networkx.algorithms.centrality.edge_betweenness_centrality¶

edge_betweenness_centrality(G, k=None, normalized=True, weight=None, seed=None)[source]¶

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 $e$

$c_B(e) =\sum_{s,t \in V} \frac{\sigma(s, t|e)}{\sigma(s, t)}$

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 $e$ [2].

Parameters:	G (graph) – A NetworkX graph. k (int, optional (default=None)) – If k is not None use k node samples to estimate betweenness. The value of k <= n where n is the number of nodes in the graph. Higher values give better approximation. 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 (default=None)) – If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight.
Returns:	edges – Dictionary of edges with betweenness centrality as the value.
Return type:	dictionary

See also

betweenness_centrality(), edge_load()

Notes

The algorithm is from Ulrik Brandes [1].

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

[1]	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

[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