Advertisement

Social Choice and Welfare

, Volume 46, Issue 3, pp 639–653 | Cite as

Axioms for centrality scoring with principal eigenvectors

  • Mitri Kitti
Original Paper

Abstract

Techniques based on using principal eigenvector decomposition of matrices representing binary relations of sets of alternatives are commonly used in social sciences, bibliometrics, and web search engines. By representing the binary relations as a directed graph the question of ranking or scoring the alternatives can be turned into the relevant question of how to score the nodes of the graph. This paper characterizes the principal eigenvector of a matrix as a scoring function with a set of axioms. Furthermore, a method of assessing individual and group centralities simultaneously is characterized by a set of axioms. A special case of this method is the hyperlink-induced topic search for ranking websites. In general, the method can be applied to aggregation of preferences or judgments to obtain a collective assessment of alternatives.

Keywords

Directed Graph Adjacency Matrix Binary Relation Principal Eigenvalue Stochastic Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

I thank Hannu Salonen, Olli Lappalainen, Matti Pihlava, Jean-Jacques Herings, the associate editor, and two anonymous referees for their comments which have greatly improved this paper. I am also grateful to seminar audiences at University of Maastricht, Vrije Universiteit Amsterdam, XXXV Finnish Economic Days in Mariehamn, SING10 conference in Krakow, and 1st EUSN conference in Barcelona. Funding from the Academy of Finland is gratefully acknowledged.

References

  1. Altman A, Tennenholtz M (2005) Ranking systems: the PageRank axioms. In: Proceedings of the 6th ACM conference on Electronic commerce (EC-05). ACM Press, New York, pp 1–8Google Scholar
  2. Austen-Smith D, Banks J (1996) Information aggregation, rationality, and the Condorcet jury theorem. Am Polit Sci Rev 90:34–45CrossRefGoogle Scholar
  3. Bonacich P (1991) Simultaneous group and individual centralities. Soc Netw 13:155–168CrossRefGoogle Scholar
  4. Borm P, van den Brink R, Slikker M (2002) An iterative procedure for evaluating digraph competitions. Ann Oper Res 109:61–75CrossRefGoogle Scholar
  5. Bozbay I, Dietrich FK, Peters HJM (2014) Judgment aggregation in search for the truth. Games Econ Behav 87:571–590CrossRefGoogle Scholar
  6. Brin S, Page L (1998) The anatomy of a large-scale hypertextual web search engine. Comput Netw ISDN Syst 30:107–117CrossRefGoogle Scholar
  7. Cvetković D, Rowlinson P, Simić S (1997) Eigenspaces of graphs. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  8. David HA (1987) Ranking from unbalanced paired-comparison data. Biometrika 74:432–436CrossRefGoogle Scholar
  9. David HA (1988) The method of paired comparisons, 2nd edn. Charles Griffin and Company, LondonGoogle Scholar
  10. Echenique F, Fryer RG (2007) A measure of segregation based on social interactions. Q J Econ 122(2):441–485CrossRefGoogle Scholar
  11. Feddersen T, Pesendorfer W (1997) Information aggregation and voting behaviour in elections. Econometrica 65(5):1029–1058CrossRefGoogle Scholar
  12. Henriet D (1985) The Copeland choice function: an axiomatic characterization. Soc Choice Welf 2:49–63CrossRefGoogle Scholar
  13. Herings JJ, van der Laan G, Talman D (2005) The positional power of nodes in digraphs. Soc Choice Welf 24:439–454CrossRefGoogle Scholar
  14. Hu X, Shapley LS (2003) On authority distributions in organizations: equilibrium. Games Econ Behav 45:132–152CrossRefGoogle Scholar
  15. Kleinberg JM (1999) Authoritative sources in a hyperlinked environment. J ACM 46(5):604–632CrossRefGoogle Scholar
  16. Laslier J-F (1997) Tournament solutions and majority voting. Springer, Berlin, Heidelberg, New YorkCrossRefGoogle Scholar
  17. List C, Pettit P (2011) Group agency: the possibility, design and status of corporate agents. Oxford University Press, OxfordCrossRefGoogle Scholar
  18. Meyer CD (2000) Matrix analysis and applied linear algebra. SIAM, PhiladelphiaCrossRefGoogle Scholar
  19. Palacios-Huerta I, Volij O (2004) The measurement of intellectual influence. Econometrica 72(3):963–977CrossRefGoogle Scholar
  20. Rubinstein A (1980) Ranking the participants in a tournament. SIAM J Appl Math 38:108–111CrossRefGoogle Scholar
  21. Slutzki G, Volij O (2005) Ranking participants in generalized tournaments. Int J Game Theory 33(2):255–270CrossRefGoogle Scholar
  22. Slutzki G, Volij O (2006) Scoring of webpages and tournaments—axiomatizations. Soc Choice Welf 26(1):75–92CrossRefGoogle Scholar
  23. van den Brink R, Gilles RP (2000) Measuring domination in directed networks. Soc Netw 22:141–157CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of TurkuTurkuFinland

Personalised recommendations