scholarly journals Sharpness in the k-Nearest-Neighbours Random Geometric Graph Model

2012 ◽  
Vol 44 (3) ◽  
pp. 617-634 ◽  
Author(s):  
Victor Falgas-Ravry ◽  
Mark Walters
Keyword(s):  
Poisson Process ◽  
Random Graph ◽  
Graph Model ◽  
Geometric Graph ◽  
Random Geometric Graph ◽  
Integer Sequence ◽  
Nearest Neighbours ◽  
Square Box

Let Sn, k denote the random graph obtained by placing points in a square box of area n according to a Poisson process of intensity 1 and joining each point to its k nearest neighbours. Balister, Bollobás, Sarkar and Walters (2005) conjectured that, for every 0 < ε < 1 and all sufficiently large n, there exists C = C(ε) such that, whenever the probability that Sn, k is connected is at least ε, then the probability that Sn, k+C is connected is at least 1 - ε. In this paper we prove this conjecture. As a corollary, we prove that there exists a constant C' such that, whenever k(n) is a sequence of integers such that the probability Sn,k(n) is connected tends to 1 as n → ∞, then, for any integer sequence s(n) with s(n) = o(logn), the probability Sn,k(n)+⌊C'slog logn⌋ is s-connected (i.e. remains connected after the deletion of any s − 1 vertices) tends to 1 as n → ∞. This proves another conjecture given in Balister, Bollobás, Sarkar and Walters (2009).

scholarly journals Sharpness in the k-Nearest-Neighbours Random Geometric Graph Model

2012 ◽  
Vol 44 (03) ◽  
pp. 617-634
Author(s):  
Victor Falgas-Ravry ◽  
Mark Walters
Keyword(s):  
Poisson Process ◽  
Random Graph ◽  
Graph Model ◽  
Geometric Graph ◽  
Random Geometric Graph ◽  
Integer Sequence ◽  
Nearest Neighbours ◽  
Square Box

Let S n, k denote the random graph obtained by placing points in a square box of area n according to a Poisson process of intensity 1 and joining each point to its k nearest neighbours. Balister, Bollobás, Sarkar and Walters (2005) conjectured that, for every 0 &lt; ε &lt; 1 and all sufficiently large n, there exists C = C(ε) such that, whenever the probability that S n, k is connected is at least ε, then the probability that S n, k+C is connected is at least 1 - ε. In this paper we prove this conjecture. As a corollary, we prove that there exists a constant C' such that, whenever k(n) is a sequence of integers such that the probability S n,k(n) is connected tends to 1 as n → ∞, then, for any integer sequence s(n) with s(n) = o(logn), the probability S n,k(n)+⌊C'slog logn⌋ is s-connected (i.e. remains connected after the deletion of any s − 1 vertices) tends to 1 as n → ∞. This proves another conjecture given in Balister, Bollobás, Sarkar and Walters (2009).

scholarly journals Connectivity of random k-nearest-neighbour graphs

2005 ◽  
Vol 37 (01) ◽  
pp. 1-24 ◽  
Author(s):  
Paul Balister ◽  
Béla Bollobás ◽  
Amites Sarkar ◽  
Mark Walters
Keyword(s):  
Poisson Process ◽  
Geometric Graph ◽  
Upper Bounds ◽  
Related Question ◽  
Nearest Neighbour ◽  
Lower And Upper Bounds ◽  
Random Geometric Graph ◽  
Nearest Neighbours ◽  
Closed Disc

Let 𝓅 be a Poisson process of intensity one in a square S n of area n. We construct a random geometric graph G n,k by joining each point of 𝓅 to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that G n, k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that G n, k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With G n, k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover S n tends to 0 as n → ∞ while, if k ≥ 0.9967 log n, then the probability that the discs cover S n tends to 1 as n → ∞.

scholarly journals A critical constant for the k nearest-neighbour model

2009 ◽  
Vol 41 (01) ◽  
pp. 1-12 ◽  
Author(s):  
Paul Balister ◽  
Béla Bollobás ◽  
Amites Sarkar ◽  
Mark Walters
Keyword(s):  
Poisson Process ◽  
Geometric Graph ◽  
Nearest Neighbour ◽  
Random Geometric Graph ◽  
Fixed Integer ◽  
Nearest Neighbours ◽  
Critical Constant

Let 𝒫 be a Poisson process of intensity 1 in a square S n of area n. For a fixed integer k, join every point of 𝒫 to its k nearest neighbours, creating an undirected random geometric graph G n,k . We prove that there exists a critical constant c crit such that, for c &lt; c crit, G n,⌊c log n⌋ is disconnected with probability tending to 1 as n → ∞ and, for c &gt; c crit, G n,⌊c log n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed in Balister et al. (2005).

scholarly journals A critical constant for the k nearest-neighbour model

2009 ◽  
Vol 41 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Paul Balister ◽  
Béla Bollobás ◽  
Amites Sarkar ◽  
Mark Walters
Keyword(s):  
Poisson Process ◽  
Geometric Graph ◽  
Nearest Neighbour ◽  
Random Geometric Graph ◽  
Fixed Integer ◽  
Nearest Neighbours ◽  
Critical Constant

