Given a graph $G = (V, E)$ and a vertex $v \in V$, calculate the betweenness centrality of vertex $v$ (or the proportion of shortest paths that go through $v$), i.e. $BC(v) := \sum\limits_{s\neq t \neq v \in V} \frac{\sigma_{st}(v)}{\sigma_{st}}$ where $\sigma_{st}(v)$ is the number of shortest paths from $s$ to $t$ that go through $v$ and $\sigma_{st}$ is the number of shortest paths from $s$ to $t$.