The edge betweenness centrality of an edge 
in a graph is a measure of how often
lies on shortest paths between pairs of vertices.
It is commonly defined by
 |
where 
is the number of shortest paths from 
to 
, and 
is the number of such paths containing 
.
Edge betweenness centrality is used, for example, in divisive community-detection algorithms that remove high-betweenness edges separating dense parts of a network (Girvan and Newman 2002). High-edge-betweenness edges often act as bridges or load-bearing links between otherwise more cohesive regions, such as important roads, power-grid connections, or metabolic interactions. It is the edge analogue of betweenness centrality, replacing the question of which vertices lie on many shortest paths by the question of which edges lie on many shortest paths.
Edge betweenness centrality is implemented in the Wolfram Language as EdgeBetweennessCentrality[g], and precomputed values for many named graphs can be obtained using GraphData[graph, "EdgeBetweennessCentralities"].
