Betweenness centrality in dense spatial networks

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.