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betweenness_centrality

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

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 \(v \in {s, t}\), \(\sigma(s, t|v) = 0\) [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 \(2/((n-1)(n-2))\) 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 of nodes with betweenness centrality as the value.
Return type:

dictionary

See also

edge_betweenness_centrality(), load_centrality()

Notes

The algorithm is from Ulrik Brandes [1]. See [4] for the original first published version and [2] 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 [3].

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]Ulrik Brandes: A Faster Algorithm for Betweenness Centrality. Journal of Mathematical Sociology 25(2):163-177, 2001. http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf
[2](1, 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
[3]Ulrik Brandes and Christian Pich: Centrality Estimation in Large Networks. International Journal of Bifurcation and Chaos 17(7):2303-2318, 2007. http://www.inf.uni-konstanz.de/algo/publications/bp-celn-06.pdf
[4]Linton C. Freeman: A set of measures of centrality based on betweenness. Sociometry 40: 35–41, 1977 http://moreno.ss.uci.edu/23.pdf