Towards Improving Brandes' Algorithm for Betweenness Centrality
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Betweenness centrality, measuring how many shortest paths pass through a vertex, is one of the most important network analysis concepts for assessing the (relative) importance of a vertex. The famous state-of-art algorithm of Brandes [2001] computes the betweenness centrality of all vertices in O(mn) worst-case time on an n-vertex and m-edge graph. In practical follow-up work, significant empirical speedups were achieved by preprocessing degree-one vertices. We extend this by showing how to also deal with degree-two vertices (turning out to be much richer in mathematical structure than the case of degree-one vertices). For our new betweenness centrality algorithm we prove the running time upper bound O(kn), where k is the size of a minimum feedback edge set of the input graph.
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