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Measuring the Power of Nodes in Digraphs


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  • P. Jean-Jacques Herings

    (University Maastricht)

  • Gerard van der Laan

    (Vrije Universiteit Amsterdam)

  • Dolf Talman

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


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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Keywords: graph; tournament; power function;

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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. Herings,P. Jean-Jacques & Laan,Gerard,van der & Talman,Dolf, 2000. "Cooperative Games in Graph Structure," Research Memorandum 011, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
  3. Bouyssou, Denis, 1992. "Ranking methods based on valued preference relations: A characterization of the net flow method," European Journal of Operational Research, Elsevier, Elsevier, vol. 60(1), pages 61-67, July.
  4. Gilles, Robert P & Owen, Guillermo & van den Brink, Rene, 1992. "Games with Permission Structures: The Conjunctive Approach," International Journal of Game Theory, Springer, Springer, vol. 20(3), pages 277-93.
  5. 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.
  6. P. Jean-Jacques Herings & Gerard van der Laan & Dolf Talman, 2000. "Cooperative Games in Graph Structure," Tinbergen Institute Discussion Papers, Tinbergen Institute 00-072/1, Tinbergen Institute.
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Cited by:
  1. Giora Slutzki & Oscar Volij, 2006. "Scoring of web pages and tournaments—axiomatizations," Social Choice and Welfare, Springer, Springer, vol. 26(1), pages 75-92, January.
  2. Borm, Peter & van den Brink, Rene & Levinsky, Rene & Slikker, Marco, 2004. "On two new social choice correspondences," Mathematical Social Sciences, Elsevier, Elsevier, vol. 47(1), pages 51-68, January.
  3. Herings,P. Jean-Jacques & Laan, van der,Gerard & Talman,Dolf, 2003. "Socially Structured Games and Their Applications," Research Memorandum 024, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).


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