scholarly journals Percolation and Connectivity in AB Random Geometric Graphs

2012 ◽  
Vol 44 (1) ◽  
pp. 21-41 ◽  
Author(s):  
Srikanth K. Iyer ◽  
D. Yogeshwaran
Keyword(s):  
Point Processes ◽  
Nearest Neighbor ◽  
Geometric Graph ◽  
Boolean Model ◽  
Unit Cube ◽  
Poisson Point Processes ◽  
Random Geometric Graphs ◽  
Asymptotic Bounds ◽  
Discrete Lattices ◽  
Critical Intensity

Given two independent Poisson point processes Φ(1), Φ(2) in , the AB Poisson Boolean model is the graph with the points of Φ(1) as vertices and with edges between any pair of points for which the intersection of balls of radius 2r centered at these points contains at least one point of Φ(2). This is a generalization of the AB percolation model on discrete lattices. We show the existence of percolation for all d ≥ 2 and derive bounds for a critical intensity. We also provide a characterization for this critical intensity when d = 2. To study the connectivity problem, we consider independent Poisson point processes of intensities n and τn in the unit cube. The AB random geometric graph is defined as above but with balls of radius r. We derive a weak law result for the largest nearest-neighbor distance and almost-sure asymptotic bounds for the connectivity threshold.

scholarly journals Percolation and Connectivity in AB Random Geometric Graphs

2012 ◽  
Vol 44 (01) ◽  
pp. 21-41 ◽  
Author(s):  
Srikanth K. Iyer ◽  
D. Yogeshwaran
Keyword(s):  
Point Processes ◽  
Nearest Neighbor ◽  
Geometric Graph ◽  
Boolean Model ◽  
Unit Cube ◽  
Poisson Point Processes ◽  
Random Geometric Graphs ◽  
Asymptotic Bounds ◽  
Discrete Lattices ◽  
Critical Intensity

Given two independent Poisson point processes Φ(1), Φ(2) in , the AB Poisson Boolean model is the graph with the points of Φ(1) as vertices and with edges between any pair of points for which the intersection of balls of radius 2r centered at these points contains at least one point of Φ(2). This is a generalization of the AB percolation model on discrete lattices. We show the existence of percolation for all d ≥ 2 and derive bounds for a critical intensity. We also provide a characterization for this critical intensity when d = 2. To study the connectivity problem, we consider independent Poisson point processes of intensities n and τn in the unit cube. The AB random geometric graph is defined as above but with balls of radius r. We derive a weak law result for the largest nearest-neighbor distance and almost-sure asymptotic bounds for the connectivity threshold.

scholarly journals Percolation on the Information-Theoretically Secure Signal to Interference Ratio Graph

2014 ◽  
Vol 51 (04) ◽  
pp. 910-920
Author(s):  
Rahul Vaze ◽  
Srikanth Iyer
Keyword(s):  
Point Processes ◽  
Interference Suppression ◽  
Percolation Model ◽  
The Other ◽  
Continuum Percolation ◽  
Directed Edge ◽  
Connected Component ◽  
Poisson Point Processes ◽  
Subcritical Regime ◽  
Critical Intensity

We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes in R 2 of intensities λ and λ E , respectively. A directed edge from one legitimate node A to another legitimate node B exists provided that the strength of the signal transmitted from node A that is received at node B is higher than that received at any eavesdropper node. The strength of the signal received at a node from a legitimate node depends not only on the distance between these nodes, but also on the location of the other legitimate nodes and an interference suppression parameter γ. The graph is said to percolate when there exists an infinitely connected component. We show that for any finite intensity λ E of eavesdropper nodes, there exists a critical intensity λ c < ∞ such that for all λ > λ c the graph percolates for sufficiently small values of the interference parameter. Furthermore, for the subcritical regime, we show that there exists a λ0 such that for all λ < λ0 ≤ λ c a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.

scholarly journals Percolation on the Information-Theoretically Secure Signal to Interference Ratio Graph

2014 ◽  
Vol 51 (4) ◽  
pp. 910-920 ◽  
Author(s):  
Rahul Vaze ◽  
Srikanth Iyer
Keyword(s):  
Point Processes ◽  
Interference Suppression ◽  
Percolation Model ◽  
The Other ◽  
Continuum Percolation ◽  
Directed Edge ◽  
Connected Component ◽  
Poisson Point Processes ◽  
Subcritical Regime ◽  
Critical Intensity

