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179
Closing the gap in the capacity of wireless networks via percolation theory
 IEEE Trans. Information Theory
, 2007
"... Abstract—An achievable bit rate per source–destination pair in a wireless network of � randomly located nodes is determined adopting the scaling limit approach of statistical physics. It is shown that randomly scattered nodes can achieve, with high probability, the same Ia � � transmission rate of ..."
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Cited by 244 (7 self)
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Abstract—An achievable bit rate per source–destination pair in a wireless network of � randomly located nodes is determined adopting the scaling limit approach of statistical physics. It is shown that randomly scattered nodes can achieve, with high probability, the same Ia � � transmission rate of arbitrarily located nodes. This contrasts with previous results suggesting that a Ia � � �� � � reduced rate is the price to pay for the randomness due to the location of the nodes. The network operation strategy to achieve the result corresponds to the transition region between order and disorder of an underlying percolation model. If nodes are allowed to transmit over large distances, then paths of connected nodes that cross the entire network area can be easily found, but these generate excessive interference. If nodes transmit over short distances, then such crossing paths do not exist. Percolation theory ensures that crossing paths form in the transition region between these two extreme scenarios. Nodes along these paths are used as a backbone, relaying data for other nodes, and can transport the total amount of information generated by all the sources. A lower bound on the achievable bit rate is then obtained by performing pairwise coding and decoding at each hop along the paths, and using a time division multiple access scheme. Index Terms—Adhoc networks, capacity, percolation theory, scaling laws, throughput, wireless networks.
Stochastic Geometry and Random Graphs for the Analysis and Design of Wireless Networks
"... Wireless networks are fundamentally limited by the intensity of the received signals and by their interference. Since both of these quantities depend on the spatial location of the nodes, mathematical techniques have been developed in the last decade to provide communicationtheoretic results accoun ..."
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Cited by 231 (43 self)
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Wireless networks are fundamentally limited by the intensity of the received signals and by their interference. Since both of these quantities depend on the spatial location of the nodes, mathematical techniques have been developed in the last decade to provide communicationtheoretic results accounting for the network’s geometrical configuration. Often, the location of the nodes in the network can be modeled as random, following for example a Poisson point process. In this case, different techniques based on stochastic geometry and the theory of random geometric graphs – including point process theory, percolation theory, and probabilistic combinatorics – have led to results on the connectivity, the capacity, the outage probability, and other fundamental limits of wireless networks. This tutorial article surveys some of these techniques, discusses their application to model wireless networks, and presents some of the main results that have appeared in the literature. It also serves as an introduction to the field for the other papers in this special issue.
Connectivity in AdHoc and Hybrid Networks
 IN PROC. IEEE INFOCOM
, 2002
"... We consider a largescale wireless network, but with a low density of nodes per unit area. Interferences are then less critical, contrary to connectivity. This paper studies the latter property for both a purely adhoc network and a hybrid network, where fixed base stations can be reached in multipl ..."
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Cited by 217 (7 self)
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We consider a largescale wireless network, but with a low density of nodes per unit area. Interferences are then less critical, contrary to connectivity. This paper studies the latter property for both a purely adhoc network and a hybrid network, where fixed base stations can be reached in multiple hops. We assume here that power constraints are modeled by a maximal distance above which two nodes are not (directly) connected. We find that
Impact of Interferences on Connectivity in Ad Hoc Networks
 in Proc. IEEE INFOCOM
, 2003
"... We study the impact of interferences on the connectivity of largescale adhoc networks, using percolation theory. We assume that a bidirectional connection can be set up between two nodes if the signal to noise ratio at the receiver is larger than some threshold. The noise is the sum of the contri ..."
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Cited by 160 (15 self)
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We study the impact of interferences on the connectivity of largescale adhoc networks, using percolation theory. We assume that a bidirectional connection can be set up between two nodes if the signal to noise ratio at the receiver is larger than some threshold. The noise is the sum of the contribution of interferences from all other nodes, weighted by a coefficient gamma, and of a background noise. We find that there is a critical value of gamma above which the network is made of disconnected clusters of nodes. We also prove that if gamma is non zero but small enough, there exist node spatial densities for which the network contains a large (theoretically infinite) cluster of nodes, enabling distant nodes to communicate in multiple hops. Since small values of gamma cannot be achieved without efficient CDMA codes, we investigate the use of a very simple TDMA scheme, where nodes can emit only every nth time slot. We show qualitatively that it even achieves a better connectivity than the previous system with a parameter gamma/n.
