The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1]^d

Abstract

On independent random points U1,· ··,Un distributed uniformly on [0, 1]d, a random graph Gn(x) is constructed in which two distinct such points are joined by an edge if the l∞-distance between them is at most some prescribed value 0 ≦ x ≦ 1. Almost-sure asymptotic rates of convergence/divergence are obtained for the maximum vertex degree of the random graph and related quantities, including the clique number, chromatic number and independence number, as the number n of points becomes large and the edge distance x is allowed to vary with n. Series and sequence criteria on edge distances {xn} are provided which guarantee the random graph to be empty of edges, a.s.

The Minimum Vertex Degree of a Graph on Uniform Points in [0, 1]d

“…This article continues an investigation begun in [2], to which the reader is referred for a detailed introduction. Let VI, V if and only if II Vi -U,II~x.…”

“…Let VI, V if and only if II Vi -U,II~x. As in [2], we take 11•11 to be the loo-norm: II Z II = max {Zl' • • • ,Zd}' This choice is made mainly for specificity and computational convenience. Other metrics may be more natural in particular modeling scenarios but the methods used herein may require modification.…”

“…For each i = 1, • • . , n, let (1) deg Vn,i(X) = L ei,j(x) l~r~i~n SGSA•583 be the degree of the vertex Vi in the graph G n (x), and let (2) be the minimum vertex degree of the random graph Gn(x). Note that 8" is non-decreasing in x for fixed n.…”

Focusing of the scan statistic and geometric clique number

“…A related concept is the clique number of a random geometric graph. Given a finite point set X ⊂ R d , given a norm • on R d and given r > 0, let G(X; r) denote the graph with vertex set X and with distinct points x, y ∈ X connected by an edge if and only if x −y ≤ h. Taking X to be a binomial or Poisson point process, many properties of such random geometric graphs are studied in [3], [14], [17], [18] and elsewhere. For general motivation in cluster analysis based on multivariate data measurements, see [12].…”

“…As observed in [3], in the special case where the scanning set C is a rectilinear cube of side r, the scan statistic on X is identical to C(G(X; r)), using the l ∞ (max-component) norm. For other norms, the scan statistic and geometric clique number are not identical, but have similar properties.…”