As stated by Wikipedia, the betweenness centrality of a node $v$ is given by the expression:
$$g(v)=\sum _{{s\neq v\neq t}}{\frac {\sigma _{{st}}(v)}{\sigma _{{st}}}}$$
where $\sigma_{st}$ is the total number of shortest paths from node s to node t and $\sigma_{st}(v)$ is the number of those paths that pass through v.
The calculation may be rescaled by $(N-1)(N-2)/2$ for undirected graphs, where $N$ is the number of nodes. Where does this normalization constant come from?
Best Answer
It's the binomial coefficient $\binom{N-1}{2}=\frac{(N-1)(N-2)}{2}$.
The binomial coefficient $\binom{N}{K}$ tells us how many ways you can take $N$ objects $K$ at a time (in any order), so $\binom{N}{2}$ tells us how many different pairs there are among $N$ objects, and $\binom{N}{3}$ tells us how many triplets there are, etc.
In the context of betweenness centrality, we use $N-1$ because this is the number of nodes in the graph, excluding node $t$. We exclude node $t$ because betweenness measures shortest paths between two nodes when those paths do not start or stop at node $t$.
Betweenness centrality uses $K=2$ because one node is the origin of the shortest path and a second node is the destination of the shortest path, so betweenness centrality considers pairs of origin/destination nodes.