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Mathew D. Penrose

Bio: Mathew D. Penrose is an academic researcher from University of Bath. The author has contributed to research in topics: Central limit theorem & Poisson distribution. The author has an hindex of 35, co-authored 133 publications receiving 4799 citations. Previous affiliations of Mathew D. Penrose include University of Edinburgh & University of California, Santa Barbara.


Papers
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Journal ArticleDOI
TL;DR: For n points placed uniformly at random on the unit square, it is known that the distribution of the minimal spanning tree on these points converges weakly to the double exponential as mentioned in this paper.
Abstract: For n points placed uniformly at random on the unit square, suppose $M_n$ (respectively, $M'_n$) denotes the longest edge-length of the nearest neighbor graph (respectively, the minimal spanning tree) on these points. It is known that the distribution of $n \pi M_n^2 - \log n$ converges weakly to the double exponential; we give a new proof of this. We show that $P[M'_n = M_n] \to 1$, so that the same weak convergence holds for $M'_n$ .

528 citations

Journal ArticleDOI
TL;DR: For n points uniformly randomly distributed on the unit cube in d dimensions, with d≥2, let ρn (respectively, σn) denote the minimum r at which the graph, obtained by adding an edge between each pair of points distant at most r apart, is k-connected.
Abstract: For n points uniformly randomly distributed on the unit cube in d dimensions, with d≥2, let ρn (respectively, σn) denote the minimum r at which the graph, obtained by adding an edge between each pair of points distant at most r apart, is k-connected (respectively, has minimum degree k). Then P[ρn=σn]1 as n∞. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 15, 145–164, 1999

453 citations

Book
26 Oct 2017
TL;DR: In this article, the authors developed the theory of the Poisson process in the setting of a general abstract measure space, establishing basic results and properties as well as certain advanced topics in the stochastic analysis of the poisson process.
Abstract: The Poisson process, a core object in modern probability, enjoys a richer theory than is sometimes appreciated. This volume develops the theory in the setting of a general abstract measure space, establishing basic results and properties as well as certain advanced topics in the stochastic analysis of the Poisson process. Also discussed are applications and related topics in stochastic geometry, including stationary point processes, the Boolean model, the Gilbert graph, stable allocations, and hyperplane processes. Comprehensive, rigorous, and self-contained, this text is ideal for graduate courses or for self-study, with a substantial number of exercises for each chapter. Mathematical prerequisites, mainly a sound knowledge of measure-theoretic probability, are kept in the background, but are reviewed comprehensively in the appendix. The authors are well-known researchers in probability theory; especially stochastic geometry. Their approach is informed both by their research and by their extensive experience in teaching at undergraduate and graduate levels.

363 citations

Journal ArticleDOI
TL;DR: In this article, a general central limit theorem for functionals of point sets was obtained by restricting a homogeneous Poisson process to the set of points in the set and then taking uniformly distributed points from the set.
Abstract: Let $(B_n)$ be an increasing sequence of regions in $d$ -dimensional space with volume $n$ and with union $\mathbb{R}^d$. We prove a general central limit theorem for functionals of point sets, obtained either by restricting a homogeneous Poisson process to $(B_n)$, or by by taking $n$ uniformly distributed points in $(B_n)$. The sets $(B_n)$ could be all cubes but a more general class of regions$(B_n)$ is considered. Using this general result we obtain central limit theorems for specific functionals suchas total edge lengthand number of components, defined in terms of graphs such as the $k$-nearest neighbors graph, the sphere of influence graph and the Voronoi graph.

239 citations

Journal ArticleDOI
TL;DR: In this article, a general weak law of large numbers for functionals of binomial point processes in d-dimensional space is established, with a limit that depends explicitly on the density of the point process.
Abstract: Using a coupling argument, we establish a general weak law of large numbers for functionals of binomial point processes in d-dimensional space, with a limit that depends explicitly on the (possibly nonuniform) density of the point process. The general result is applied to the minimal spanning tree, the k-nearest neighbors graph, the Voronoi graph and the sphere of influence graph. Functionals of interest include total edge length with arbitrary weighting, number of vertices of specified degree and number of components. We also obtain weak laws of large numbers functionals of marked point processes, including statistics of Boolean models.

224 citations


Cited by
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Journal ArticleDOI
TL;DR: In this article, the authors present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches, and discuss the advantages and disadvantages of these algorithms.
Abstract: In recent years, spectral clustering has become one of the most popular modern clustering algorithms. It is simple to implement, can be solved efficiently by standard linear algebra software, and very often outperforms traditional clustering algorithms such as the k-means algorithm. On the first glance spectral clustering appears slightly mysterious, and it is not obvious to see why it works at all and what it really does. The goal of this tutorial is to give some intuition on those questions. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Advantages and disadvantages of the different spectral clustering algorithms are discussed.

9,141 citations

Proceedings ArticleDOI
22 Jan 2006
TL;DR: Some of the major results in random graphs and some of the more challenging open problems are reviewed, including those related to the WWW.
Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

7,116 citations

Journal ArticleDOI

6,278 citations

Journal ArticleDOI
TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

5,689 citations

Journal ArticleDOI
TL;DR: This work proposes a framework for analyzing and comparing distributions, which is used to construct statistical tests to determine if two samples are drawn from different distributions, and presents two distribution free tests based on large deviation bounds for the maximum mean discrepancy (MMD).
Abstract: We propose a framework for analyzing and comparing distributions, which we use to construct statistical tests to determine if two samples are drawn from different distributions. Our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel Hilbert space (RKHS), and is called the maximum mean discrepancy (MMD).We present two distribution free tests based on large deviation bounds for the MMD, and a third test based on the asymptotic distribution of this statistic. The MMD can be computed in quadratic time, although efficient linear time approximations are available. Our statistic is an instance of an integral probability metric, and various classical metrics on distributions are obtained when alternative function classes are used in place of an RKHS. We apply our two-sample tests to a variety of problems, including attribute matching for databases using the Hungarian marriage method, where they perform strongly. Excellent performance is also obtained when comparing distributions over graphs, for which these are the first such tests.

3,792 citations