We consider a random graph G(n, p) whose vertex set V has been randomly embedded in the unit square and whose edges are given weight equal to the geometric distance between their end vertices. Then each pair {u, v} of vertices have a distance in the weighted graph, and a Euclidean distance. The stretch factor of the embedded graph is defined as the maximum ratio of these two distances, over all...

The dilation of a geometric graph is the maximum, over all pairs of points in the graph, of the ratio of the Euclidean length of the shortest path between them in the graph and their Euclidean distance. We consider a generalized version of this notion, where the nodes of the graph are not points but axis-parallel rectangles in the plane. The arcs in the graph are horizontal or vertical segments...

We formulate the problem of graph inference where part of the graph is known as a supervised learning problem, and propose an algorithm to solve it. The method involves the learning of a mapping of the vertices to a Euclidean space where the graph is easy to infer, and can be formulated as an optimization problem in a reproducing kernel Hilbert space. We report encouraging results on the proble...

In this paper we propose an encoding scheme and ad hoc operators for a genetic approach to graph clustering. Given a connected graph whose vertices correspond to points within a Euclidean space and a fitness function, a hierarchy of graphs in which each vertex corresponds to a connected subgraph of the graph below is generated. Both the number of clustering levels and the number of clusters on ...

We improve by an exponential factor the best known asymptotic upper bound for the density of sets avoiding 1 in Euclidean space. This result is obtained by a combination of an analytic bound that is an analogue of Lovász theta number and of a combinatorial argument involving finite subgraphs of the unit distance graph. In turn, we straightforwardly obtain an asymptotic improvement for the measu...

In this paper we investigate the relations between spanners, weak spanners, and power spanners in R D for any dimension D and apply our results to topology control in wireless networks. For c ∈ R, a c-spanner is a subgraph of the complete Euclidean graph satisfying the condition that between any two vertices there exists a path of length at most c-times their Euclidean distance. Based on this a...

For achieving seamless global coverage and real-time communications while providing intelligent applications with increased quality of service (QoS), AI-enabled space–air–ground integrated networks (SAGINs) have attracted widespread attention from all walks life. However, high-intensity interactions pose fundamental challenges for resource orchestration security issues. Meanwhile, virtual netwo...

Given any two vertices u, v of a random geometric graph, denote by dE(u, v) their Euclidean distance and by dG(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) in terms of dE(u, v) has received a lot of attention in the literature [1, 2, 6, 8]. In this paper, we improve these upper bounds for values of r = ω( √ logn) (i.e. for r above the connectivity threshold). Our ...

We study the problem how to obtain a small drawing of a 3-polytope with Euclidean distance between any two points at least 1. The problem can be reduced to a one-dimensional problem, since it is sufficient to guarantee distinct integer x-coordinates. We develop an algorithm that yields an embedding with the desired property such that the polytope is contained inside a 2(n − 2) × 2 × 1 box. The ...