One of the most famous network centrality measures is the degree measure which assigns to every position in a weighted network the sum of the weights of all links with its neighbours. We show that this degree measure can be seen as a von Neumann-Morgenstern utility function. We do this in three steps. First, we characterize the degree measure as a centrality measure for weighted networks using four natural axioms (anonymity, the isolated node property, scale invariance and additivity). Second, we relate these network centrality axioms to properties of preference relations over positions in networks. In particular, we consider the property of neutrality to ordinary risk. Third, we prove that the utility function is equal to a multiple of the degree measure if and only if it represents a preference relation that is neutral to ordinary risk. In this way we build a bridge between the social network literature on network centrality, and the economic literature on preferences and utility.