title: Random geometric graphs with general connection functions.
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.
authors.affiliation: School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom.
authors.forename: Carl P
authors.initials: CP
authors.lastname: Dettmann
authors.affiliation: Toshiba Telecommuncations Research Laboratory, 32 Queens Square, Bristol BS1 4ND, United Kingdom.
authors.forename: Orestis
authors.initials: O
authors.lastname: Georgiou