Limit theory for the Gilbert graph

Abstract

For a given homogeneous Poisson point process in R d two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behaviour of the resulting random graph, the Gilbert graph or random geometric graph, is investigated as the intensity of the Poisson point process is increased and the distance parameter goes to zero. The asymptotic expectation and covariance structure of a class of length-power functionals are computed. Distributional limit theorems are derived that have a Gaussian, a stable or a compound Poisson limiting distribution. Finally, concentration inequalities are provided using a concentration inequality for the convex distance.

Concentration for Poisson U-statistics: Subgraph counts in random geometric graphs

“…This is actually best possible as will be pointed out later. Note also that the asymptotic exponent of the estimates in previous results from [21,15] are 1/k for both the upper and the lower tail which is compatible with our upper tail bounds but worse than our lower tail inequalities.…”

“…In the context of concentration properties, a natural question is whether concentration inequalities also hold in these situations. We emphasize that Theorem 1.1 only requires N < ∞ almost surely and hence covers such cases, as opposed to previous results from [21,15] where only finite intensity measures are considered.…”

Poisson Point Process Convergence and Extreme Values in Stochastic Geometry

“…Consequently, 2R(V η t ) is also the same as the shortest edge length of a random geometric graph based on η t as discussed in the introduction (cf. [26] and [23] for an exhaustive reference on random geometric graphs) or as the shortest edge length of a Delaunay graph (see [7,28] for background material on Delaunay graphs or tessellations). A similar comment applies if η t is replaced by a binomial point process ζ n .…”

“…Taking η t as vertex set of a random graph, we connect two different points of η t by an edge if and only if their Euclidean distance does not exceed θ t . The so-constructed random geometric graph, or Gilbert graph, is among the most prominent random graph models (see [26] for some recent developments and [23] for an exhaustive reference). We now consider the order statistic ξ t = {M (m) t : m ∈ N} defined by the edge-lengths of the random geometric graph, that is, M (1) t is the length of the shortest edge, M (2) t is the length of the second-shortest edge etc.…”

A multivariate generalization of Prony's method

“…, j) , imply sep(Ω) ≈ CM −1 , i.e., both terms are of similar size, iii) and finally parameters t j ∈ [0, 1) d chosen at random from the uniform distribution, imply E sep(Ω) = C d M −2 , see e.g. [29], and thus max{2dq −1 ,…”