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Brian D. O. Anderson - One of the best experts on this subject based on the ideXlab platform.
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critical density for connectivity in 2d and 3d wireless multi hop networks
IEEE Transactions on Wireless Communications, 2013Co-Authors: Seh Chun Ng, Brian D. O. AndersonAbstract:In this paper we investigate the critical node density required to ensure that an arbitrary node in a large-scale wireless multi-hop network is connected (via multi-hop path) to infinitely many other nodes with a positive probability. Specifically we consider a wireless multi-hop network where nodes are distributed in \mathbb{R}^d (d = 2,3) following a homogeneous Poisson point process. The establishment of a direct Connection between any two nodes is independent of Connections between other pairs of nodes and its probability satisfies some intuitively reasonable conditions, viz. rotational and translational invariance, non-increasing monotonicity, and integral boundedness. Under the above random Connection Model we first obtain analytically the upper and lower bounds for the critical density. Then we compare the new bounds with other existing bounds in the literature under the unit disk Model and the log-normal Model which are special cases of the random Connection Model. The comparison shows that our bounds are either close to or tighter than the known ones. To the best of our knowledge, this is the first result for the random Connection Model in both 2D and 3D networks. The result is of practical use for designing large-scale wireless multi-hop networks such as wireless sensor networks.
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Connectivity of Large Wireless Networks Under A General Connection Model
IEEE Transactions on Information Theory, 2013Co-Authors: Guoqiang Mao, Brian D. O. AndersonAbstract:This paper studies networks where all nodes are distributed on a unit square A=Δ [- [1/2], [1/2]]2 following a Poisson distribution with known density ρ and a pair of nodes separated by an Euclidean distance x are directly connected with probability grρ(x)=Δg(x/rρ), independent of the event that any other pair of nodes are directly connected. Here, g:[0,∞)→ [0,1] satisfies the conditions of rotational invariance, nonincreasing monotonicity, integral boundedness, and g(x)=o(1/(x2log2x)) ; further, rρ=√{(logρ+b)/(Cρ)} where C=∫ℜ2g(||x||)dx and b is a constant. Denote the aforementioned network by G(Xρ,grρ,A). We show that as ρ→ ∞, 1) the distribution of the number of isolated nodes in G(Xρ,grρ,A) converges to a Poisson distribution with mean e-b ; 2) asymptotically almost surely (a.a.s.) there is no component in G(Xρ,grρ,A) of fixed and finite order k >; 1; c) a.a.s. the number of components with an unbounded order is one. Therefore, as ρ→ ∞, the network a.a.s. contains a unique unbounded component and isolated nodes only; a sufficient and necessary condition for G(Xρ,grρ,A) to be a.a.s. connected is that there is no isolated node in the network, which occurs when b→ ∞ as ρ→ ∞. These results expand recent results obtained for connectivity of random geometric graphs from the unit disk Model and the fewer results from the log-normal Model to the more general and more practical random Connection Model.
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Connectivity of Large Wireless Networks under A Generic Connection Model
arXiv: Networking and Internet Architecture, 2012Co-Authors: Guoqiang Mao, Brian D. O. AndersonAbstract:This paper provides a necessary and sufficient condition for a random network with nodes Poissonly distributed on a unit square and a pair of nodes directly connected following a generic random Connection Model to be asymptotically almost surely connected. The results established in this paper expand recent results obtained for connectivity of random geometric graphs from the unit disk Model and the fewer results from the log-normal Model to the more generic and more practical random Connection Model.
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on the asymptotic connectivity of random networks under the random Connection Model
International Conference on Computer Communications, 2011Co-Authors: Guoqiang Mao, Brian D. O. AndersonAbstract:Consider a network where all nodes are distributed on a unit square following a Poisson distribution with known density ρ and a pair of nodes separated by an Euclidean distance x are directly connected with probability g(x over r ρ ), where g : [0,∞) → [0,1] satisfies three conditions: rotational invariance, qnon-increasing monotonicity and integral boundedness, equation and b is a constant, independent of the event that another pair of nodes are directly connected. In this paper, we analyze the asymptotic distribution of the number of isolated nodes in the above network using the Chen-Stein technique and the impact of the boundary effect on the number of isolated nodes as ρ → ∞. On that basis we derive a necessary condition for the above network to be asymptotically almost surely connected. These results form an important link in expanding recent results on the connectivity of the random geometric graphs from the commonly used unit disk Model to the more generic and more practical random Connection Model.
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GLOBECOM - Analytical Bounds on the Critical Density for Percolation in Wireless Multi-Hop Networks
2011 IEEE Global Telecommunications Conference - GLOBECOM 2011, 2011Co-Authors: Guoqiang Mao, Brian D. O. AndersonAbstract:In this paper we develop analytical bounds on the critical density for percolation in wireless multi- hop networks, but in contrast to other studies, under a random Connection Model and with nodes Poissonly distributed in the plane $\mathbb{R}^2$. The establishment of a direct Connection between any two nodes follows a random Connection Model satisfying some intuitively reasonable conditions, i.e. rotational and translational invariance, non- increasing monotonicity and integral boundedness. It is well known that under the above network Model and Connection Model there exists a critical density below which almost surely a fixed but arbitrary node is connected (via single or multi-hop path) to finite number of other nodes only, and above which the node is connected to an infinite number of other nodes with a positive probability. In this paper we investigate the bounds on the critical density. The result is compared with the existing results under a specific Connection Model, i.e. the unit disk communication Model, and it is shown that our method generates bounds close to the known ones. The result provides valuable insight into the design of large- scale wireless multi-hop networks.
