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arXiv · 1411.3617

Random geometric graphs with general connection functions

Carl P. Dettmann·Orestis Georgiou
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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.

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Carl P. Dettmann, Orestis Georgiou. 2016-01-20. Random geometric graphs with general connection functions. https://doi.org/10.1103/physreve.93.032313

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