Centrality of an edge of a graph is proposed to be viewed as a degree of global sensitivity of a graph distance function (i. e., a graph metric) on the weight of the considered edge. For different choices of distance function, contact is made with several previous ideas of centrality, whence their different characteristics are clarified, and strengths or short-comings are indicated, via selected examples. The centrality based on "resistance distance" exhibits several nice features, and might be termed "amongness" centrality. 2010 Springer Science+Business Media, LLC.
