Structural Complexity of One-Dimensional Random Geometric Graphs

Connectivity of Random High Dimensional Geometric Graphs

We consider graphs obtained by placing n points at random on a unit sphere in R, and connecting two points by an edge if they are close to each other (e.g., the angle at the origin that their corresponding unit vectors make is at most π/3). We refer to these graphs as geometric graphs. We also consider a complement family of graphs in which two points are connected by an edge if they are far aw...

Connectivity in One-Dimensional Geometric Random Graphs: Poisson Approximations, Zero- One Laws and Phase Transitions

Consider n points (or nodes) distributed uniformly and independently on the unit interval [0, 1]. Two nodes are said to be adjacent if their distance is less than some given threshold value. For the underlying random graph we derive zero-one laws for the property of graph connectivity and give the asymptotics of the transition widths for the associated phase transition. These results all flow f...

On the zero-one law for connectivity in one-dimensional geometric random graphs

We consider the geometric random graph where n points are distributed uniformly and independently on the unit interval [0, 1]. Using the method of first and second moments, we provide a simple proof of the “zero-one” law for the property of graph connectivity under the asymptotic regime created by having n become large and the transmission range scaled appropriately with n.

Topological properties of the one dimensional exponential random geometric graph

Infinite Random Geometric Graphs

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...