Skip to main content Accessibility help
×
Home
Hostname: page-component-cf9d5c678-5tm97 Total loading time: 0.794 Render date: 2021-07-30T02:32:12.510Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Spectral ranking

Published online by Cambridge University Press:  21 November 2016

SEBASTIANO VIGNA
Affiliation:
Dipartimento di informatica, Università degli Studi di Milano, Milano, 20122, Italy (e-mail: sebastiano.vigna@unimi.it)
Corresponding
E-mail address:
Rights & Permissions[Opens in a new window]

Abstract

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.

Type
Research Article
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), 1143311434.CrossRefGoogle Scholar
Boldi, P., Santini, M., & Vigna, S. (2009). PageRank: Functional dependencies. ACM Transactions on Information Systems, 27 (4), 123.CrossRefGoogle Scholar
Bonacich, P. (1972). Factoring and weighting approaches to status scores and clique identification. Journal of Mathematical Sociology, 2 (1), 113120.CrossRefGoogle Scholar
Bonacich, P. (1987). Power and centrality: A family of measures. The American Journal of Sociology, 92 (5), 11701182.CrossRefGoogle Scholar
Bonacich, P. (1991). Simultaneous group and individual centralities. Social Networks, 13 (2), 155168.CrossRefGoogle Scholar
Bonacich, P., & Lloyd, P. (2001). Eigenvector-like measures of centrality for asymmetric relations. Social Networks, 23 (3), 191201.CrossRefGoogle Scholar
Brauer, A. (1952). Limits for the characteristic roots of a matrix. IV: Applications to stochastic matrices. Duke Mathematical Journal, 19 (1), 7591.CrossRefGoogle Scholar
Drazin, M. P. (1958). Pseudo-inverses in associative rings and semigroups. The American Mathematical Monthly, 65 (7), 506514.CrossRefGoogle Scholar
Festinger, L. (1949). The analysis of sociograms using matrix algebra. Human Relations, 2 (2), 153–9.CrossRefGoogle Scholar
Franceschet, M. (2011). PageRank: Standing on the shoulders of giants. Communications of the ACM, 54 (6), 92101.CrossRefGoogle Scholar
French, J. R. P. Jr. (1956). A formal theory of social power. Psychological Review, 63 (3), 181194.CrossRefGoogle ScholarPubMed
Friedkin, N. E. (1991). Theoretical foundations for centrality measures. The American Journal of Sociology, 96 (6), 14781504.CrossRefGoogle Scholar
Frisse, M. E. (1988). Searching for information in a hypertext medical handbook. Communications of the ACM, 31 (7), 880886.CrossRefGoogle Scholar
Geller, N. L. (1978). On the citation influence methodology of Pinski and Narin. Information Processing & Management, 14 (2), 9395.CrossRefGoogle Scholar
Gleich, D. F. (2015). PageRank beyond the web. SIAM Review, 57 (3), 321363.CrossRefGoogle Scholar
Gould, P. R. (1967). On the geographical interpretation of eigenvalues. Transactions of the Institute of British Geographers, 42, 5386.CrossRefGoogle 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), 377399.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
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), 3943.CrossRefGoogle Scholar
Keener, J. P. (1993). The Perron–Frobenius theorem and the ranking of football teams. SIAM Review, 35 (1), 8093.CrossRefGoogle Scholar
Kendall, M. G. (1955). Further contributions to the theory of paired comparisons. Biometrics, 11 (1), 4362.CrossRefGoogle Scholar
Kleinberg, J. M. (1999). Authoritative sources in a hyperlinked environment. Journal of the ACM, 46 (5), 604632.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), 95116.CrossRefGoogle Scholar
Marchiori, M. (1997). The quest for correct information on the Web: Hyper search engines. Computer Networks and ISDN Systems, 29 (8), 12251235.CrossRefGoogle Scholar
Markov, A. A. (1906). Rasprostranenie zakona bolshih chisel na velichiny, zavisyaschie drug ot druga. Izvestiya Fiziko-matematicheskogo obschestva pri Kazanskom universitete, 2 (15), 135156.Google Scholar
Meyer, C. D. Jr. (1974). Limits and the index of a square matrix. SIAM Journal on Applied Mathematics, 26 (3), 469478.CrossRefGoogle 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), 297312.CrossRefGoogle Scholar
Rothblum, U. G. (1981). Expansions of sums of matrix powers. SIAM Review, 23 (2), 143164.CrossRefGoogle 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), 211213.CrossRefGoogle Scholar
Seeley, J. R. (1949). The net of reciprocal influence: A problem in treating sociometric data. Canadian Journal of Psychology, 3 (4), 234240.CrossRefGoogle Scholar
Wei, T.-H. (1952). The algebraic foundations of ranking theory. Ph.D. thesis, University of Cambridge.Google Scholar
You have Access
12
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Spectral ranking
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Spectral ranking
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Spectral ranking
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *