# 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
Ggraph

A NetworkX graph.

kint, 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.

normalizedbool, 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.

weightNone or string, optional (default=None)

If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. Weights are used to calculate weighted shortest paths, so they are interpreted as distances.

seedinteger, random_state, or None (default)

Indicator of random number generation state. See Randomness. Note that this is only used if k is not None.

Returns
edgesdictionary

Dictionary of edges with betweenness centrality as the value.

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. https://doi.org/10.1080/0022250X.2001.9990249

2

Ulrik Brandes: On Variants of Shortest-Path Betweenness Centrality and their Generic Computation. Social Networks 30(2):136-145, 2008. https://doi.org/10.1016/j.socnet.2007.11.001