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Cover Art
PRINTED BOOKS
Author Penrose, Mathew.

Title Random geometric graphs / Mathew Penrose.

Published Oxford ; New York : Oxford University Press, 2003.

Copies

Location Call No. Status
 UniM ERC  511.5 PENR    AVAILABLE
Physical description xiii, 330 pages : illustrations ; 25 cm.
Series Oxford studies in probability ; 5.
Oxford studies in probability ; 5.
Bibliography Includes bibliographical references and index.
Contents 1.1 Motivation and history 1 -- 1.2 Statistical background 4 -- 1.3 Computer science background 7 -- 1.4 Outline of results 9 -- 1.5 Some basic definitions 11 -- 1.6 Elements of probability 14 -- 1.7 Poissonization 18 -- 2 Probabilistic ingredients 22 -- 2.1 Dependency graphs and Poisson approximation 22 -- 2.2 Multivariate Poisson approximation 25 -- 2.3 Normal approximation 27 -- 2.4 Martingale theory 33 -- 2.5 De-Poissonization 37 -- 3 Subgraph and component counts 47 -- 3.1 Expectations 48 -- 3.2 Poisson approximation 52 -- 3.3 Second moments in a Poisson process 55 -- 3.4 Normal approximation for Poisson processes 60 -- 3.5 Normal approximation: de-Poissonization 65 -- 3.6 Strong laws of large numbers 69 -- 4 Typical vertex degrees 74 -- 4.1 Setup 75 -- 4.2 Laws of large numbers 76 -- 4.3 Asymptotic covariances 78 -- 4.4 Moments for de-Poissonization 82 -- 4.5 Finite-dimensional central limit theorems 87 -- 4.6 Convergence in Skorohod space 91 -- 5 Geometrical ingredients 95 -- 5.1 Consequences of the Lebesgue density theorem 95 -- 5.2 Covering, packing, and slicing 97 -- 5.3 Brunn-Minkowski inequality 102 -- 5.4 Expanding sets in the orthant 104 -- 6 Maximum degree, cliques, and colourings 109 -- 6.1 Focusing 110 -- 6.2 Subconnective laws of large numbers 118 -- 6.3 More laws of large numbers for maximum degree 120 -- 6.4 Laws of large numbers for clique number 126 -- 6.5 Chromatic number 130 -- 7 Minimum degree: laws of large numbers 136 -- 7.1 Thresholds in smoothly bounded regions 136 -- 7.2 Strong laws for thresholds in the cube 145 -- 7.3 Strong laws for the minimum degree 151 -- 8 Minimum degree: convergence in distribution 155 -- 8.1 Uniformly distributed points I 156 -- 8.2 Uniformly distributed points II 160 -- 8.3 Normally distributed points I 167 -- 8.4 Normally distributed points II 173 -- 9 Percolative ingredients 177 -- 9.1 Unicoherence 177 -- 9.2 Connectivity and Peierls arguments 177 -- 9.3 Bernoulli percolation 180 -- 9.4 k-Dependent percolation 186 -- 9.5 Ergodic theory 187 -- 9.6 Continuum percolation: fundamentals 188 -- 10 Percolation and the largest component 194 -- 10.1 Subcritical regime 195 -- 10.2 Existence of a crossing component 200 -- 10.3 Uniqueness of the giant component 205 -- 10.4 Sub-exponential decay for supercritical percolation 210 -- 10.5 Second-largest component 216 -- 10.6 Large deviations in the supercritical regime 220 -- 10.7 Fluctuations of the giant component 224 -- 11 Largest component for a binomial process 231 -- 11.1 Subcritical case 231 -- 11.2 Supercritical case on the cube 234 -- 11.3 Fractional consistency of single-linkage clustering 240 -- 11.4 Consistency of the RUNT test for unimodality 247 -- 11.5 Fluctuations of the giant component 252 -- 12 Ordering and partitioning problems 259 -- 12.1 Background on layout problems 259 -- 12.2 Subcritical case 262 -- 12.3 Supercritical case 268 -- 12.4 Superconnectivity regime 275 -- 13 Connectivity and the number of components 281 -- 13.1 Multiple connectivity 282 -- 13.2 Strong laws for points in the cube or torus 283 -- 13.3 SLLN in smoothly bounded regions 289 -- 13.4 Convergence in distribution 295 -- 13.5 Further results on points in the cube 302 -- 13.6 Normally distributed points 306 -- 13.7 Component count in the thermodynamic limit 309.
Summary This monograph provides and explains the mathematics behind geometric graph theory, which studies the properties of a graph that consists of nodes placed in Euclidean space so that edges can be added to connect points that are close to one another. For example, a collection of trees scattered in a forest and the disease that is passed between them, a set of nests of animals or birds on a region and the communication between them or communication between communications stations or nerve cells. Aimed at graduate students and researchers in probability, statistics, combinatorics and graph theory including computer scientists, it covers topics such as: technical tools, edge and component counts, vertex degrees, clique and chromatic number, and connectivity. Applications of this theory are used in the study of neural networks, spread of disease, astrophysics and spatial statistics.
Subject Random graphs.
ISBN 0198506260