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Measuring the Power of Nodes in Digraphs

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  • Herings, P.J.J.
  • Laan, G. van der
  • Talman, A.J.J.

    (Tilburg University, Center for Economic Research)

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 Tilburg University, Center for Economic Research in its series Discussion Paper with number 2001-72.

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Date of creation: 2001
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Handle: RePEc:dgr:kubcen:200172

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Keywords: graphs;

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References

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  1. Brink, J.R. van den & Borm, P.E.M., 1994. "Digraph competitions and cooperative games," Discussion Paper 1994-24, Tilburg University, Center for Economic Research.
  2. 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.
  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. 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.
  5. 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.
  6. 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.
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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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