Guoqiang Mao - One of the best experts on this subject based on the ideXlab platform.
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Connectivity of Large Wireless Networks: Sufficient and Necessary Conditions
Connectivity of Communication Networks, 2017Co-Authors: Guoqiang MaoAbstract:This chapter studies the sufficient and necessary condition for a large wireless network to be asymptotically almost surely connected. Consider a dense network Model, we show that as the node density approaches infinity, (a) the distribution of the number of isolated nodes converges to a Poisson distribution; (b) asymptotically almost surely (a.a.s.) there is no component of fixed and finite order k > 1; (c) a.a.s. the number of components with an unbounded order is one. Therefore, as the node density approaches infinity, the network a.a.s. contains a unique unbounded component and isolated nodes only; a sufficient and necessary condition for the network to be a.a.s. connected is that there is no isolated node in the network. These results, established assuming a general random Connection Model, readily incorporate existing results established assuming the unit disk Connection Model and the fewer results assuming the log-normal Connection Model as its special cases.
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Critical Density for Percolation
Connectivity of Communication Networks, 2017Co-Authors: Guoqiang MaoAbstract:In this chapter we investigate the critical node density required to ensure that an arbitrary node in a large wireless network is connected (via multi-hop paths) to infinitely many other nodes with a positive probability, known as the percolation probability. A network is said to percolate if there exists a component of infinite order in the network. Specifically, assuming the infinite network Model and the random Connection Model, we first obtain an upper and a lower bounds for the critical density. Then, we compare the bounds with other existing bounds in the literature under the unit disk Connection Model and the log-normal Connection Model, which are special cases of the random Connection Model. Percolation is an important subject in the study of connectivity of large random networks.
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Connectivity of Large Wireless Networks Under A General Connection Model
IEEE Transactions on Information Theory, 2013Co-Authors: Guoqiang Mao, Brian D. O. AndersonAbstract:This paper studies networks where all nodes are distributed on a unit square A=Δ [- [1/2], [1/2]]2 following a Poisson distribution with known density ρ and a pair of nodes separated by an Euclidean distance x are directly connected with probability grρ(x)=Δg(x/rρ), independent of the event that any other pair of nodes are directly connected. Here, g:[0,∞)→ [0,1] satisfies the conditions of rotational invariance, nonincreasing monotonicity, integral boundedness, and g(x)=o(1/(x2log2x)) ; further, rρ=√{(logρ+b)/(Cρ)} where C=∫ℜ2g(||x||)dx and b is a constant. Denote the aforementioned network by G(Xρ,grρ,A). We show that as ρ→ ∞, 1) the distribution of the number of isolated nodes in G(Xρ,grρ,A) converges to a Poisson distribution with mean e-b ; 2) asymptotically almost surely (a.a.s.) there is no component in G(Xρ,grρ,A) of fixed and finite order k >; 1; c) a.a.s. the number of components with an unbounded order is one. Therefore, as ρ→ ∞, the network a.a.s. contains a unique unbounded component and isolated nodes only; a sufficient and necessary condition for G(Xρ,grρ,A) to be a.a.s. connected is that there is no isolated node in the network, which occurs when b→ ∞ as ρ→ ∞. These results expand recent results obtained for connectivity of random geometric graphs from the unit disk Model and the fewer results from the log-normal Model to the more general and more practical random Connection Model.
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Connectivity of Large Wireless Networks under A Generic Connection Model
arXiv: Networking and Internet Architecture, 2012Co-Authors: Guoqiang Mao, Brian D. O. AndersonAbstract:This paper provides a necessary and sufficient condition for a random network with nodes Poissonly distributed on a unit square and a pair of nodes directly connected following a generic random Connection Model to be asymptotically almost surely connected. The results established in this paper expand recent results obtained for connectivity of random geometric graphs from the unit disk Model and the fewer results from the log-normal Model to the more generic and more practical random Connection Model.
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on the asymptotic connectivity of random networks under the random Connection Model
International Conference on Computer Communications, 2011Co-Authors: Guoqiang Mao, Brian D. O. AndersonAbstract:Consider a network where all nodes are distributed on a unit square following a Poisson distribution with known density ρ and a pair of nodes separated by an Euclidean distance x are directly connected with probability g(x over r ρ ), where g : [0,∞) → [0,1] satisfies three conditions: rotational invariance, qnon-increasing monotonicity and integral boundedness, equation and b is a constant, independent of the event that another pair of nodes are directly connected. In this paper, we analyze the asymptotic distribution of the number of isolated nodes in the above network using the Chen-Stein technique and the impact of the boundary effect on the number of isolated nodes as ρ → ∞. On that basis we derive a necessary condition for the above network to be asymptotically almost surely connected. These results form an important link in expanding recent results on the connectivity of the random geometric graphs from the commonly used unit disk Model to the more generic and more practical random Connection Model.
