  A formula based on a vertex contributions of the Steiner $k$-Wiener index is
induced by a newly introduced $k$-Steiner betweenness centrality, which
measures the number of $k$-Steiner trees that include a particular vertex as a
non-terminal vertex. This generalizes \v{S}krekovski and Gutman's Vertex
version of the Wiener Theorem and a result of Gago on the average betweenness
centrality and the average distance in general graphs.
