Measuring the Power of Nodes in Digraphs

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.