# The positional power of nodes in digraphs.

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## Abstract

The positional power of nodes in digraphs P. Jean-Jacques Herings1, Gerard van der Laan2, Dolf Talman3 1 Department of Economics and METEOR, Universiteit Maastricht, P.O. Box 616, 6200 MD Maastricht, The Netherlands (e-mail: [email protected]) 2 Department of Econometrics and Tinbergen Institute, Free University, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands (e-mail: [email protected]) 3 Department of Econometrics & Operations Research and CentER, Tilburg University, P.O. Box 90153, 5000 LE, Tilburg, The Netherlands (e-mail: [email protected]) Received: 21 January 2003/Accepted: 10 November 2003 Abstract. Many economic and social situations can be represented by a di- graph. Both local and global methods to determine the strength or power of all the nodes in a digraph have been proposed in the literature. We propose a new method, where the power of a node is determined by both the number of its successors and the powers of its successors. Our method, called the posi- tional power function, determines a full ranking of the nodes for any digraph. The positional power function can either be determined as the unique solu- tion to a nonhomogeneous system of equations, or as the limit point of an iterative process. The solution can easily be obtained explicitly, which enables us to derive a number of interesting properties of the positional power function. We also consider the Copeland variant of the positional power function. Finally, we extend our method to the class of all weighted graphs. 1 Introduction Many economic and social situations can be modelled by means of a digraph. A digraph is an irreflexive directed graph consisting of a finite set of nodes and a collection of ordered pairs of these nodes, called arcs or arrows, e.g. see Behzad et al. [1]. An arc from one node to another node represents a domi- nance relation of the former node over the latter node. For instance, in a Soc Choice Welfare (2005) 24: 439–454 DOI: 10.1007/s00355-003-0308-9 The authors like to

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