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Merging Percolation on Z(d) and Classical Random Graphs: Phase Transition

Turova, Tatyana LU and Vallier, Thomas LU (2010) In Random Structures & Algorithms 36(2). p.185-217
Abstract
We study a random graph model which is a superposition of bond percolation on Z(d) with parameter p, and a classical random graph G(n,c/n). We show that this model, being a homogeneous random graph, has a natural relation to the so-called "rank I case" of inhomogeneous random graphs. This allows us to use the newly developed theory of inhomogeneous random graphs to describe the phase diagram on the set of parameters c >= 0 and 0 <= p < p(c), where p(c) = p(c)(d) is the critical probability for the bond percolation on Z(d). The phase transition is of second order as in the classical random graph. We find the scaled size of the largest connected component in the supercirtical regime. We also provide a sharp upper bound for the... (More)
We study a random graph model which is a superposition of bond percolation on Z(d) with parameter p, and a classical random graph G(n,c/n). We show that this model, being a homogeneous random graph, has a natural relation to the so-called "rank I case" of inhomogeneous random graphs. This allows us to use the newly developed theory of inhomogeneous random graphs to describe the phase diagram on the set of parameters c >= 0 and 0 <= p < p(c), where p(c) = p(c)(d) is the critical probability for the bond percolation on Z(d). The phase transition is of second order as in the classical random graph. We find the scaled size of the largest connected component in the supercirtical regime. We also provide a sharp upper bound for the largest connected component in the subcritical regime. The latter is a new result for inhomogeneous random graphs with unbounded kernels. (C) 2009 Wiley Periodicals, Inc. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
inhomogeneous random graphs, percolation, phase transition
in
Random Structures & Algorithms
volume
36
issue
2
pages
185 - 217
publisher
John Wiley & Sons
external identifiers
  • wos:000274458400003
  • scopus:74349124266
ISSN
1098-2418
DOI
10.1002/rsa.20287
language
English
LU publication?
yes
id
5e94ac8d-e283-4c85-8700-afc8d7bccdb0 (old id 1568788)
date added to LUP
2010-03-17 12:25:11
date last changed
2018-05-29 11:09:14
@article{5e94ac8d-e283-4c85-8700-afc8d7bccdb0,
  abstract     = {We study a random graph model which is a superposition of bond percolation on Z(d) with parameter p, and a classical random graph G(n,c/n). We show that this model, being a homogeneous random graph, has a natural relation to the so-called "rank I case" of inhomogeneous random graphs. This allows us to use the newly developed theory of inhomogeneous random graphs to describe the phase diagram on the set of parameters c &gt;= 0 and 0 &lt;= p &lt; p(c), where p(c) = p(c)(d) is the critical probability for the bond percolation on Z(d). The phase transition is of second order as in the classical random graph. We find the scaled size of the largest connected component in the supercirtical regime. We also provide a sharp upper bound for the largest connected component in the subcritical regime. The latter is a new result for inhomogeneous random graphs with unbounded kernels. (C) 2009 Wiley Periodicals, Inc.},
  author       = {Turova, Tatyana and Vallier, Thomas},
  issn         = {1098-2418},
  keyword      = {inhomogeneous random graphs,percolation,phase transition},
  language     = {eng},
  number       = {2},
  pages        = {185--217},
  publisher    = {John Wiley & Sons},
  series       = {Random Structures & Algorithms},
  title        = {Merging Percolation on Z(d) and Classical Random Graphs: Phase Transition},
  url          = {http://dx.doi.org/10.1002/rsa.20287},
  volume       = {36},
  year         = {2010},
}