*Wilfrid S. Kendall and Ilya Molchanov (eds)*

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.001.0001
- Subject:
- Mathematics, Geometry / Topology

Stochastic geometry is a subject with roots stretching back at least 300 years, but one which has only been formed as an academic area in the last 50 years. It covers the study of random patterns, ...
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Stochastic geometry is a subject with roots stretching back at least 300 years, but one which has only been formed as an academic area in the last 50 years. It covers the study of random patterns, their probability theory, and the challenging problems raised by their statistical analysis. It has grown rapidly in response to challenges in all kinds of applied science, from image analysis through to materials science. Recently, still more stimulus has arisen from exciting new links with rapidly developing areas of mathematics, from fractals through percolation theory to randomized allocation schemes. Coupled with many ongoing developments arising from all sorts of applications, the area is changing and developing rapidly. This book is intended to lay foundations for future research directions by collecting together seventeen chapters contributed by leading researchers in the field, both theoreticians and people involved in applications, surveying these new developments both in theory and in applications. It will introduce and lay foundations for appreciating the fresh perspectives, new ideas, and interdisciplinary connections now arising from stochastic geometry and from other areas of mathematics now connecting to this area.Less

Stochastic geometry is a subject with roots stretching back at least 300 years, but one which has only been formed as an academic area in the last 50 years. It covers the study of random patterns, their probability theory, and the challenging problems raised by their statistical analysis. It has grown rapidly in response to challenges in all kinds of applied science, from image analysis through to materials science. Recently, still more stimulus has arisen from exciting new links with rapidly developing areas of mathematics, from fractals through percolation theory to randomized allocation schemes. Coupled with many ongoing developments arising from all sorts of applications, the area is changing and developing rapidly. This book is intended to lay foundations for future research directions by collecting together seventeen chapters contributed by leading researchers in the field, both theoreticians and people involved in applications, surveying these new developments both in theory and in applications. It will introduce and lay foundations for appreciating the fresh perspectives, new ideas, and interdisciplinary connections now arising from stochastic geometry and from other areas of mathematics now connecting to this area.

*Rolf Schneider and Wolfgang Weil*

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.003.0001
- Subject:
- Mathematics, Geometry / Topology

The aim of this chapter is to introduce the basic tools and structures of stochastic geometry and thus to lay the foundations for much of the book. Before this, a brief historic account will reflect ...
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The aim of this chapter is to introduce the basic tools and structures of stochastic geometry and thus to lay the foundations for much of the book. Before this, a brief historic account will reflect the development from elementary geometric probabilities over heuristic principles in applications to the advanced models employed in modern stochastic geometry. After the basic geometric and stochastic concepts have been presented, their interplay will be demonstrated by typical examples.Less

The aim of this chapter is to introduce the basic tools and structures of stochastic geometry and thus to lay the foundations for much of the book. Before this, a brief historic account will reflect the development from elementary geometric probabilities over heuristic principles in applications to the advanced models employed in modern stochastic geometry. After the basic geometric and stochastic concepts have been presented, their interplay will be demonstrated by typical examples.

*Mathew Penrose*

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198506263
- eISBN:
- 9780191707858
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506263.001.0001
- Subject:
- Mathematics, Probability / Statistics

This book sets out a body of rigorous mathematical theory for finite graphs with nodes placed randomly in Euclidean d-space according to a common probability density, and edges added to connect ...
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This book sets out a body of rigorous mathematical theory for finite graphs with nodes placed randomly in Euclidean d-space according to a common probability density, and edges added to connect points that are close to each other. As an alternative to classical random graph models, these geometric graphs are relevant to the modelling of real networks having spatial content, arising for example in wireless communications, parallel processing, classification, epidemiology, astronomy, and the internet. Their study illustrates numerous techniques of modern stochastic geometry, including Stein's method, martingale methods, and continuum percolation. Typical results in the book concern properties of a graph G on n random points with edges included for interpoint distances up to r, with the parameter r dependent on n and typically small for large n. Asymptotic distributional properties are derived for numerous graph quantities. These include the number of copies of a given finite graph embedded in G, the number of isolated components isomorphic to a given graph, the empirical distributions of vertex degrees, the clique number, the chromatic number, the maximum and minimum degree, the size of the largest component, the total number of components, and the connectivity of the graph.Less

This book sets out a body of rigorous mathematical theory for finite graphs with nodes placed randomly in Euclidean *d*-space according to a common probability density, and edges added to connect points that are close to each other. As an alternative to classical random graph models, these geometric graphs are relevant to the modelling of real networks having spatial content, arising for example in wireless communications, parallel processing, classification, epidemiology, astronomy, and the internet. Their study illustrates numerous techniques of modern stochastic geometry, including Stein's method, martingale methods, and continuum percolation. Typical results in the book concern properties of a graph *G* on *n* random points with edges included for interpoint distances up to *r*, with the parameter *r* dependent on *n* and typically small for large *n*. Asymptotic distributional properties are derived for numerous graph quantities. These include the number of copies of a given finite graph embedded in *G*, the number of isolated components isomorphic to a given graph, the empirical distributions of vertex degrees, the clique number, the chromatic number, the maximum and minimum degree, the size of the largest component, the total number of components, and the connectivity of the graph.