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# 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) – If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. edges – Dictionary of edges with betweenness centrality as the value. dictionary

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