betweenness_centrality_subset(G, sources, targets, normalized=False, weight=None)
$$c_B(v) =\sum_{s\in S, t \in T} \frac{\sigma(s, t|v)}{\sigma(s, t)}$$
where $S$ is the set of sources, $T$ is the set of targets, $\sigma(s, t)$ 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 $v \in {s, t}$, $\sigma(s, t|v) = 0$ .
The basic algorithm is from .
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 normalization might seem a little strange but it is designed to make betweenness_centrality(G) be the same as betweenness_centrality_subset(G,sources=G.nodes(),targets=G.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.
A NetworkX graph.
Nodes to use as sources for shortest paths in betweenness
Nodes to use as targets for shortest paths in betweenness
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.
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.
Dictionary of nodes with betweenness centrality as the value.
Compute betweenness centrality for a subset of nodes.
Hover to see nodes names; edges to Self not shown, Caped at 50 nodes.
Using a canvas is more power efficient and can get hundred of nodes ; but does not allow hyperlinks; , arrows or text (beyond on hover)
SVG is more flexible but power hungry; and does not scale well to 50 + nodes.
All aboves nodes referred to, (or are referred from) current nodes; Edges from Self to other have been omitted (or all nodes would be connected to the central node "self" which is not useful). Nodes are colored by the library they belong to, and scaled with the number of references pointing them