Sebastian Ziesche - One of the best experts on this subject based on the ideXlab platform.
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On the Ornstein–Zernike equation for stationary cluster processes and the random Connection Model
Advances in Applied Probability, 2017Co-Authors: Sebastian ZiescheAbstract:Abstract In the first part of this paper we consider a general stationary subcritical cluster Model in ℝd. The associated pair-connectedness function can be defined in terms of two-point Palm probabilities of the underlying point process. Using Palm calculus and Fourier theory we solve the Ornstein–Zernike equation (OZE) under quite general distributional assumptions. In the second part of the paper we discuss the analytic and combinatorial properties of the OZE solution in the special case of a Poisson-driven random Connection Model.
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On the Ornstein-Zernike equation for stationary cluster processes and the random Connection Model
arXiv: Probability, 2016Co-Authors: Sebastian ZiescheAbstract:In the first part of this paper we consider a general stationary subcritical cluster Model in $\mathbb{R}^d$. The associated pair-connectedness function can be defined in terms of two-point Palm probabilities of the underlying point process. Using Palm calculus and Fourier theory we solve the Ornstein-Zernike equation (OZE) under quite general distributional assumptions. In the second part of the paper we discuss the analytic and combinatorial properties of the OZE-solution in the special case of a Poisson driven random Connection Model.
Nina Taft - One of the best experts on this subject based on the ideXlab platform.
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an independent Connection Model for traffic matrices
Internet Measurement Conference, 2006Co-Authors: Vijayi Erramill, Mark Crovella, Nina TaftAbstract:A common assumption made in traffic matrix (TM) Modeling and estimation is independence of a packet's network ingress and egress. We argue that in real IP networks, this assumption should not and does not hold. The fact that most traffic consists of two-way exchanges of packets means that traffic streams flowing in opposite directions at any point in the network are not independent. In this paper we propose a Model for traffic matrices based on independence of Connections rather than packets. We argue that the independent-Connection (IC) Model is more intuitive, and has a more direct Connection to underlying network phenomena than the gravity Model. To validate the IC Model, we show that it fits real data better than the gravity Model and that it works well as a prior in the TM estimation problem. We study the Model's parameters empirically and identify useful stability properties. This justifies the use of the simpler versions of the Model for TM applications. To illustrate the utility of the Model we focus on two such applications: synthetic TM generation and TM estimation. To the best of our knowledge this is the first traffic matrix Model that incorporates properties of bidirectional traffic.
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Internet Measurement Conference - An independent-Connection Model for traffic matrices
Proceedings of the 6th ACM SIGCOMM on Internet measurement - IMC '06, 2006Co-Authors: Vijayi Erramill, Mark Crovella, Nina TaftAbstract:A common assumption made in traffic matrix (TM) Modeling and estimation is independence of a packet's network ingress and egress. We argue that in real IP networks, this assumption should not and does not hold. The fact that most traffic consists of two-way exchanges of packets means that traffic streams flowing in opposite directions at any point in the network are not independent. In this paper we propose a Model for traffic matrices based on independence of Connections rather than packets. We argue that the independent-Connection (IC) Model is more intuitive, and has a more direct Connection to underlying network phenomena than the gravity Model. To validate the IC Model, we show that it fits real data better than the gravity Model and that it works well as a prior in the TM estimation problem. We study the Model's parameters empirically and identify useful stability properties. This justifies the use of the simpler versions of the Model for TM applications. To illustrate the utility of the Model we focus on two such applications: synthetic TM generation and TM estimation. To the best of our knowledge this is the first traffic matrix Model that incorporates properties of bidirectional traffic.
Seh Chun Ng - One of the best experts on this subject based on the ideXlab platform.
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critical density for connectivity in 2d and 3d wireless multi hop networks
IEEE Transactions on Wireless Communications, 2013Co-Authors: Seh Chun Ng, Brian D. O. AndersonAbstract:In this paper we investigate the critical node density required to ensure that an arbitrary node in a large-scale wireless multi-hop network is connected (via multi-hop path) to infinitely many other nodes with a positive probability. Specifically we consider a wireless multi-hop network where nodes are distributed in \mathbb{R}^d (d = 2,3) following a homogeneous Poisson point process. The establishment of a direct Connection between any two nodes is independent of Connections between other pairs of nodes and its probability satisfies some intuitively reasonable conditions, viz. rotational and translational invariance, non-increasing monotonicity, and integral boundedness. Under the above random Connection Model we first obtain analytically the upper and lower bounds for the critical density. Then we compare the new bounds with other existing bounds in the literature under the unit disk Model and the log-normal Model which are special cases of the random Connection Model. The comparison shows that our bounds are either close to or tighter than the known ones. To the best of our knowledge, this is the first result for the random Connection Model in both 2D and 3D networks. The result is of practical use for designing large-scale wireless multi-hop networks such as wireless sensor networks.