, 23 (2), 143164.
Saaty, T. L. (1980). The analytical hierarchy process. New York: McGraw-Hill.
Saaty, T. L. (1987). Rank according to Perron: A new insight. Mathematics Magazine, 60 (4), 211213.
Seeley, J. R. (1949). The net of reciprocal influence: A problem in treating sociometric data. Canadian Journal of Psychology, 3 (4), 234240.
Wei, T.-H. (1952). The algebraic foundations of ranking theory. Ph.D. thesis, University of Cambridge.


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