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 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 \(e\) [R189].

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.