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Measuring the Power of Nodes in Digraphs

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Author Info

  • P. Jean-Jacques Herings

    (University Maastricht)

  • Gerard van der Laan

    ()
    (Vrije Universiteit Amsterdam)

  • Dolf Talman

    (Department of Econometrics & Operations Research, and CentER, Tilburg University)

Abstract

Many economic and social situations can be represented by a digraph. Both axiomatic and iterativemethods to determine the strength or power of all the nodes in a digraph have been proposed inthe literature. We propose a new method, where the power of a node is determined by both thenumber of its successors, as in axiomatic methods, and the powers of its successors, as initerative methods. Contrary to other iterative methods, we obtain a full ranking of the nodes forany digraph. The new power function, called the positional power function, can either bedetermined as the unique solution to a system of equations, or as the limit point of an iterativeprocess. The solution is also explicitly characterized. This characterization enables us to derive anumber of interesting properties of the positional power function. Next we consider a number ofextensions, like the positional weakness function and the position function.

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Bibliographic Info

Paper provided by Tinbergen Institute in its series Tinbergen Institute Discussion Papers with number 01-096/1.

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Date of creation: 16 Oct 2001
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Handle: RePEc:dgr:uvatin:20010096

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Web page: http://www.tinbergen.nl

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Keywords: graph; tournament; power function;

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References

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  1. Brink, J.R. van den & Borm, P.E.M., 2002. "Digraph competitions and cooperative games," Open Access publications from Tilburg University urn:nbn:nl:ui:12-123145, Tilburg University.
  2. Borm, P.E.M. & Brink, J.R. van den & Slikker, M., 2002. "An iterative procedure for evaluating digraph competitions," Open Access publications from Tilburg University urn:nbn:nl:ui:12-90188, Tilburg University.
  3. Herings, P.J.J. & Laan, G. van der & Talman, A.J.J., 2000. "Cooperative Games in Graph Structure," Discussion Paper 2000-90, Tilburg University, Center for Economic Research.
  4. Gilles, R.P. & Owen, G. & Brink, J.R. van den, 1991. "Games with permission structures: The conjunctive approach," Discussion Paper 1991-14, Tilburg University, Center for Economic Research.
  5. P. Jean-Jacques Herings & Gerard van der Laan & Dolf Talman, 2000. "Cooperative Games in Graph Structure," Tinbergen Institute Discussion Papers 00-072/1, Tinbergen Institute.
  6. Bouyssou, Denis, 1992. "Ranking methods based on valued preference relations: A characterization of the net flow method," European Journal of Operational Research, Elsevier, vol. 60(1), pages 61-67, July.
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Cited by:
  1. Borm, P.E.M. & Brink, J.R. van den & Levinsky, R. & Slikker, M., 2000. "On Two New Social Choice Correspondences," Discussion Paper 2000-125, Tilburg University, Center for Economic Research.
  2. Herings, P.J.J. & Laan, G. van der & Talman, A.J.J., 2003. "Socially Structured Games and their Applications," Discussion Paper 2003-40, Tilburg University, Center for Economic Research.
  3. Giora Slutzki & Oscar Volij, 2006. "Scoring of web pages and tournaments—axiomatizations," Social Choice and Welfare, Springer, vol. 26(1), pages 75-92, January.

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