betweenness_centrality¶

betweenness_centrality
(G, k=None, normalized=True, weight=None, endpoints=False, seed=None)[source]¶ Compute the shortestpath betweenness centrality for nodes.
Betweenness centrality of a node \(v\) is the sum of the fraction of allpairs shortest paths that pass through \(v\)
\[c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, tv)}{\sigma(s, t)}\]where \(V\) is the set of nodes, \(\sigma(s, t)\) is the number of shortest \((s, t)\)paths, and \(\sigma(s, tv)\) is the number of those paths passing through some node \(v\) other than \(s, t\). If \(s = t\), \(\sigma(s, t) = 1\), and if \(v \in {s, t}\), \(\sigma(s, tv) = 0\) [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 \(2/((n1)(n2))\) for graphs, and \(1/((n1)(n2))\) 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
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):163177, 2001. http://www.inf.unikonstanz.de/algo/publications/bfabc01.pdf [2] (1, 2) Ulrik Brandes: On Variants of ShortestPath Betweenness Centrality and their Generic Computation. Social Networks 30(2):136145, 2008. http://www.inf.unikonstanz.de/algo/publications/bvspbc08.pdf [3] Ulrik Brandes and Christian Pich: Centrality Estimation in Large Networks. International Journal of Bifurcation and Chaos 17(7):23032318, 2007. http://www.inf.unikonstanz.de/algo/publications/bpceln06.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