[1] Entrainment processes in convective clouds often occur stochastically and entrainment rate estimates depend on the distance from the cloud from which the dry air is entrained. However, no observational studies exist on either the distance dependence or probability density function of entrainment rate, hindering understanding and the parameterization of convection. Here entrainment rate in c...

We prove a central limit theorem for the uctuations of the size of the Giant Component in a random graph with small edge probability.

The j-State General Markov Model of evolution (due to Steel) is a stochastic model concerned with the evolution of strings over an alphabet of size j. In particular, the Two-State General Markov Model of evolution generalises the well-known Cavender-Farris-Neyman model of evolution by removing the symmetry restriction (which requires that the probability that a ‘0’ turns into a ‘1’ along an edg...

The j-State General Markov Model of evolution (due to Steel) is a stochastic model concerned with the evolution of strings over an alphabet of size j. In particular, the Two-State General Markov Model of evolution generalises the well-known Cavender-Farris-Neyman model of evolution by removing the symmetry restriction (which requires that the probability that a `0' turns into a `1' along an edg...

The NP-hard graph bisection problem is to partition the nodes of an undirected graph into two equal-sized groups so as to minimize the number of edges that cross the partition. The more general graph l-partition problem is to partition the nodes of an undirected graph into l equal-sized groups so as to minimize the total number of edges that cross between groups. We present a simple, linear-tim...

The following probabilistic process models the generation of noisy clustering data: Clusters correspond to disjoint sets of vertices in a graph. Each two vertices from the same set are connected by an edge with probability p, and each two vertices from different sets are connected by an edge with probability r < p. The goal of the clustering problem is to reconstruct the clusters from the graph...

In this paper, we study the treewidth of the random geometric graph, obtained by dropping n points onto the square [0, √ n] and connect pairs of points by an edge if their distance is at most r = r(n). We prove a conjecture of Mitsche and Perarnau [19] stating that, with probability going to one as n→∞, the treewidth the random geometric graph is Θ(r √ n) when lim inf r > rc, where rc is the th...

1 Executive Summary We started with an introduction to the notion of percolation over random graphs with a 2-D lattice as an example. We then studied branching processes over random trees and proved conditions under which the such a process continues forever with positive probability. Finally, we defined the probability space of a random graph over a 2-D lattice. 2 Introduction to Percolation C...

Let G be a given graph (modelling a communication network) which we assume suuers from static edge faults: That is we let each edge of G be present independently with probability p (or absent with fault probability f = 1 ? p). In particular we are interested in ro-bustness results for the case that the graph G itself is a random member of the class of all regular graphs with given degree d. Her...

We extend the jigsaw percolation model to analyze graphs where both underlying people and puzzle graphs are Erdös-Rényi random graphs. Let pppl and ppuz denote the probability that an edge exists in the respective people and puzzle graphs and define peff = ppplppuz, the effective probability. We show for constants c1 > 1 and c2 > π /6 and c3 < e −5 if min(pppl, ppuz) > c1 logn/n the critical ef...