The radial spanning tree of a Poisson point process

Baccelli, François; Bordenave, Charles

doi:10.1214/105051606000000826

Cited by 65 publications

(168 citation statements)

References 36 publications

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“…Define p = p(M, N, ε) to be the probability that some point x ∈ C ∞ ∩ S(0) is joined to some point y outside the square [−M + N, M) 2 Therefore, by Lemma 3.7 we have p(M, N, ε) → 0 as M − N → ∞. So the corresponding lemmas in [8] are all proved and the proof is thus complete.…”

Section: L(ρ( )))mentioning

confidence: 82%

“…Then some new FPP models in a random environment were studied. Based on homogeneous Poisson point processes, VahidiAsl and Wierman [16] introduced a class of FPP models for the Poisson-Voronoi tessellations, Howard and Newman [10], [11] established a Euclidean FPP, and Baccelli and Bordenave [2] analyzed a class of spatial random spanning trees built on the Poisson point processes of the plane. The authors proved shape theorems for these continuum FPP models.…”

Section: Introductionmentioning

confidence: 99%

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Large Deviations for the Graph Distance in Supercritical Continuum Percolation

2011

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Denote the Palm measure of a homogeneous Poisson process H λ with two points 0 and x by P 0,x . We prove that there exists a constant µ ≥ 1 such that (0, x) is the graph distance between 0 and x in the infinite component C ∞ of the random geometric graph G(H λ ; 1). We derive a large deviation inequality for an asymptotic shape result. Our results have applications in many fields and especially in wireless sensor networks.decreases when x 2 tends to ∞, where D

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Section: L(ρ( )))mentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Large Deviations for the Graph Distance in Supercritical Continuum Percolation

2011

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“…Later on, in Section 7, we will make a more specific choice for g n . For n ≥ 1, locally finite X ⊂ R d , and x ∈ X, we define the scaled-up version of ξ restricted to n by ξ n (x; X) : 4) using the fact that ξ as given by (4.1) is homogeneous of order α. We employ the following notion of stabilization (see [21] and [22]).…”

Section: Limit Theorems Away From the Boundarymentioning

confidence: 99%

“…Examples considered previously are the 'coordinatewise' (or 'south-west') partial ordering on point sets in (0, 1) 2 [7], [23], [24] or in (0, 1) d [5], and the radial spanning tree [4] on point sets in R 2 . Also, laws of large numbers for the MDST on a class of partial orders of R 2 were given in [34].…”

Section: Introductionmentioning

confidence: 99%

Limit theorems for random spatial drainage networks

Penrose

Wade

2010

Adv. Appl. Probab.

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Suppose that, under the action of gravity, liquid drains through the unit d-cube via a minimal-length network of channels constrained to pass through random sites and to flow with nonnegative component in one of the canonical orthogonal basis directions of R d , d ≥ 2. The resulting network is a version of the so-called minimal directed spanning tree. We give laws of large numbers and convergence in distribution results on the large-sample asymptotic behaviour of the total power-weighted edge length of the network on uniform random points in (0, 1) d . The distributional results exhibit a weightdependent phase transition between Gaussian and boundary-effect-derived distributions. These boundary contributions are characterized in terms of limits of the so-called on-line nearest-neighbour graph, a natural model of spatial network evolution, for which we also present some new results. Also, we give a convergence in distribution result for the length of the longest edge in the drainage network; when d = 2, the limit is expressed in terms of Dickman-type variables.

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“…In this case, the optimal algorithm consists in choosing as next hop the closest node of the current node but which is closer to the destination than the current node. This routing algorithm has been recently studied in [8].…”

Section: Modelsmentioning

confidence: 99%

Energy aware unicast geographic routing

Busson¹,

Chelius

Fleury

2006 4th International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks

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Abstract-In this paper, we are investigating the optimal radio range minimizing the energy globally consummed by a geographical routing process. Considering a geographical greedy routing protocol and a uniform distribution of nodes in the network area, we analytically evaluate the energy cost of a multi-hop communication. This cost evaluation corresponds to the asymptotic behavior of the routing protocol and turns out to be very accurate compared to the results obtained by simulations. We show that this cost is function of the node intensity and we use this result to deduce the optimal radio range. We evaluate this range with two energy consumption models, the first one considering the energy consumed by transmission operations only and the second one considering both transmission and reception operations. These results can be used in two ways. First, the nodes range can be tuned in advance as a function of the expected node intensity during an off-line planning. Second, we propose an adaptative algorithm where nodes tune their powers according to an on-line evaluation of the local node intensity.

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The radial spanning tree of a Poisson point process

Cited by 65 publications

References 36 publications

Large Deviations for the Graph Distance in Supercritical Continuum Percolation

Large Deviations for the Graph Distance in Supercritical Continuum Percolation

Limit theorems for random spatial drainage networks

Energy aware unicast geographic routing

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