betweenness_centrality

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 \(v \in {s, t}\), \(\sigma(s, t|v) = 0\) [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-1)(n-2)) for graphs, and 1/((n-1)(n-2)) 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.

endpointsbool, optional

If True include the endpoints in the shortest path counts.

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:

nodesdictionary

Dictionary of nodes with betweenness centrality as the value.

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.

The total number of paths between source and target is counted differently for directed and undirected graphs. Directed paths are easy to count. Undirected paths are tricky: should a path from “u” to “v” count as 1 undirected path or as 2 directed paths?

For betweenness_centrality we report the number of undirected paths when G is undirected.

For betweenness_centrality_subset the reporting is different. If the source and target subsets are the same, then we want to count undirected paths. But if the source and target subsets differ – for example, if sources is {0} and targets is {1}, then we are only counting the paths in one direction. They are undirected paths but we are counting them in a directed way. To count them as undirected paths, each should count as half a path.

References

[1]

Ulrik Brandes: A Faster Algorithm for Betweenness Centrality. Journal of Mathematical Sociology 25(2):163-177, 2001. https://doi.org/10.1080/0022250X.2001.9990249

[2] (1,2)

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

[3]

Ulrik Brandes and Christian Pich: Centrality Estimation in Large Networks. International Journal of Bifurcation and Chaos 17(7):2303-2318, 2007. https://dx.doi.org/10.1142/S0218127407018403

[4]

Linton C. Freeman: A set of measures of centrality based on betweenness. Sociometry 40: 35–41, 1977 https://doi.org/10.2307/3033543