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 $v \in {s, t}$ , $\sigma(s, t|v) = 0$ [R172].

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.