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Mark Walters - One of the best experts on this subject based on the ideXlab platform.
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A Critical Constant for the k nearest-neighbour model
Advances in Applied Probability, 2009Co-Authors: Paul Balister, Béla Bollobás, Amites Sarkar, Mark WaltersAbstract:Let
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A Critical Constant for the k nearest neighbour model
Advances in Applied Probability, 2009Co-Authors: Paul Balister, Béla Bollobás, Amites Sarkar, Mark WaltersAbstract:Let P be a Poisson process of intensity one 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,⌊clog n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed by the authors in [1]. Let P be a Poisson process of intensity one in a square Sn of area n. For a fixed integer k, we join every point of P to its k nearest neighbours, creating an undirected random geometric graph GSn,k = Gn,k in which every vertex has degree at least k. The connectivity of these graphs was studied by the present authors in [1]. It is not hard to see that Gn,k becomes connected around k = �(log n), and we proved in [1] that if k(n) ≤ 0.3043log n then the probability that Gn,k(n) is connected tends to zero as n → ∞, while if k(n) ≥ 0.5139log n then the probability that Gn,k(n) is connected tends to one as n → ∞. However, we were unable to prove the natural conjecture that there exists a Critical Constant ccrit such that for c ccrit, P(Gn,⌊clog n⌋ is connected) → 1 as n → ∞. In this paper we prove this conjecture.
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A Critical Constant for the k nearest neighbour model
arXiv: Probability, 2007Co-Authors: Paul Balister, Béla Bollobás, Amites Sarkar, Mark WaltersAbstract:Let P be a Poisson process of intensity one 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 such that for c' c G_{n,c'\log n} is connected with probability tending to 1 as n tends to infinity. This answers a question previously posed by the authors.
Paul Balister - One of the best experts on this subject based on the ideXlab platform.
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A Critical Constant for the k nearest-neighbour model
Advances in Applied Probability, 2009Co-Authors: Paul Balister, Béla Bollobás, Amites Sarkar, Mark WaltersAbstract:Let
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A Critical Constant for the k nearest neighbour model
Advances in Applied Probability, 2009Co-Authors: Paul Balister, Béla Bollobás, Amites Sarkar, Mark WaltersAbstract:Let P be a Poisson process of intensity one 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,⌊clog n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed by the authors in [1]. Let P be a Poisson process of intensity one in a square Sn of area n. For a fixed integer k, we join every point of P to its k nearest neighbours, creating an undirected random geometric graph GSn,k = Gn,k in which every vertex has degree at least k. The connectivity of these graphs was studied by the present authors in [1]. It is not hard to see that Gn,k becomes connected around k = �(log n), and we proved in [1] that if k(n) ≤ 0.3043log n then the probability that Gn,k(n) is connected tends to zero as n → ∞, while if k(n) ≥ 0.5139log n then the probability that Gn,k(n) is connected tends to one as n → ∞. However, we were unable to prove the natural conjecture that there exists a Critical Constant ccrit such that for c ccrit, P(Gn,⌊clog n⌋ is connected) → 1 as n → ∞. In this paper we prove this conjecture.
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A Critical Constant for the k nearest neighbour model
arXiv: Probability, 2007Co-Authors: Paul Balister, Béla Bollobás, Amites Sarkar, Mark WaltersAbstract:Let P be a Poisson process of intensity one 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 such that for c' c G_{n,c'\log n} is connected with probability tending to 1 as n tends to infinity. This answers a question previously posed by the authors.
Amites Sarkar - One of the best experts on this subject based on the ideXlab platform.
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A Critical Constant for the k nearest-neighbour model
Advances in Applied Probability, 2009Co-Authors: Paul Balister, Béla Bollobás, Amites Sarkar, Mark WaltersAbstract:Let
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A Critical Constant for the k nearest neighbour model
Advances in Applied Probability, 2009Co-Authors: Paul Balister, Béla Bollobás, Amites Sarkar, Mark WaltersAbstract:Let P be a Poisson process of intensity one 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,⌊clog n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed by the authors in [1]. Let P be a Poisson process of intensity one in a square Sn of area n. For a fixed integer k, we join every point of P to its k nearest neighbours, creating an undirected random geometric graph GSn,k = Gn,k in which every vertex has degree at least k. The connectivity of these graphs was studied by the present authors in [1]. It is not hard to see that Gn,k becomes connected around k = �(log n), and we proved in [1] that if k(n) ≤ 0.3043log n then the probability that Gn,k(n) is connected tends to zero as n → ∞, while if k(n) ≥ 0.5139log n then the probability that Gn,k(n) is connected tends to one as n → ∞. However, we were unable to prove the natural conjecture that there exists a Critical Constant ccrit such that for c ccrit, P(Gn,⌊clog n⌋ is connected) → 1 as n → ∞. In this paper we prove this conjecture.
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A Critical Constant for the k nearest neighbour model
arXiv: Probability, 2007Co-Authors: Paul Balister, Béla Bollobás, Amites Sarkar, Mark WaltersAbstract:Let P be a Poisson process of intensity one 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 such that for c' c G_{n,c'\log n} is connected with probability tending to 1 as n tends to infinity. This answers a question previously posed by the authors.
Béla Bollobás - One of the best experts on this subject based on the ideXlab platform.
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A Critical Constant for the k nearest-neighbour model
Advances in Applied Probability, 2009Co-Authors: Paul Balister, Béla Bollobás, Amites Sarkar, Mark WaltersAbstract:Let
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A Critical Constant for the k nearest neighbour model
Advances in Applied Probability, 2009Co-Authors: Paul Balister, Béla Bollobás, Amites Sarkar, Mark WaltersAbstract:Let P be a Poisson process of intensity one 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,⌊clog n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed by the authors in [1]. Let P be a Poisson process of intensity one in a square Sn of area n. For a fixed integer k, we join every point of P to its k nearest neighbours, creating an undirected random geometric graph GSn,k = Gn,k in which every vertex has degree at least k. The connectivity of these graphs was studied by the present authors in [1]. It is not hard to see that Gn,k becomes connected around k = �(log n), and we proved in [1] that if k(n) ≤ 0.3043log n then the probability that Gn,k(n) is connected tends to zero as n → ∞, while if k(n) ≥ 0.5139log n then the probability that Gn,k(n) is connected tends to one as n → ∞. However, we were unable to prove the natural conjecture that there exists a Critical Constant ccrit such that for c ccrit, P(Gn,⌊clog n⌋ is connected) → 1 as n → ∞. In this paper we prove this conjecture.
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A Critical Constant for the k nearest neighbour model
arXiv: Probability, 2007Co-Authors: Paul Balister, Béla Bollobás, Amites Sarkar, Mark WaltersAbstract:Let P be a Poisson process of intensity one 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 such that for c' c G_{n,c'\log n} is connected with probability tending to 1 as n tends to infinity. This answers a question previously posed by the authors.
Russ M Thompson - One of the best experts on this subject based on the ideXlab platform.
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Critical Constants for Recurrence on Groups of Polynomial Growth
Electronic Journal of Probability, 2010Co-Authors: David Revelle, Russ M ThompsonAbstract:The Critical Constant for recurrence, $c_{rt}$, is an invariant of the quotient space $H/G$ of a finitely generated group. The Constant is determined by the largest moment a probability measure on $G$ can have without the induced random walk on $H/G$ being recurrent. We present a description of which subgroups of groups of polynomial volume growth are recurrent. Using this we show that for such recurrent subgroups $c_{rt}$ corresponds to the relative growth rate of $H$ in $G$, and in particular $c_{rt}$ is either $0$, $1$ or $2$.