Related papers: Several Topics in Experimental Mathematics

Given a graphical degree sequence ${\bf d}=(d_1,\ldots, d_n)$, let $G(n, {\bf d})$ denote a uniformly random graph on vertex set $[n]$ where vertex $ i$ has degree $d_i$ for every $1\le i\le n$. We give upper and lower bounds on the joint…

Continuous mixtures of distributions are widely employed in the statistical literature as models for phenomena with highly divergent outcomes; in particular, many familiar heavy-tailed distributions arise naturally as mixtures of…

We study some properties of graphs (or, rather, graph sequences) defined by demanding that the number of subgraphs of a given type, with vertices in subsets of given sizes, approximatively equals the number expected in a random graph. It…

This PhD thesis deals with a number of different problems in mathematical physics with the common thread that they have probabilistic aspects. The problems all stem from mathematical studies of lattice systems in statistical and quantum…

We consider the following problem arising from the study of human problem solving: Let $G$ be a vertex-weighted graph with marked "in" and "out" vertices. Suppose a random walker begins at the in-vertex, steps to neighbors of vertices with…

We develop a theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes. Our approach generalizes the usual (co)variance to the setting of…

Graph theory provides fundamental concepts for many fields of science like statistical physics, network analysis and theoretical computer science. Here we give a pedagogical introduction to graph theory, divided into three sections. In the…

We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics. The three main types of problems we…

We examine a number of results of infinite combinatorics using the techniques of reverse mathematics. Our results are inspired by similar results in recursive combinatorics. Theorems included concern colorings of graphs and bounded graphs,…

This work addresses the problem of simulating Gaussian random fields that are continuously indexed over a class of metric graphs, termed graphs with Euclidean edges, being more general and flexible than linear networks. We introduce three…

In this thesis, which is supervised by Dr. David Penman, we examine random interval graphs. Recall that such a graph is defined by letting $X_{1},\ldots X_{n},Y_{1},\ldots Y_{n}$ be $2n$ independent random variables, with uniform…

Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the…

Solving optimization problems leads to elegant and practical solutions in a wide variety of real-world applications. In many of those real-world applications, some of the information required to specify the relevant optimization problem is…

We develop a statistical mechanics approach for random networks with uncorrelated vertices. We construct equilibrium statistical ensembles of such networks and obtain their partition functions and main characteristics. We find simple…

In this paper, we investigate the problem of assessing statistical methods and effectively summarizing results from simulations. Specifically, we consider problems of the type where multiple methods are compared on a reasonably large test…

Here I will present an introduction to the results that have been recently obtained in constraint optimization of random problems using statistical mechanics techniques. After presenting the general results, in order to simplify the…

We review mathematically tractable models for connected networks on random points in the plane, emphasizing the class of proximity graphs which deserves to be better known to applied probabilists and statisticians. We introduce and motivate…

This thesis is devoted to the study of extreme value statistics in stochastic processes and their applications. In the first part, we obtain exact analytical results on the extreme value statistics of both discrete-time and continuous-time…

Determining the crossing numbers of Cartesian products of small graphs with arbitrarily large paths has been an ongoing topic of research since the 1970s. Doing so requires the establishment of coincident upper and lower bounds; the former…

We revisit the problem of designing sublinear algorithms for estimating the average degree of an $n$-vertex graph. The standard access model for graphs allows for the following queries: sampling a uniform random vertex, the degree of a…