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edge_betweenness_centrality¶

edge_betweenness_centrality
(G, normalized=True, weight=None)[source]¶ Compute betweenness centrality for edges.
Betweenness centrality of an edge \(e\) is the sum of the fraction of allpairs shortest paths that pass through \(e\):
\[c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, te)}{\sigma(s, t)}\]where \(V\) is the set of nodes,`sigma(s, t)` is the number of shortest \((s, t)\)paths, and \(\sigma(s, te)\) is the number of those paths passing through edge \(e\) [R189].
Parameters: G : graph
A NetworkX graph
normalized : bool, optional
If True the betweenness values are normalized by \(2/(n(n1))\) for graphs, and \(1/(n(n1))\) for directed graphs where \(n\) is the number of nodes in G.
weight : None or string, optional
If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight.
Returns: edges : dictionary
Dictionary of edges with betweenness centrality as the value.
See also
Notes
The algorithm is from Ulrik Brandes [R188].
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
[R188] (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 [R189] (1, 2) 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