In this paper we introduce a depth function for distributions on graphs that is analogous to recent multivariate definitions. Using the property of geodesic convexity on graphs, a median-like center for distributions on graphs is constructed and applied to ranking data as well as multivariate data spanned by the minimal spanning tree.
"Multidimensional medians arising from geodesics on graphs." Ann. Statist. 25 (2) 478 - 494, April 1997. https://doi.org/10.1214/aos/1031833660