Maintaining the duality of closeness and betweenness centrality

We show that the first of these ideas - the duality - is not only true in a general conceptual sense but also in precise mathematical terms. This becomes apparent when the two indices are expressed in terms of a shared dyadic dependency relation. We also show that the second idea - the shortest paths - is false because it is not preserved when the indices are generalized using the standard definition of shortest paths in valued graphs. This unveils that closeness-as-independence is in fact different from closeness-as-efficiency, and we propose a variant notion of distance that maintains the duality of closeness-as-independence with betweenness also on valued relations.