On the variance of the random sphere of influence graph

Hitczenko, Paweł; Janson, Svante; Yukich, J. E.

doi:10.1002/(sici)1098-2418(199903)14:2<139::aid-rsa2>3.0.co;2-e

Cited by 6 publications

(8 citation statements)

References 13 publications

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“…We show that H satisfies the following CLT. In this way we recover most of the results of [10], we show a de-Poissonized version of their central limit theorem and we show convergence of the variance of the total number of edges.…”

Section: Lemma 65 With Probability 1 the Values Of R X Are Finite supporting

confidence: 75%

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Central Limit Theorems for Some Graphs in Computational Geometry

Penrose¹,

Yukich²

2001

Ann. Appl. Probab.

168

455

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Let B n be an increasing sequence of regions in d-dimensional space with volume n and with union d . We prove a general central limit theorem for functionals of point sets, obtained either by restricting a homogeneous Poisson process to B n , or by by taking n uniformly distributed points in B n . The sets B n could be all cubes but a more general class of regions B n is considered. Using this general result we obtain central limit theorems for specific functionals such as total edge length and number of components, defined in terms of graphs such as the k-nearest neighbors graph, the sphere of influence graph and the Voronoi graph.1. Introduction. The purpose of this paper is to develop a general methodology to establish central limit theorems (CLTs) for functionals of graphs in computational geometry. Functionals of interest include total edge length, total number of edges, total number of components and total number of vertices of fixed degree. Graphs of interest include the k-nearest neighbors graph, the Voronoi and Delaunay tessellations, the sphere of influence graph, the Gabriel graph and the relative neighbor graph. These graphs are formally defined later on. In each case, the graph or its dual graph (as with the Voronoi graph) is constructed as follows: given a finite vertex set in d d ≥ 1, undirected edges are drawn from each vertex to various nearby vertices, the choice of edges to include being determined by the local point configuration according to some specified rule. Sometimes such graphs are called proximity graphs; see [3] for a precise definition.Our graphs are random in the sense that the vertex set is a random point set in d d ≥ 1. We establish CLTs for two related types of random point sets: the homogeneous Poisson point process on a large region or "window" of d and the point set consisting of a large independent sample of nonrandom sample size from the uniform distribution on such a region. By scaling, these often yield a CLT for Poisson processes of high intensity on a fixed set such as the unit cube 0 1 d , or for large independent samples of nonrandom size from the uniform distribution on a fixed set. As a by-product, we also prove the convergence of the (scaled) variance of our functionals of interest.One of our more interesting new results is a CLT for the total number of components of the k-nearest neighbors graph, either on a Poisson process or on a sample of nonrandom size, and likewise for the sphere of influence graph.

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Section: Lemma 65 With Probability 1 the Values Of R X Are Finite supporting

confidence: 75%

“…These latter results add to existing results for their Poisson counterparts [1,8,10]. We believe that the CLTs established here, particularly those for nonrandom sample sizes, may have uses in the statistical analysis of data and may lead to useful tests for clustering.…”

supporting

confidence: 54%

Central Limit Theorems for Some Graphs in Computational Geometry

Penrose¹,

Yukich²

2001

Ann. Appl. Probab.

168

455

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“…Moreover, there are many cases where the objective method quickly gives one the essential limit theory, yet subadditive methods appear to be awkward to apply. Here the list can be made as long as one likes, but one should certainly include the limit theory for Voronoi regions [32] and the sphere of influence graphs ( [27], [33]). …”

Section: Further Comparison To Subadditive Methodsmentioning

confidence: 97%

“…Theorem 8 applies to many problems of computational and it provides limit laws for the minimal spanning tree, the k-nearest neighbors graph ( [34], [20]), the Voronoi graph [32], the Delaunay graph, and the sphere of influence graphs ( [27], [33]). Nevertheless, before Theorem 8 can be applied, one needs to prove the associated summands ξ(x, X ) are indeed stable on P τ for all 0 ≤ τ < ∞.…”

Section: Confirmation Of Stabilitymentioning

confidence: 99%

The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence

Aldous¹,

Steele²

2004

Encyclopaedia of Mathematical Sciences

309

684

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“…pp. 142-43 of [13]) we obtain for any φ with polynomial growth (e) Proximity graphs. Devroye [7] defines a proximity graph on X to be one in which each {x, y} is included as an edge if a specified set S(x, y) is empty.…”

Section: Applicationsmentioning

confidence: 96%

Weak laws of large numbers in geometric probability

Penrose¹,

Yukich²

2003

Ann. Appl. Probab.

Self Cite

160

346

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Using a coupling argument, we establish a general weak law of large numbers for functionals of binomial point processes in d-dimensional space, with a limit that depends explicitly on the (possibly non-uniform) density of the point process. The general result is applied to the minimal spanning tree, the k-nearest neighbors graph, the Voronoi graph, and the sphere of influence graph. Functionals of interest include total edge length with arbitrary weighting, number of vertices of specifed degree, and number of components. We also obtain weak laws for functionals of marked point processes, including statistics of Boolean models.

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On the variance of the random sphere of influence graph

Cited by 6 publications

References 13 publications

Central Limit Theorems for Some Graphs in Computational Geometry

Central Limit Theorems for Some Graphs in Computational Geometry

The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence

Weak laws of large numbers in geometric probability

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