Let 𝒫 be a Poisson process of intensity 1 in a square Sn of area n. For a fixed integer k, join every point of 𝒫 to its k nearest neighbours, creating an undirected random geometric graph Gn,k. We prove that there exists a critical constant ccrit such that, for c < ccrit, Gn,⌊c log n⌋ is disconnected with probability tending to 1 as n → ∞ and, for c > ccrit, Gn,⌊c log n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed in Balister et al. (2005).

scholarly journals Connectivity of random k-nearest-neighbour graphs

2005 ◽  
Vol 37 (1) ◽  
pp. 1-24 ◽  
Author(s):  
Paul Balister ◽  
Béla Bollobás ◽  
Amites Sarkar ◽  
Mark Walters
Keyword(s):  
Poisson Process ◽  
Geometric Graph ◽  
Upper Bounds ◽  
Related Question ◽  
Nearest Neighbour ◽  
Lower And Upper Bounds ◽  
Random Geometric Graph ◽  
Nearest Neighbours ◽  
Closed Disc

Let 𝓅 be a Poisson process of intensity one in a square Sn of area n. We construct a random geometric graph Gn,k by joining each point of 𝓅 to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that Gn, k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that Gn, k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With Gn, k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover Sn tends to 0 as n → ∞ while, if k ≥ 0.9967 log n, then the probability that the discs cover Sn tends to 1 as n → ∞.

scholarly journals АЛГОРИТМ ОЦЕНКИ ПРОПУСКНОЙ СПОСОБНОСТИ ПРИ УПРАВЛЕНИИ ТРАФИКОМ БЕСПИЛОТНЫХ ЛЕТАТЕЛЬНЫХ АППАРАТОВ

2018 ◽  
pp. 69-75
Author(s):  
Ольга Константиновна Погудина ◽  
Ирина Васильевна Вайленко
Keyword(s):  
Poisson Process ◽  
Unmanned Aerial Vehicles ◽  
Traffic Control ◽  
Graph Model ◽  
Geometric Graph ◽  
Time Interval ◽  
Random Geometric Graph ◽  
Technical Requirements ◽  
Aerial Vehicles ◽  
The Law

The subject of the study in the article is the processes of assessing the airship throughput in controlling the unmanned aerial vehicles (UAV) traffic management. The goal is to improve the quality of air traffic control, taking into account the avoidance of conflicts involving three or more UAV. Problems: to develop a mathematical model of the probabilistic traffic map, as well as to formalize the construction of a random geometric graph model for the estimation of alleged UAVs conflicts and collisions; To implement algorithms given models construction for airship throughput automation. The models used: Poisson process whose intensity model is used for building a probabilistic traffic map, random geometric graph model is used for calculate the number of possible conflicts involving the UAV. The following results are obtained. A formalized model of the UAV location map has been created taking into account: the given region with the specified population density and the expected number of operations during the specified time interval. This model was used in the construction of a random geometric graph, in which, taking into account the minimum distance possible for the approximation of two UAVs, an estimation of the probability of conflicts and collisions was conducted. The model is the basis for obtaining an algorithm for estimating the factors limiting the capacity of the airspace, as a result of the occurrence of difficult solvable conflicts. The scientific novelty of the obtained results is as follows: The random geometric graph model is improved by formalizing the position of the vertices. The vertices, taking into account the law of the Poisson process, are placed in the cells of a given region. This allows us to obtain an objective picture of the location of the UAV in the city's airspace. Two-dimensional models of probabilistic traffic maps (Dutch model "Metropolis", model Cal) have been further developed, due to the formalization of the initial UAV placement, taking into account the law of the Poisson process. This will help to determine the technical requirements for ensuring uninterrupted operation of small unmanned aerial vehicles in the urban airspace

scholarly journals Directed random geometric graphs

2019 ◽  
Vol 7 (5) ◽  
pp. 792-816
Author(s):  
Jesse Michel ◽  
Sushruth Reddy ◽  
Rikhav Shah ◽  
Sandeep Silwal ◽  
Ramis Movassagh
Keyword(s):  
Real World ◽  
Word Association ◽  
Graph Model ◽  
Small World ◽  
Geometric Graph ◽  
Clustering Coefficient ◽  
Random Geometric Graph ◽  
Random Geometric Graphs ◽  
Scale Free ◽  
Small World Property

Abstract Many real-world networks are intrinsically directed. Such networks include activation of genes, hyperlinks on the internet and the network of followers on Twitter among many others. The challenge, however, is to create a network model that has many of the properties of real-world networks such as power-law degree distributions and the small-world property. To meet these challenges, we introduce the Directed Random Geometric Graph (DRGG) model, which is an extension of the random geometric graph model. We prove that it is scale-free with respect to the indegree distribution, has binomial outdegree distribution, has a high clustering coefficient, has few edges and is likely small-world. These are some of the main features of aforementioned real-world networks. We also empirically observed that word association networks have many of the theoretical properties of the DRGG model.

scholarly journals On the One Dimensional Poisson Random Geometric Graph

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
L. Decreusefond ◽  
E. Ferraz
Keyword(s):  
Poisson Process ◽  
Laplace Transforms ◽  
Geometric Graph ◽  
Connected Components ◽  
Random Geometric Graph ◽  
One Dimensional ◽  
Bounded Interval ◽  
The One

Given a Poisson process on a bounded interval, its random geometric graph is the graph whose vertices are the points of the Poisson process, and edges exist between two points if and only if their distance is less than a fixed given threshold. We compute explicitly the distribution of the number of connected components of this graph. The proof relies on inverting some Laplace transforms.

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