Central Limit Theorems for Some Graphs in Computational Geometry

Abstract

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.

Limit theory for the random on‐line nearest‐neighbor graph

“…whereG 1 has distribution given by (16 (iii) For α > 1, the distribution of the limit W (1, α) of (4) is given by…”

“…(e) Figure 3 is a plot of the estimated probability density function ofG 1 given by (16). This was obtained by performing 10 5 repeated simulations of the ONG on a sequence of 10 3 uniform (simulated) random points on (0, 1).…”

“…given by (9), (11), (16) respectively. The first three moments ofJ 1 ,H 1 andG 1 are given in Table 2.…”

“…Many aspects of the large-sample asymptotic theory for such graphs, which are locally determined in a certain sense, are by now quite well understood. See for example [10,13,14,16,17,23,26].…”

Concentration of measures via size-biased couplings

“…The random variables V and S are of fundamental interest in stochastic geometry, see [16] and [21]. If n → ∞ and ρ remains fixed, both V and S satisfy a central limit theorem [16,20,23]. The L 1 distance of V , properly standardized, to the normal is studied in [7] using Stein's method.…”