I'm working with graphs of a very large size (> 60k vertices), and want to speed up B.C. measurements. It is defined here: http://en.wikipedia.org/wiki/Betweenness_centrality The algorithm that I am using to compute B.C. is Brandes' algorithm: http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf
For undirected graphs, would running the Betweenness centrality algorithm on each of its connected components (and then combine the results) give the exact same answer as computing on the whole graph at once? I would think so (but don't have a proof) because different connected components are not related.