Betweenness Centrality in Dense Random Geometric Networks

TL;DR

This paper derives an analytical expression for betweenness centrality in dense random geometric networks, applicable to convex shapes, with potential applications in wireless network management.

Contribution

It provides a new analytical formula for betweenness centrality in dense geometric networks with convex boundaries, extending previous work to more general shapes.

Findings

Derived an explicit integral formula for betweenness centrality

Validated the formula through numerical simulations

Discussed applications in wireless ad hoc networks

Abstract

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.