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.

Isolation and Connectivity in Random Geometric Graphs with Self-similar Intensity Measures

“…Free propagation leads to (inverse square law); more cluttered environments have and recovers the RGG. There are many more complicated fading models leading to H ( r ) involving a number of special functions [11], however these can often be approximated by the above form [7]. Virtually all previous literature uses a uniform (Lebesgue) intensity measure, perhaps on a finite domain.…”

“…Connection functions can be constructed empirically with any spatial network for which the links can be assumed to be random [46]; several functions that have arisen from mathematical or physical arguments are given in Ref. [11]. …”

“…[13] considers rectangular domains, and Refs. [9, 11] consider arbitrary convex polygons and some domains that are polyhedral or have curved boundaries. Poisson distribution of isolated nodes was proved in Refs.…”

Spatial Transmission Models: A Taxonomy and Framework

“…Random movement of nodes on the geometric space has been successfully used to model a sexual interaction network where collisions between nodes generate connections (González, Lind, & Herrmann, ). Further models of connection mechanisms, weakening the deterministic connection when another node is within distance d of the other, and instead modeling “soft” or “probabilistic” connections, have been reviewed by Dettmann and Gerogiou (). A random geometric graph is shown in Fig.…”

“…; Riley, ), but also in fields as diverse as connectivity on wireless networks (Andrews, Ganti, Haenggi, Jindal, & Weber, ), the spread of fire (Rothermel, ), marketing (Bradlow et al., ), and the diffusion of technology (Berger, ). Risk analysis models that model the transmission of threats around a system include analytical models for epidemiological studies (Eisenberg, Seto, Olivieri, & Spear, ; Moreno & Alvar, ; Zagmutt, Schoenbaum, & Hill, ), cellular automata models for nuclear terrorism (Atkinson, Cao, & Wein, ), soil contamination (Cox, ) and species invasion (Sikder, Mal‐Sarkar, & Mal, ), domain‐based studies of exposure to ozone (Fann et al., ), network models for communication (Dettmann & Georgiou, ), power networks (Zio & Sansavini, ), and in the social amplification of risk (Kasperson et al., ).…”