Maintaining the duality of closeness and betweenness centrality

Betweenness centrality is generally regarded as a measure of others' dependence on a given node, and therefore as a measure of potential control. Closeness centrality is usually interpreted either as a measure of access efficiency or of independence from potential control by intermediaries. Betweenness and closeness are commonly assumed to be related for two reasons: first, because of their conceptual duality with respect to dependency, and second, because both are defined in terms of shortest paths. 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.