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

Abstract

This article continues an investigation begun in [2]. A random graph Gn(x) is constructed on independent random points U1, · ··, Un distributed uniformly on [0, 1]d, d ≧ 1, 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 results are obtained for the convergence/divergence of the minimum vertex degree of the random graph, as the number n of points becomes large and the edge distance x is allowed to vary with n. The largest nearest neighbor link dn, the smallest x such that Gn(x) has no vertices of degree zero, is shown to satisfy Series and sequence criteria on edge distances {xn} are provided which guarantee the random graph to be complete, a.s. These criteria imply a.s. limiting behavior of the diameter of the vertex set.

Performance Analysis of Connectivity and Localization in Multi-Hop Underwater Optical Wireless Sensor Networks

“…Random sector directed graphs and random geometric graphs are identical in case of φ = 2π [13], [14], [48], [49].…”

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“…The graph G(ᐂ, d) is called a scaled unit disk graph, since each disk has the same diameter. (It is often easier to work with the supremum norm; see, for example, [1,2], but it is the Euclidean norm that is appropriate here. )…”

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“…Then, with a very high probability, the resulting graph becomes (k + 1)-connected at the same edge length r * (n) at which the minimum degree of the graph becomes k + 1, for k ≥ 0. With k = 0, this result means that the graph becomes connected with high probability at the same time that the isolated vertices disappear from the graph [1][2][3] is a similar study for the l ∞ norm. The best introduction to the study of RGGs via their asymptotic properties is [17].…”

“…[17] derives similar results for dimensions d ≥ 2 and for general densities having bounded support. [2,3] show such strong laws for the uniform RGG for d ≥ 1. [13] obtains strong law results for the one-dimensional uniform RGG using the graph with independent exponential spacings of [15].…”