The degree measure as utility function over positions in graphs and digraphs

We explore the possibility to compare positions in different directed and undirected graphs. assume an agent have a preference relation over weighted (directed undirected) graphs, stating pairwise comparisons between these positions. Ideally, such can be expressed by utility function, where are evaluated their assigned ‘utility’. Extending relations mixture set containing all lotteries graph positions, we specify axioms on preferences that allow them represented von Neumann–Morgenstern expected functions. For show only vNM function satisfies certain risk neutrality, is assigns every position same linear combination of its outdegree indegree. this degree measure degree. In way, our results provide foundation for centrality as function. obtain following approach Shapley value cooperative transferable games Roth (1977b), noticing graphs form subclass Deng Papadimitriou (1994). extend result class generalized games. Using networks, apply some applications Economics Operations Research.