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Asymptotics for the size of the largest component scaled to "logn" in inhomogeneous random graphs

Turova, Tatyana LU (2013) In Arkiv för matematik 51(2). p.371-403
Abstract
We study inhomogeneous random graphs in the subcritical case. Among other results, we derive an exact formula for the size of the largest connected component scaled by logn, with n being the size of the graph. This generalizes a result for the "rank-1 case". We also investigate branching processes associated with these graphs. In particular, we discover that the same well-known equation for the survival probability, whose positive solution determines the asymptotics of the size of the largest component in the supercritical case, also plays a crucial role in the subcritical case. However, now it is the negative solutions that come into play. We disclose their relationship to the distribution of the progeny of the branching process.
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Arkiv för matematik
volume
51
issue
2
pages
371 - 403
publisher
Springer
external identifiers
  • wos:000323247000010
  • scopus:84883824997
ISSN
0004-2080
DOI
10.1007/s11512-012-0178-4
language
English
LU publication?
yes
id
0c86109f-74e9-41fa-ac51-86c7c6913a73 (old id 4027116)
date added to LUP
2016-04-01 14:45:36
date last changed
2020-01-12 17:04:22
@article{0c86109f-74e9-41fa-ac51-86c7c6913a73,
  abstract     = {We study inhomogeneous random graphs in the subcritical case. Among other results, we derive an exact formula for the size of the largest connected component scaled by logn, with n being the size of the graph. This generalizes a result for the "rank-1 case". We also investigate branching processes associated with these graphs. In particular, we discover that the same well-known equation for the survival probability, whose positive solution determines the asymptotics of the size of the largest component in the supercritical case, also plays a crucial role in the subcritical case. However, now it is the negative solutions that come into play. We disclose their relationship to the distribution of the progeny of the branching process.},
  author       = {Turova, Tatyana},
  issn         = {0004-2080},
  language     = {eng},
  number       = {2},
  pages        = {371--403},
  publisher    = {Springer},
  series       = {Arkiv för matematik},
  title        = {Asymptotics for the size of the largest component scaled to "logn" in inhomogeneous random graphs},
  url          = {http://dx.doi.org/10.1007/s11512-012-0178-4},
  doi          = {10.1007/s11512-012-0178-4},
  volume       = {51},
  year         = {2013},
}