  The betweenness centrality (BC) is an important quantity for understanding
the structure of complex large networks. However, its calculation is in general
difficult and known in simple cases only. In particular, the BC has been
exactly computed for graphs constructed over a set of $N$ points in the
infinite density limit, displaying a universal behavior. We reconsider this
calculation and propose an expansion for large and finite densities. We compute
the lowest non-trivial order and show that it encodes how straight are shortest
paths and is therefore non-universal and depends on the graph considered. We
compare our analytical result to numerical simulations obtained for various
graphs such as the minimum spanning tree, the nearest neighbor graph, the
relative neighborhood graph, the random geometric graph, the Gabriel graph, or
the Delaunay triangulation. We show that in most cases the agreement with our
analytical result is excellent even for densities of points that are relatively
low. This method and our results provide a framework for understanding and
computing this important quantity in large spatial networks.