We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes in R2 of intensities λ and λE, respectively. A directed edge from one legitimate node A to another legitimate node B exists provided that the strength of the signal transmitted from node A that is received at node B is higher than that received at any eavesdropper node. The strength of the signal received at a node from a legitimate node depends not only on the distance between these nodes, but also on the location of the other legitimate nodes and an interference suppression parameter γ. The graph is said to percolate when there exists an infinitely connected component. We show that for any finite intensity λE of eavesdropper nodes, there exists a critical intensity λc < ∞ such that for all λ > λc the graph percolates for sufficiently small values of the interference parameter. Furthermore, for the subcritical regime, we show that there exists a λ0 such that for all λ < λ0 ≤ λc a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.

scholarly journals Continuum AB percolation and AB random geometric graphs

2014 ◽  
Vol 51 (A) ◽  
pp. 333-344 ◽  
Author(s):  
Mathew D. Penrose
Keyword(s):  
Point Processes ◽  
Law Of Large Numbers ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Strong Law ◽  
Random Geometric Graphs ◽  
Large Numbers ◽  
The One ◽  
Distance Parameter

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ and μ. We show for d ≥ 2 that if λ is supercritical for the one-type random geometric graph with distance parameter 2r, there exists μ such that (λ, μ) is supercritical (this was previously known for d = 2). For d = 2, we also consider the restriction of this graph to points in the unit square. Taking μ = τ λ for fixed τ, we give a strong law of large numbers as λ → ∞ for the connectivity threshold of this graph.

scholarly journals On the critical threshold for continuum AB percolation

2018 ◽  
Vol 55 (4) ◽  
pp. 1228-1237
Author(s):  
David Dereudre ◽  
Mathew Penrose
Keyword(s):  
Percolation Threshold ◽  
Point Processes ◽  
Critical Value ◽  
Geometric Graph ◽  
Boolean Model ◽  
Critical Threshold ◽  
Random Geometric Graph ◽  
Fixed Constant ◽  
Distance Parameter ◽  
Poisson Boolean Model

Abstract Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ,μ. For any λ>0 we consider the percolation threshold μc(λ) associated to the parameter μ. Denoting by λc the percolation threshold for the standard Poisson Boolean model with radii r, we show the lower bound μc(λ)≥clog(c∕(λ−λc)) for any λ>λc with c>0 a fixed constant. In particular, there is no phase transition in μ at the critical value of λ, that is, μc(λc) =∞.

scholarly journals Continuum AB percolation and AB random geometric graphs

2014 ◽  
Vol 51 (A) ◽  
pp. 333-344 ◽  
Author(s):  
Mathew D. Penrose
Keyword(s):  
Point Processes ◽  
Law Of Large Numbers ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Strong Law ◽  
Random Geometric Graphs ◽  
Large Numbers ◽  
The One ◽  
Distance Parameter

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ and μ. We show for d ≥ 2 that if λ is supercritical for the one-type random geometric graph with distance parameter 2r, there exists μ such that (λ, μ) is supercritical (this was previously known for d = 2). For d = 2, we also consider the restriction of this graph to points in the unit square. Taking μ = τ λ for fixed τ, we give a strong law of large numbers as λ → ∞ for the connectivity threshold of this graph.

scholarly journals Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

2020 ◽  
pp. 1-14
Author(s):  
SHOTA OSADA
Keyword(s):  
Point Processes ◽  
Duke Math ◽  
Random Point ◽  
Determinantal Point Processes ◽  
Fredholm Determinants ◽  
Poisson Point Processes ◽  
Translation Invariant ◽  
Ergodic Properties ◽  
Tree Representations ◽  
Invariant Kernels

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

scholarly journals A generalization of Matérn hard-core processes with applications to max-stable processes

2020 ◽  
Vol 57 (4) ◽  
pp. 1298-1312
Author(s):  
Martin Dirrler ◽  
Christopher Dörr ◽  
Martin Schlather
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Hard Core ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Poisson Point Processes ◽  
Original Process ◽  
Mixed Moving Maxima

AbstractMatérn hard-core processes are classical examples for point processes obtained by dependent thinning of (marked) Poisson point processes. We present a generalization of the Matérn models which encompasses recent extensions of the original Matérn hard-core processes. It generalizes the underlying point process, the thinning rule, and the marks attached to the original process. Based on our model, we introduce processes with a clear interpretation in the context of max-stable processes. In particular, we prove that one of these processes lies in the max-domain of attraction of a mixed moving maxima process.

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