Compute the shortestpath betweenness centrality for nodes.
Betweenness centrality of a node is the sum of the fraction of allpairs shortest paths that pass through :
where is the set of nodes, is the number of shortest paths, and is the number of those paths passing through some node other than . If , , and if , [R155].
Parameters :  G : graph
k : int, optional (default=None)
normalized : bool, optional
weight : None or string, optional
endpoints : bool, optional


Returns :  nodes : dictionary

See also
Notes
The algorithm is from Ulrik Brandes [R154]. See [R155] for details on algorithms for variations and related metrics.
For approximate betweenness calculations set k=#samples to use k nodes (“pivots”) to estimate the betweenness values. For an estimate of the number of pivots needed see [R156].
For weighted graphs the edge weights must be greater than zero. Zero edge weights can produce an infinite number of equal length paths between pairs of nodes.
References
[R154]  (1, 2) A Faster Algorithm for Betweenness Centrality. Ulrik Brandes, Journal of Mathematical Sociology 25(2):163177, 2001. http://www.inf.unikonstanz.de/algo/publications/bfabc01.pdf 
[R155]  (1, 2, 3) Ulrik Brandes: On Variants of ShortestPath Betweenness Centrality and their Generic Computation. Social Networks 30(2):136145, 2008. http://www.inf.unikonstanz.de/algo/publications/bvspbc08.pdf 
[R156]  (1, 2) Ulrik Brandes and Christian Pich: Centrality Estimation in Large Networks. International Journal of Bifurcation and Chaos 17(7):23032318, 2007. http://www.inf.unikonstanz.de/algo/publications/bpceln06.pdf 