Measuring the Power of Nodes in Digraphs

Many economic and social situations can be represented by a digraph. Both axiomatic and iterative 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, as in axiomatic methods, and the powers of its successors, as in iterative methods. Contrary to other iterative methods, we obtain a full ranking of the nodes for any digraph. The new power function, called the positional power function, can either be determined as the unique solution to a system of equations, or as the limit point of an iterative process. The solution is also explicitly characterized. This characterization enables us to derive a number of interesting properties of the positional power function. Next we consider a number of extensions, like the positional weakness function and the position function.