Random geometric graphs (RGG) contain vertices whose points are uniformly distributed in a given plane and an edge between two distinct nodes exists when their distance is less than a given positive value. RGGs are appropriate for modeling multi-hop wireless networks consisting of n mobile devices with transmission radius r unit length that are independently and uniformly distributed randomly i...

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 ...

This paper presents a method for fitting a digital line (resp. plane) to a given set of points in a 2D (resp. 3D) image in the presence of outliers. One of the most widely used methods is Random Sample Consensus (RANSAC). However it is also known that RANSAC has a drawback: as maximum iteration number must be set, the solution may not be optimal. To overcome this problem, we present a new metho...

We provide the first analytical results for the connectivity of dynamic random geometric graphs-a model of mobile wireless networks in which vertices move in random (and periodically updated) directions, and an edge exists between two vertices if their Euclidean distance is below a given threshold. We provide precise asymptotic results for the expected length of the connectivity and disconnecti...

We show that a variant of the random-edge pivoting rule results in a strongly polynomial time simplex algorithm for linear programs max{cTx : x ∈ R, Ax 6 b}, whose constraint matrix A satisfies a geometric property introduced by Brunsch and Röglin: The sine of the angle of a row of A to a hyperplane spanned by n− 1 other rows of A is at least δ. This property is a geometric generalization of A ...

We introduce a new class of countably infinite random geometric graphs, whose vertices V are points in a metric space, and vertices are adjacent independently with probability p ∈ (0, 1) if the metric distance between the vertices is below a given threshold. If V is a countable dense set in R equipped with the metric derived from the L∞-norm, then it is shown that with probability 1 such infini...

We study the expected topological properties of Čech and Vietoris-Rips complexes built on randomly sampled points in R. These are, in some cases, analogues of known results for connectivity and component counts for random geometric graphs. However, an important difference in this setting is that homology is not monotone in the underlying parameter. In the sparse range, we compute the expectatio...

We analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present. The critical connectivity is found numerically by examining the size of the largest cluster. We derive an analytical expression for the cluster coefficient, which shows that the graphs are distinctly different from standard ra...

For the theoretical study of real-world networks, we propose a model of scale-free randomgraphs with underlying geometry that we call geometric inhomogeneous random graphs (GIRGs).GIRGs generalize hyperbolic random graphs, which are a popular model to test algorithms forsocial and technological networks. Our generalization overcomes some limitations of hyperbolicrandom graphs, w...