Poisson Point Process Of Intensity Research Articles

Community detection on Euclidean random graphs

AbstractWe study the problem of community detection on Euclidean random geometric graphs where each vertex has two latent variables: a binary community label and a $\mathbb{R}^d$ valued location label which forms the support of a Poisson point process of intensity $\lambda $. A random graph is then drawn with edge probabilities dependent on both the community and location labels. In contrast to the stochastic block model (SBM) that has no location labels, the resulting random graph contains many more short loops due to the geometric embedding. We consider the recovery of the community labels, partial and exact, using the random graph and the location labels. We establish phase transitions for both sparse and logarithmic degree regimes and provide bounds on the location of the thresholds, conjectured to be tight in the case of exact recovery. We also show that the threshold of the distinguishability problem, i.e., the testing between our model and the null model without community labels exhibits no phase transition and in particular, does not match the weak recovery threshold (in contrast to the SBM).

Poisson–Voronoi Tessellation on a Riemannian Manifold

Abstract In this paper, we consider a Riemannian manifold $M$ and the Poisson–Voronoi tessellation generated by the union of a fixed point $x_0$ and a Poisson point process of intensity $\lambda $ on $M$. We obtain a two-term asymptotic expansion, when $\lambda $ goes to infinity, of the mean number of vertices of the Voronoi cell associated with $x_0$. The 1st term of the estimate is equal to the mean number of vertices in the Euclidean setting, while the 2nd term involves the scalar curvature of $M$ at $x_0$. This settles with the proper and rigorous frame the former 2D statement from [ 19] and extends it to higher dimension. The key tool for proving this result is a new change of variables formula of Blaschke–Petkantschin type in the Riemannian setting, which brings out the Ricci curvatures of the manifold.

Distribution of Cell Area in Bounded Poisson Voronoi Tessellations with Application to Secure Local Connectivity

Poisson Voronoi tessellations have been used in modeling many types of systems across different sciences, from geography and astronomy to telecommunications. The existing literature on the statistical properties of Poisson Voronoi cells is vast, however, little is known about the properties of Voronoi cells located close to the boundaries of a compact domain. In a domain with boundaries, some Voronoi cells would be naturally clipped by the boundary, and the cell area falling inside the deployment domain would have different statistical properties as compared to those of non-clipped Voronoi cells located in the bulk of the domain. In this paper, we consider the planar Voronoi tessellation induced by a homogeneous Poisson point process of intensity $\lambda\!>\!0$ in a quadrant, where the two half-axes represent boundaries. We show that the mean cell area is less than $\lambda^{-1}$ when the seed is located exactly at the boundary, and it can be larger than $\lambda^{-1}$ when the seed lies close to the boundary. In addition, we calculate the second moment of cell area at two locations for the seed: (i) at the corner of a quadrant, and (ii) at the boundary of the half-plane. We illustrate that the two-parameter Gamma distribution, with location-dependent parameters calculated using the method of moments, can be of use in approximating the distribution of cell area. As a potential application, we use the Gamma approximations to study the degree distribution for secure connectivity in wireless sensor networks deployed over a domain with boundaries.

Walking in a Planar Poisson–Delaunay Triangulation: Shortcuts in the Voronoi Path

Let [Formula: see text] be a planar Poisson point process of intensity [Formula: see text]. We give a new proof that the expected length of the Voronoi path between [Formula: see text] and [Formula: see text] in the Delaunay triangulation associated with [Formula: see text] is [Formula: see text] when [Formula: see text] goes to infinity; and we also prove that the variance of this length is [Formula: see text]. We investigate the length of possible shortcuts in this path, and define a shortened Voronoi path whose expected length can be expressed as an integral that is numerically evaluated to [Formula: see text]. The shortened Voronoi path has the property to be locally defined; and is shorter than the previously known locally defined paths in Delaunay triangulation such as the upper path whose expected length is [Formula: see text].

Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on $\mathbb{R}^{d}$

Nous considérons le modèle Booléen $Z$ sur $\mathbb{R}^{d}$ avec des grains compacts aléatoires de diamètres bornés, c’est-à-dire $Z:=\bigcup_{i\in\mathbb{N}}(Z_{i}+X_{i})$ où $\{X_{1},X_{2},\dots\}$ est un processus de Poisson d’intensité $t$ et $(Z_{1},Z_{2},\dots)$ est une suite i.i.d. de grains compacts (non nécessairement des boules) de diamètres p.s. bornés par une constante. Nous montrons une décroissance exponentielle dans le régime sous-critique, ce qui veut dire que le volume et le rayon du cluster d’un grain typique dans $Z$ a une queue exponentielle. Pour cela, nous adaptons des résultats de (A new proof of the sharpness of the phase transition for Bernoulli percolation on $\mathbb{Z}^{d}$ (2015) Preprint) et appliquons une nouvelle construction du cluster d’un grain typique avec des arguments issus des processus de branchement. Dans la seconde partie du papier, nous obtenons de nouvelles bornes inférieures pour le modèle booléen avec grains déterministes. Certaines des ces bornes sont rigoureuses, alors que d’autres sont obtenues par simulation. Les bornes obtenues par simulation sont fournies avec des intervalles de confiance et sont beaucoup plus précises que celles obtenues rigoureusement. Elles améliorent les résultats connus (J. Chem. Phys. 137 (2012) 074106) en dimension 6 et plus.

