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betweenness_centrality(G, k=None, normalized=True, weight=None, endpoints=False, seed=None)

Compute the shortest-path betweenness centrality for nodes.

Betweenness centrality of a node v is the sum of the fraction of all-pairs shortest paths that pass through v:

c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, t|v)}{\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|v) is the number of those paths passing through some node v other than s, t. If s = t, \sigma(s, t) = 1, and if v \in {s, t}, \sigma(s, t|v) = 0 [R155].

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-1)(n-2)) for graphs, and 1/((n-1)(n-2)) 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.

endpoints : bool, optional

If True include the endpoints in the shortest path counts.

Returns :

nodes : dictionary

Dictionary of nodes with betweenness centrality as the value.


The algorithm is from Ulrik Brandes [R154]. See [R155] for details on algorithms for variations and related metrics.

For approximate betweenness calculations set k=#samples to use k nodes (“pivots”) to estimate the betweenness values. For an estimate of the number of pivots needed see [R156].

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.


[R154](1, 2) A Faster Algorithm for Betweenness Centrality. Ulrik Brandes, Journal of Mathematical Sociology 25(2):163-177, 2001.
[R155](1, 2, 3) Ulrik Brandes: On Variants of Shortest-Path Betweenness Centrality and their Generic Computation. Social Networks 30(2):136-145, 2008.
[R156](1, 2) Ulrik Brandes and Christian Pich: Centrality Estimation in Large Networks. International Journal of Bifurcation and Chaos 17(7):2303-2318, 2007.