Mathew D. Penrose

Random geometric graphs

1. Introduction 2. Probabilistic ingredients 3. Subgraph and component counts 4. Typical vertex degrees 5. Geometrical ingredients 6. Maximum degree, cliques and colourings 7. Minimum degree: laws of large numbers 8. Minimum degree: convergence in distribution 9. Percolative ingredients 10. Percolation and the largest component 11. The largest component for a binomial process 12. Ordering and partitioning problems 13. Connectivity and the number of components References Index

On a continuum percolation model

Consider particles placed in space by a Poisson process. Pairs of particles are bonded together, independently of other pairs, with a probability that depends on their separation, leading to the formation of clusters of particles. We prove the existence of a non-trivial critical intensity at which percolation occurs (that is, an infinite cluster forms). We then prove the continuity of the cluster density, or free energy. Also, we derive a formula for the probability that an arbitrary Poisson particle lies in a cluster consisting of k particles (or equivalently, a formula for the density of such clusters), and show that at high Poisson intensity, the probability that an arbitrary Poisson particle is isolated, given that it lies in a finite cluster, approaches 1. POISSON PROCESS; CLUSTER DENSITY; LARGE DEVIATIONS AT HIGH DENSITY

Approximating Layout Problems on Random Geometric Graphs

In this paper, we study the approximability of several layout problems on a family of random geometric graphs. Vertices of random geometric graphs are randomly distributed on the unit square and are connected by edges whenever they are closer than some given parameter. The layout problems that we consider are bandwidth, minimum linear arrangement, minimum cut width, minimum sum cut, vertex separation, and edge bisection. We first prove that some of these problems remain NP-complete even for geometric graphs. Afterwards, we compute lower bounds that hold, almost surely, for random geometric graphs. Then, we present two heuristics that, almost surely, turn out to be constant approximation algorithms for our layout problems on random geometric graphs. In fact, for the bandwidth and vertex separation problems, these heuristics are asymptotically optimal. Finally, we use the theoretical results in order to empirically compare these and other well-known heuristics.

Large deviations for discrete and continuous percolation

Motivated by a statistical application, we consider continuum percolation in two or more dimensions, restricted to a large finite box, when above the critical point. We derive surface order large deviation estimates for the volume of the largest cluster and for its intersection with the boundary of the box. We also give some natural extensions to known, analogous results on lattice percolation.

Laws of large numbers in stochastic geometry with statistical applications

Given

A Strong Law for the Largest Nearest-Neighbour Link between Random Points

n

Moments and central limit theorems for some multivariate Poisson functionals

independent random marked

Convergence Theorems for Some Layout Measures on Random Lattice and Random Geometric Graphs

d

CONNECTIVITY OF SOFT RANDOM GEOMETRIC GRAPHS

-vectors (points)

Gaussian limits for generalized spacings

X_i