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“Maintaining the Duality of Closeness and Betweenness Centrality”

Ulrik Brandes, Stephen Borgatti, and Linton Freeman have an interesting paper in the latest volume of Social Networks: “Maintaining the Duality of Closeness and Betweenness Centrality.” Here’s the abstract:

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.

Ulrik Brandes, Stephen P. Borgatti, and Linton C. Freeman, “Maintaining the Duality of Closeness and Betweenness Centrality,” Social Networks 44 (2016): 153-159.

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