Network models are widely employed in many areas of science and technology. Mathematical analysis of their properties includes various methods to characterize, rank and compare network nodes. A key concept here is centrality, a numerical value of node importance in the whole network. As the links in a network represent interactions between the nodes, non-additive measures can serve to evaluate characteristics of sets of nodes considering these interactions, and thus to define new centrality measures. In this work, we investigate variants of network centralities based on non-additive measures, calculated as the Choquet integral of a function of the distance between pairs of vertices. We illustrate the applications of the constructed centrality measures on three examples: a social network, a chemical space network and a transportation network. The proposed centrality measures can complement existing methods of node ranking in different applications and serve as a starting point for developing new algorithms.