Random geometric graphs with general connection functions

TL;DR

This paper analyzes the connectivity of random geometric graphs with general connection functions, providing formulas for connection probability in dense networks and validating results with simulations.

Contribution

It introduces a general framework for calculating connection probabilities in dense random geometric graphs with arbitrary connection functions, extending previous models.

Findings

Connection probability expressed via boundary contributions and moments of the connection function

Good agreement with special cases and numerical simulations

Framework applicable to wireless networks and other applications

Abstract

In the original (1961) Gilbert model of random geometric graphs, nodes are placed according to a Poisson point process, and links formed between those within a fixed range. Motivated by wireless ad-hoc networks "soft" or "probabilistic" connection models have recently been introduced, involving a "connection function" H(r) that gives the probability that two nodes at distance r are linked (directly connect). In many applications (not only wireless networks), it is desirable that the graph is connected, that is every node is linked to every other node in a multihop fashion. Here, the connection probability of a dense network in a convex domain in two or three dimensions is expressed in terms of contributions from boundary components, for a very general class of connection functions. It turns out that only a few quantities such as moments of the connection function appear. Good agreement is found with special cases from previous studies and with numerical simulations.