In this paper, we provide degree distributions for $AB$ random geometric graphs, in which points of type $A$ connect to the closest $k$ $B$. The motivating example derive such is 5G wireless networks with multi-connectivity, where users their base stations. It important know how many a particular station serves, gives that station. To obtain these distributions, investigate distribution area si...

We give a characterization of vertex-monotone properties with sharp thresholds in a Poisson random geometric graph or hypergraph. As an application we show that a geometric model of random k-SAT exhibits a sharp threshold for satisfiability.

Entropy and—maybe even more so—mutual information are invaluable tools for analyzing properties of probability distributions, especially in coding theory. While there are general definitions for both concepts available for arbitrary probability distributions, these tend to be hard to work with and the literature (e.g., [CT06]) focuses on either discrete, or continuous random variables. In this ...

In this paper, we study the vertex pursuit game of Cops and Robbers, in which cops try to capture a robber on the vertices of a graph. The minimum number of cops required to win on a given graph G is called the cop number of G. We focus on G(n, r, p), a percolated random geometric graph in which n vertices are chosen uniformly at random and independently from [0, 1]. Two vertices are adjacent w...