*Mathew Penrose*

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506263.003.0001
- Subject:
- Mathematics, Probability / Statistics

This introductory chapter contains a general discussion of both the historical and the applied background behind the study of random geometric graphs. A brief overview is presented, along with some ...
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This introductory chapter contains a general discussion of both the historical and the applied background behind the study of random geometric graphs. A brief overview is presented, along with some standard definitions in graph theory and probability theory. Specific terminology is introduced for two limiting regimes in the choice of r=r(n) (namely the thermodynamic limit where the mean vertex degree is made to approach a finite constant) and the connectivity regime (where it grows logarithmically with n). Some elementary probabilistic results are given on large deviations for the binomial and Poisson distribution, and on Poisson point processes.Less

This introductory chapter contains a general discussion of both the historical and the applied background behind the study of random geometric graphs. A brief overview is presented, along with some standard definitions in graph theory and probability theory. Specific terminology is introduced for two limiting regimes in the choice of *r=r(n)* (namely the thermodynamic limit where the mean vertex degree is made to approach a finite constant) and the connectivity regime (where it grows logarithmically with *n*). Some elementary probabilistic results are given on large deviations for the binomial and Poisson distribution, and on Poisson point processes.

*Andrew Goodall*

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.003.0007
- Subject:
- Mathematics, Probability / Statistics

This article reviews basic techniques of Fourier analysis on a finite abelian group Q, with subsequent applications in graph theory. These include evaluations of the Tutte polynomial of a graph G in ...
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This article reviews basic techniques of Fourier analysis on a finite abelian group Q, with subsequent applications in graph theory. These include evaluations of the Tutte polynomial of a graph G in terms of cosets of the Q-flows of G. Other applications to spanning trees of Cayley graphs and to group-valued models on phylogenetic trees are also presented to illustrate methods.Less

This article reviews basic techniques of Fourier analysis on a finite abelian group *Q*, with subsequent applications in graph theory. These include evaluations of the Tutte polynomial of a graph *G* in terms of cosets of the *Q*-flows of *G*. Other applications to spanning trees of Cayley graphs and to group-valued models on phylogenetic trees are also presented to illustrate methods.

*Peter Grindrod CBE*

- Published in print:
- 2014
- Published Online:
- March 2015
- ISBN:
- 9780198725091
- eISBN:
- 9780191792526
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198725091.003.0002
- Subject:
- Mathematics, Analysis, Probability / Statistics

This chapter considers how a range of ideas from linear algebra, nonnegative matrices, random matrices and modern graph theory may assist in the understanding of observed pairwise similarities and ...
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This chapter considers how a range of ideas from linear algebra, nonnegative matrices, random matrices and modern graph theory may assist in the understanding of observed pairwise similarities and networks over a large population of people or objects. In particular it examines some well-known different types of network models and considers (inverse problems concerning) how best to represent any observed social and friendship networks within such classes. This provides methods for inverse problems and soft clustering. Example applications are given to social networks and to clinical data.Less

This chapter considers how a range of ideas from linear algebra, nonnegative matrices, random matrices and modern graph theory may assist in the understanding of observed pairwise similarities and networks over a large population of people or objects. In particular it examines some well-known different types of network models and considers (inverse problems concerning) how best to represent any observed social and friendship networks within such classes. This provides methods for inverse problems and soft clustering. Example applications are given to social networks and to clinical data.