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edge_betweenness_centrality¶
- edge_betweenness_centrality(G, normalized=True, weight=None)¶
Compute betweenness centrality for edges.
Betweenness centrality of an edge \(e\) is the sum of the fraction of all-pairs shortest paths that pass through \(e\):
\[c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, t|e)}{\sigma(s, t)}\]where \(V\) is the set of nodes,`sigma(s, t)` is the number of shortest \((s, t)\)-paths, and \(\sigma(s, t|e)\) is the number of those paths passing through edge \(e\) [R186].
Parameters : G : graph
A NetworkX graph
normalized : bool, optional
If True the betweenness values are normalized by \(2/(n(n-1))\) for graphs, and \(1/(n(n-1))\) 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 [R185].
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
[R185] (1, 2) A Faster Algorithm for Betweenness Centrality. Ulrik Brandes, Journal of Mathematical Sociology 25(2):163-177, 2001. http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf [R186] (1, 2) Ulrik Brandes: On Variants of Shortest-Path Betweenness Centrality and their Generic Computation. Social Networks 30(2):136-145, 2008. http://www.inf.uni-konstanz.de/algo/publications/b-vspbc-08.pdf