Connectivity of a Gaussian network

Abstract

Following Etherington, Hoge and Parkes, we consider a network consisting of (approximately) N transceivers in the plane R 2 distributed randomly with density given by a Gaussian distribution about the origin, and assume each transceiver can communicate with all other transceivers within distance s. We give bounds for the distance from the origin to the furthest transceiver connected to the origin, and that of the closest transceiver that is not connected to the origin.

Which Sampling Densities are Suitable for Spectral Clustering on Unbounded Domains?

“…Theorem 4.3 below indicates that there exists an asymptotic threshold such that, for values of r n decaying faster than this threshold, the graph G(P n , r n ) is disconnected w.h.p., while for values of r n decaying slower than this same threshold, concentration inequalities hold on small cubes partitioning R d (which imply in particular the connectivity of the graph). This threshold agrees with the results found in [14] and [3], which studied the special case where the density is a Gaussian.…”

“…Note in particular that the above concentration inequalities imply connectivity of the graph G(P n , r n ), which could already be deduced by Theorem 4.5 of [11]. Note also that if the sampling density is a Gaussian, i.e., up to multiplicative constants, say ψ(z) ∼ z 2 , ψ ′ (z) ∼ z and ψ ← (z) ∼ z 1/2 , then the connectivity threshold obtained from Theorem 4.3, i.e., τ ∼ log log n √ log n , agrees with the results of [14] and [3]. The authors of [11] establish the contractibility of the union of the random balls ∪ x∈Pn B(x, r n ) under the same asymptotic condition as in Theorem 4.3 above:…”

“…See [14] for a classic exposition on random geometric graphs, and [2] for a survey on the topic. In the setting where the domain is R d , both [14] and [3] investigate the special case where the sampling density is a Gaussian, and identify sharp connectivity thresholds. However, to our knowledge, random geometric graphs with other sampling densities on R d have not been studied, and it is not known how the connectivity of the graphs are affected by the choice of sampling density on R d .…”

Percolation, Connectivity, Coverage and Colouring of Random Geometric Graphs

“…This model G Gauss r (n) was analyzed in detail by Balister, Bollobás, Sarkar and Walters [8], who determined the threshold for connectivity.…”

The Influence of Communication Range on Connectivity for Resilient Wireless Sensor Networks Using a Probabilistic Approach

“…random variable with normal probability density ri~Nfalse(μ,σnormal2false). Reference [32] adopts the model that Poisson intensity is given by a normal distribution; then it obtains the asymptotic bound of range that all nodes in this area are connected to the origin. Reference [33] considers nodes are placed according to a shot-noise Cox process rather than uniform deployment.…”