The random connection model: Connectivity, edge lengths, and degree distributions

AbstractConsider the random graph whose vertex set is a Poisson point process of intensity n on . Any two vertices are connected by an edge with probability , independently of all other edges, and independent of the other points of . d is the toroidal metric, r > 0 and is non‐increasing and . Under suitable conditions on g, almost surely, the critical parameter Mn for which does not have any isolated nodes satisfies . Let , and θ be the volume of the unit ball in . Then for all , is connected with probability approaching one as . The bound can be seen to be tight for the usual random geometric graph obtained by setting . We also prove some useful results on the asymptotic behavior of the length of the edges and the degree distribution in the connectivity regime. The results in this paper work for connection functions g that are not necessarily compactly supported but satisfy .

Surface order scaling in stochastic geometry

Let ${{ \mathcal{P}}}_{{\lambda}}:={{ \mathcal{P}}}_{{\lambda}{\kappa}}$ denote a Poisson point process of intensity ${\lambda}{\kappa}$ on $[0,1]^{d}$, $d\geq2$, with ${\kappa}$ a bounded density on $[0,1]^{d}$ and ${\lambda}\in(0,\infty)$. Given a closed subset ${ \mathcal{M}}\subset[0,1]^{d}$ of Hausdorff dimension $(d-1)$, we consider general statistics $\sum_{x\in{{ \mathcal{P}}}_{{\lambda}}}\xi(x,{{ \mathcal{P}}}_{{\lambda}},{ \mathcal{M}})$, where the score function $\xi$ vanishes unless the input $x$ is close to ${ \mathcal{M}}$ and where $\xi$ satisfies a weak spatial dependency condition. We give a rate of normal convergence for the rescaled statistics $\sum_{x\in{{ \mathcal{P}}}_{{\lambda}}}\xi({\lambda}^{1/d}x,{\lambda}^{1/d}{{ \mathcal{P}}}_{{\lambda}},{\lambda}^{1/d}{ \mathcal{M}})$ as ${\lambda}\to\infty$. When ${ \mathcal{M}}$ is of class $C^{2}$, we obtain weak laws of large numbers and variance asymptotics for these statistics, showing that growth is surface order, that is, of order $\mathrm{Vol} ({\lambda}^{1/d}{ \mathcal{M}})$. We use the general results to deduce variance asymptotics and central limit theorems for statistics arising in stochastic geometry, including Poisson–Voronoi volume and surface area estimators, answering questions in Heveling and Reitzner [Ann. Appl. Probab. 19 (2009) 719–736] and Reitzner, Spodarev and Zaporozhets [Adv. in Appl. Probab. 44 (2012) 938–953]. The general results also yield the limit theory for the number of maximal points in a sample.

Phase transition for finite-speed detection among moving particles

Consider the model where particles are initially distributed on Zd,d≥2, according to a Poisson point process of intensity λ>0, and are moving in continuous time as independent simple symmetric random walks. We study the escape versus detection problem, in which the target, initially placed at the origin of Zd,d≥2, and changing its location on the lattice in time according to some rule, is said to be detected if at some finite time its position coincides with the position of a particle. For any given S>0, we consider the case where the target can move with speed at most S, according to any continuous function and can adapt its motion based on the location of the particles. We show that, for any S>0, there exists a sufficiently small λ∗>0, so that if the initial density of particles λ<λ∗, then the target can avoid detection forever.

Asymptotics of Visibility in the Hyperbolic Plane

At each point of a Poisson point process of intensity λ in the hyperbolic plane, center a ball of bounded random radius. Consider the probability Pr that, from a fixed point, there is some direction in which one can reach distance r without hitting any ball. It is known (see Benjamini, Jonasson, Schramm and Tykesson (2009)) that if λ is strictly smaller than a critical intensity λgv thenPr does not go to 0 as r → ∞. The main result in this note shows that in the case λ=λgv, the probability of reaching a distance larger than r decays essentially polynomially, while if λ&gt;λgv, the decay is exponential. We also extend these results to various related models and we finally obtain asymptotic results in several situations.

