  Random geometric networks consist of 1) a set of nodes embedded randomly in a
bounded domain $\mathcal{V} \subseteq \mathbb{R}^d$ and 2) links formed
probabilistically according to a function of mutual Euclidean separation. We
quantify how often all paths in the network characterisable as topologically
`shortest' contain a given node (betweenness centrality), deriving an
expression in terms of a known integral whenever 1) the network boundary is the
perimeter of a disk and 2) the network is extremely dense. Our method shows how
similar formulas can be obtained for any convex geometry. Numerical
corroboration is provided, as well as a discussion of our formula's potential
use for cluster head election and boundary detection in densely deployed
wireless ad hoc networks.
