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Random Geometric Graphs$
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Mathew Penrose

Print publication date: 2003

Print ISBN-13: 9780198506263

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198506263.001.0001

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SUBGRAPH AND COMPONENT COUNTS

SUBGRAPH AND COMPONENT COUNTS

Chapter:
(p.47) 3 SUBGRAPH AND COMPONENT COUNTS
Source:
Random Geometric Graphs
Author(s):

Mathew Penrose (Contributor Webpage)

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198506263.003.0003

This chapter is concerned with the number of embedded copies of a given finite graph γ in the random geometric graph G(n,r) (for example the number of edges or triangles). It is shown that if γ has k vertices and if r is chosen so that the number of copies of γ has mean value approaching a constant, then it is asymptotically Poisson distributed; if its mean tends to infinity, then after scaling and centring it is asymptotically normal. Similar results are given for the number of isolated components of G(n,r) isomorphic to γ. Multivariate extensions are also presented, with explicit formulae for limiting means and covariances.

Keywords:   edges, embedded copies, isomorphic components, asymptotically Poisson, asymptotically normal, multivariate extensions, explicit formulae

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