Betweenness centrality in dense random geometric networks

Random geometric networks are mathematical structures consisting of a set of nodes placed randomly within a bounded set V ⊆ ℝ
^d
mutually coupled with a probability dependent on their Euclidean separation, and are the classic model used within the expanding field of ad hoc wireless networks. In order to rank the importance of the network's communicating nodes, we consider the well established `betweenness' centrality measure (quantifying how often a node is on a shortest path of links between any pair of nodes), providing an analytic treatment of betweenness within a random graph model by deriving a closed form expression for the expected betweenness of a node placed within a dense random geometric network formed inside a disk of radius R. We confirm this with numerical simulations, and discuss the importance of the formula for mitigating the `boundary effect' connectivity phenomenon, for cluster head node election protocol design and for detecting the location of a network's `vulnerability backbone'.