Distribution of components in the $k$-nearest neighbour random geometric graph for $k$ below the connectivity threshold

Let $S_{n,k}$ denote the random geometric graph obtained by placing points inside a square of area $n$ according to a Poisson point process of intensity $1$ and joining each such point to the $k=k(n)$ points of the process nearest to it. In this paper we show that if $\mathbb{P}(S_{n,k} \textrm{ connected})>n^{-\gamma_1}$ then the probability that $S_{n,k}$ contains a pair of `small' components `close' to each other is $o(n^{-c_1})$ (in a precise sense of `small' and 'close'), for some absolute constants $\gamma_1>0$ and $c_1 >0$. This answers a question of Walters. (A similar result was independently obtained by Balister.) As an application of our result, we show that the distribution of the connected components of $S_{n,k}$ below the connectivity threshold is asymptotically Poisson.

Multicast Capacity Scaling Laws for Multihop Cognitive Networks

In this paper, we study multicast capacity for cognitive networks. We consider the cognitive network model consisting of two overlapping ad hoc networks, called the primary ad hoc network (PaN) and secondary ad hoc network (SaN), respectively. PaN and SaN operate on the same space and spectrum. For PaN (or SaN, respectively), we assume that primary (or secondary, respectively) nodes are placed according to a Poisson point process of intensity n (or m, respectively) over a unit square region. We randomly choose ns (or ms, respectively) nodes as the sources of multicast sessions in PaN (or SaN, respectively), and for each primary source vp (or secondary source vs, respectively), we pick uniformly at random nd primary nodes (or md secondary nodes, respectively) as the destinations of vp (or vs, respectively). Above all, we assume that PaN can adopt the optimal protocol in terms of the throughput. Our main work is to design the multicast strategy for SaN by which the optimal throughput can be achieved, without any negative impact on the throughput for PaN in order sense. Depending on nd and n, we choose the optimal one for PaN from two strategies called percolation strategy and connectivity strategy, respectively. Subsequently, we design the corresponding throughput-optimal strategy for SaN. We derive the regimes in terms of n, nd, m, and md in which the upper bounds on multicast capacities for PaN and SaN can be achieved simultaneously. Unicast and broadcast capacities for the cognitive network can be derived by our results as the special cases by letting nd = 1 (or md = 1) and nd = n - 1 (or md = m - 1), respectively, which enhances the generality of this work.

The scaling limit of Poisson-driven order statistics with applications in geometric probability

Let ηt be a Poisson point process of intensity t≥1 on some state space Y and let f be a non-negative symmetric function on Yk for some k≥1. Applying f to all k-tuples of distinct points of ηt generates a point process ξt on the positive real half-axis. The scaling limit of ξt as t tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the m-th smallest point of ξt is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener–Itô chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen–Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as k-flats, random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry.

Geometry of the Poisson Boolean model on a region of logarithmic width in the plane

Consider the region L = {(x, y): 0 ≤ y ≤ Clog(1 + x), x &amp;gt; 0} for a constant C &amp;gt; 0. We study the percolation and coverage properties of this region. For the coverage properties, we place a Poisson point process of intensity λ on the entire half space R + x R and associated with each Poisson point we place a box of a random side length ρ. Depending on the tail behaviour of the random variable ρ we exhibit a phase transition in the intensity for the eventual coverage of the region L. For the percolation properties, we place a Poisson point process of intensity λ on the region R 2. At each point of the process we centre a box of a random side length ρ. In the case ρ ≤ R for some fixed R &amp;gt; 0 we study the critical intensity λc of the percolation on L.

Geometry of the Poisson Boolean model on a region of logarithmic width in the plane

Consider the region L = {(x, y): 0 ≤ y ≤ Clog(1 + x), x &gt; 0} for a constant C &gt; 0. We study the percolation and coverage properties of this region. For the coverage properties, we place a Poisson point process of intensity λ on the entire half space R+ x R and associated with each Poisson point we place a box of a random side length ρ. Depending on the tail behaviour of the random variable ρ we exhibit a phase transition in the intensity for the eventual coverage of the region L. For the percolation properties, we place a Poisson point process of intensity λ on the region R2. At each point of the process we centre a box of a random side length ρ. In the case ρ ≤ R for some fixed R &gt; 0 we study the critical intensity λc of the percolation on L.

