Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T13:56:00.343Z Has data issue: false hasContentIssue false
  • English
  • Français

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: sebastiano.vigna@unimi.it)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Berge, C. (1958). Théorie des graphes et ses applications. Paris, France: Dunod.Google Scholar
Bergstrom, C. T., West, J. D., & Wiseman, M. A. (2008). The Eigenfactor™ metrics. Journal of Neuroscience, 28 (45), 11433–11434.Google Scholar
Boldi, P., Santini, M., & Vigna, S. (2009). PageRank: Functional dependencies. ACM Transactions on Information Systems, 27 (4), 1–23.CrossRefGoogle Scholar
Bonacich, P. (1972). Factoring and weighting approaches to status scores and clique identification. Journal of Mathematical Sociology, 2 (1), 113–120.Google Scholar
Bonacich, P. (1987). Power and centrality: A family of measures. The American Journal of Sociology, 92 (5), 1170–1182.CrossRefGoogle Scholar
Bonacich, P. (1991). Simultaneous group and individual centralities. Social Networks, 13 (2), 155–168.Google Scholar
Bonacich, P., & Lloyd, P. (2001). Eigenvector-like measures of centrality for asymmetric relations. Social Networks, 23 (3), 191–201.Google Scholar
Brauer, A. (1952). Limits for the characteristic roots of a matrix. IV: Applications to stochastic matrices. Duke Mathematical Journal, 19 (1), 75–91.Google Scholar
Drazin, M. P. (1958). Pseudo-inverses in associative rings and semigroups. The American Mathematical Monthly, 65 (7), 506–514.CrossRefGoogle Scholar
Festinger, L. (1949). The analysis of sociograms using matrix algebra. Human Relations, 2 (2), 153–9.Google Scholar
Franceschet, M. (2011). PageRank: Standing on the shoulders of giants. Communications of the ACM, 54 (6), 92–101.Google Scholar
French, J. R. P. Jr. (1956). A formal theory of social power. Psychological Review, 63 (3), 181–194.Google Scholar
Friedkin, N. E. (1991). Theoretical foundations for centrality measures. The American Journal of Sociology, 96 (6), 1478–1504.Google Scholar
Frisse, M. E. (1988). Searching for information in a hypertext medical handbook. Communications of the ACM, 31 (7), 880–886.Google Scholar
Geller, N. L. (1978). On the citation influence methodology of Pinski and Narin. Information Processing & Management, 14 (2), 93–95.Google Scholar
Gleich, D. F. (2015). PageRank beyond the web. SIAM Review, 57 (3), 321–363.Google Scholar
Gould, P. R. (1967). On the geographical interpretation of eigenvalues. Transactions of the Institute of British Geographers, 42, 53–86.Google Scholar
Hoede, C. (1978). A new status score for actors in a social network. Memorandum 243. Twente University Department of Applied Mathematics.Google Scholar
Hubbell, C. H. (1965). An input-output approach to clique identification. Sociometry, 28 (4), 377–399.Google Scholar
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.Google Scholar
Jeh, G., & Widom, J. (2003). Scaling personalized web search. In Proceedings of the 12th International World Wide Web Conference. New York, NY: ACM Press.Google Scholar
Katz, L. (1953). A new status index derived from sociometric analysis. Psychometrika, 18 (1), 39–43.Google Scholar
Keener, J. P. (1993). The Perron–Frobenius theorem and the ranking of football teams. SIAM Review, 35 (1), 80–93.CrossRefGoogle Scholar
Kendall, M. G. (1955). Further contributions to the theory of paired comparisons. Biometrics, 11 (1), 43–62.Google Scholar
Kleinberg, J. M. (1999). Authoritative sources in a hyperlinked environment. Journal of the ACM, 46 (5), 604–632.CrossRefGoogle Scholar
Leontief, W. W. (1941). The structure of American economy, 1919-1929: An empirical application of equilibrium analysis. Cambridge, MA: Harvard University Press.Google Scholar
Luce, R. D., & Perry, A. D. (1949). A method of matrix analysis of group structure. Psychometrika, 14 (2), 95–116.CrossRefGoogle ScholarPubMed
Marchiori, M. (1997). The quest for correct information on the Web: Hyper search engines. Computer Networks and ISDN Systems, 29 (8), 1225–1235.Google Scholar
Markov, A. A. (1906). Rasprostranenie zakona bolshih chisel na velichiny, zavisyaschie drug ot druga. Izvestiya Fiziko-matematicheskogo obschestva pri Kazanskom universitete, 2 (15), 135–156.Google Scholar
Meyer, C. D. Jr. (1974). Limits and the index of a square matrix. SIAM Journal on Applied Mathematics, 26 (3), 469–478.Google Scholar
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.Google Scholar
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), 297–312.Google Scholar
Rothblum, U. G. (1981). Expansions of sums of matrix powers. SIAM Review, 23 (2), 143–164.Google Scholar
Saaty, T. L. (1980). The analytical hierarchy process. New York: McGraw-Hill.Google Scholar
Saaty, T. L. (1987). Rank according to Perron: A new insight. Mathematics Magazine, 60 (4), 211–213.Google Scholar
Seeley, J. R. (1949). The net of reciprocal influence: A problem in treating sociometric data. Canadian Journal of Psychology, 3 (4), 234–240.Google Scholar
Wei, T.-H. (1952). The algebraic foundations of ranking theory. Ph.D. thesis, University of Cambridge.Google Scholar
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