Connectivity vs Capacity in Dense Ad Hoc Networks
, 2004
"... We study the connectivity and capacity of finite area ad hoc wireless networks, with an increasing number of nodes (dense networks). We find that the properties of the network strongly depend on the shape of the attenuation function. For power law attenuation functions, connectivity scales, and the ..."
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Cited by 85 (1 self)
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We study the connectivity and capacity of finite area ad hoc wireless networks, with an increasing number of nodes (dense networks). We find that the properties of the network strongly depend on the shape of the attenuation function. For power law attenuation functions, connectivity scales, and the available rate per node is known to decrease like 1/ # n.Onthe contrary, if the attenuation function does not have a singularity at the origin and is uniformly bounded, we obtain bounds on the percolation domain for large node densities, which show that either the network becomes disconnected, or the available rate per node decreases like 1/n.
The design of calamari: an adhoc localization system for sensor networks
, 2002
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Percolation in the signal to interference ratio graph
 in space. Problemy Peredachi Informatsii 14(1), 3–25. in
, 2006
"... Continuum percolation models in which pairs of points of a twodimensional Poisson point process are connected if they are within some range to each other have been extensively studied. This paper considers a variation in which a connection between two points depends not only on their Euclidean dist ..."
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Cited by 45 (4 self)
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Continuum percolation models in which pairs of points of a twodimensional Poisson point process are connected if they are within some range to each other have been extensively studied. This paper considers a variation in which a connection between two points depends not only on their Euclidean distance, but also on the positions of all other points of the point process. This model has been recently proposed to model interference in radio communication networks. Our main result shows that, despite the infinite range dependencies, percolation occurs in the model when the density λ of the Poisson point process is greater than the critical density value λc of the independent model, provided that interference from other nodes can be sufficiently reduced (without vanishing). 1
Extremal properties of threedimensional sensor networks with applications
 IEEE Transactions on Mobile Computing
"... In this paper, we analyze various critical transmitting/sensing ranges for connectivity and coverage in threedimensional sensor networks. As in other largescale complex systems, many global parameters of sensor networks undergo phase transitions: For a given property of the network, there is a cri ..."
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Cited by 42 (3 self)
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In this paper, we analyze various critical transmitting/sensing ranges for connectivity and coverage in threedimensional sensor networks. As in other largescale complex systems, many global parameters of sensor networks undergo phase transitions: For a given property of the network, there is a critical threshold, corresponding to the minimum amount of the communication effort or power expenditure by individual nodes, above (resp. below) which the property exists with high (resp. a low) probability. For sensor networks, properties of interest include simple and multiple degrees of connectivity/coverage. First, we investigate the network topology according to the region of deployment, the number of deployed sensors and their transmitting/sensing ranges. More specifically, we consider the following problems: Assume that n nodes, each capable of sensing events within a radius of r, are randomly and uniformly distributed in a 3dimensional region R of volume V, how large must the sensing range rSense be to ensure a given degree of coverage of the region to monitor? For a given transmission range rTrans, what is the minimum (resp. maximum) degree of the network? What is then the typical hopdiameter of the underlying network? Next, we show how these results affect algorithmic aspects of the network by designing specific distributed protocols for sensor networks. Keywords Sensor networks, ad hoc networks; coverage, connectivity; hopdiameter; minimum/maximum degrees; transmitting/sensing ranges; analytical methods; energy consumption; topology control. I.
Connectivity of random knearest neighbour graphs
, 2006
"... 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 zero as n→∞, wh ..."
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Cited by 40 (8 self)
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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 zero as n→∞, while if k ≥ 5.1774 logn then the probability that Gn,k is connected tends to one as n→∞. They conjectured that the threshold for connectivity is k = (1+ o(1)) logn. 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, 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 zero as n → ∞, while if k ≥ 0.9967 log n then the probability that the discs cover Sn tends to one as n→∞.