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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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THE LARGEST COMPONENT FOR A BINOMIAL PROCESS

THE LARGEST COMPONENT FOR A BINOMIAL PROCESS

Chapter:
(p.231) 11 THE LARGEST COMPONENT FOR A BINOMIAL PROCESS
Source:
Random Geometric Graphs
Author(s):

Mathew Penrose (Contributor Webpage)

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198506263.003.0011

This chapter is mainly concerned with the largest component of G(n,r) in the thermodynamic limit (with a non-random number of points). In the subcritical case, the order of the largest component grows logarithmically in n. If the underlying density f is non-uniform and the limiting mean degree has supercritical ‘islands’, then there is high probability that there is a large component (i.e., one of order proportional to n) associated with each island, which can be interpreted as consistency of single linkage clustering. A consistency result is also given for the ‘runt’ test, a test for unimodality of f based on the order of the second-largest cluster.

Keywords:   thermodynamic limit, non-random, single linkage, runt test, unimodality

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