Multicast Throughput for Hybrid Wireless Networks under Gaussian Channel Model

We study the multicast capacity for hybrid wireless networks consisting of ordinary ad hoc nodes and base stations under Gaussian Channel model, which generalizes both the unicast and broadcast capacities for hybrid wireless networks. Assume that all ordinary ad hoc nodes transmit at a constant power P, and the power decays along the path, with attenuation exponent α >; 2. The data rate of a transmission is determined by the Signal to Interference plus Noise Ratio (SINR) at the receiver as Blog(1 + SINR). The ordinary ad hoc nodes are placed in the square region A(α) of area a according to a Poisson point process of intensity n/a. Then, m additional base stations (BSs) acting as the relaying communication gateways are placed regularly in the region A(a), and are connected by a high-band width wired network. Let a = n and a = 1, we construct the hybrid extended network (HEN) and hybrid dense network (HON), respectively. We choose randomly and independently ns ordinary ad hoc nodes to be the sources of multicast sessions. We assume that each multicast session has nd randomly chosen terminals. Three broad categories of multicast strategies are proposed. The first one is the hybrid strategy, i.e., the multihop scheme with BS-supported, which further consists of two types of strategies called connectivity strategy and percolation strategy, respectively. The second one is the ordinary ad hoc strategy, i.e., the multihop scheme without any BS-supported. The third one is the classical BS-based strategy under which any communication between two ordinary ad hoc nodes is relayed by some specific BSs. According to the different scenarios in terms of m, n, and nd, we select the optimal scheme from the three categories of strategies, and derive the achievable multicast throughput based on the optimal decision.

Lower bounds on lifetime of ultra wide band wireless sensor networks

The asymptotic lower bounds on the lifetime of time hopping impulse radio ultra wide band (TH-IR UWB) wireless sensor networks are derived using percolation theory arguments. It is shown that for static dense TH-IR UWB wireless sensor network, which sensor nodes are distributed in a square of unit area according to a Poisson point process of intensity n, the lower bound on the lifetime is $$ \Upomega \left( {\left( {{{\sqrt n } \mathord{\left/ {\vphantom {{\sqrt n } {\log \sqrt n }}} \right. \kern-\nulldelimiterspace} {\log \sqrt n }}} \right)^{\alpha - 2} } \right) $$ , where ? > 2 is the path loss exponent, thus dense TH-IR UWB wireless sensor network is fit to be employed in large-scale network. For static extended TH-IR UWB wireless sensor network which sensor nodes are distributed in a square $$ \left[ {0,\sqrt n } \right] \times \left[ {0,\sqrt n } \right] $$ according to a Poisson point process of unit intensity, the lower bound on the lifetime is $$ \Upomega \left( {{{\left( {\log \sqrt n } \right)^{2 - \alpha } } \mathord{\left/ {\vphantom {{\left( {\log \sqrt n } \right)^{2 - \alpha } } n}} \right. \kern-\nulldelimiterspace} n}} \right) $$ , therefore large-scale extended network will lead to shorten network lifetime. The results also indicate that the lower bound on the lifetime in the ideal case is longer than that of a static network by a factor of $$ n^{1/2} \left( {\log \sqrt n } \right)^{\alpha - 4} $$ . Hence mobility of sensor nodes can improve network lifetime.

Multivariate Normal Approximation in Geometric Probability

Consider a measure $\mu_\lambda = \sum_x \xi_x \delta_x$ where the sum is over points $x$ of a Poisson point process of intensity $\lambda$ on a bounded region in $d$-space, and $\xi_x$ is a functional determined by the Poisson points near to $x$, i.e. satisfying an exponential stabilization condition, along with a moments condition (examples include statistics for proximity graphs, germ-grain models and random sequential deposition models). A known general result says the $\mu_\lambda$-measures (suitably scaled and centred) of disjoint sets in $R^d$ are asymptotically independent normals as $\lambda \to \infty$; here we give an $O(\lambda^{-1/(2d + \epsilon)})$ bound on the rate of convergence. We illustrate our result with an explicit multivariate central limit theorem for the nearest-neighbour graph on Poisson points on a finite collection of disjoint intervals.

On a random directed spanning tree

We study the asymptotic properties of a minimal spanning tree formed by n points uniformly distributed in the unit square, where the minimality is amongst all rooted spanning trees with a direction of growth. We show that the number of branches from the root of this tree, the total length of these branches, and the length of the longest branch each converges weakly. This model is related to the study of record values in the theory of extreme-value statistics and this relation is used to obtain our results. The results also hold when the tree is formed from a Poisson point process of intensity n in the unit square.

On a random directed spanning tree

We study the asymptotic properties of a minimal spanning tree formed by n points uniformly distributed in the unit square, where the minimality is amongst all rooted spanning trees with a direction of growth. We show that the number of branches from the root of this tree, the total length of these branches, and the length of the longest branch each converges weakly. This model is related to the study of record values in the theory of extreme-value statistics and this relation is used to obtain our results. The results also hold when the tree is formed from a Poisson point process of intensity n in the unit square.

Continuity and convergence of the percolation function in continuum percolation

We consider a percolation model on the d-dimensional Euclidean space which consists of spheres centred at the points of a Poisson point process of intensity ?. The radii of the spheres are random and are chosen independently and identically according to a distribution of a positive random variable. We show that the percolation function is continuous everywhere except perhaps at the critical point. Further, we show that the percolation functions converge to the appropriate percolation function except at the critical point when the radius random variables are uniformly bounded and converge weakly to another bounded random variable.