We analyze a minimal model of a growing network. At each time step, a new
vertex is added; then, with probability delta, two vertices are chosen
uniformly at random and joined by an undirected edge. This process is repeated
for t time steps. In the limit of large t, the resulting graph displays
surprisingly rich characteristics. In particular, a giant component emerges in
an infinite-order phase transition at delta = 1/8. At the transition, the
average component size jumps discontinuously but remains finite. In contrast, a
static random graph with the same degree distribution exhibits a second-order
phase transition at delta = 1/4, and the average component size diverges there.
These dramatic differences between grown and static random graphs stem from a
positive correlation between the degrees of connected vertices in the grown
graph--older vertices tend to have higher degree, and to link with other
high-degree vertices, merely by virtue of their age. We conclude that grown
graphs, however randomly they are constructed, are fundamentally different from
their static random graph counterparts.
%0 Journal Article
%1 Callaway2001Are
%A Callaway, Duncan S.
%A Hopcroft, John E.
%A Kleinberg, Jon M.
%A Newman, M. E. J.
%A Strogatz, Steven H.
%D 2001
%I American Physical Society
%J Physical Review E
%K critical-phenomena network-growth
%N 4
%P 041902+
%R 10.1103/physreve.64.041902
%T Are randomly grown graphs really random?
%U http://dx.doi.org/10.1103/physreve.64.041902
%V 64
%X We analyze a minimal model of a growing network. At each time step, a new
vertex is added; then, with probability delta, two vertices are chosen
uniformly at random and joined by an undirected edge. This process is repeated
for t time steps. In the limit of large t, the resulting graph displays
surprisingly rich characteristics. In particular, a giant component emerges in
an infinite-order phase transition at delta = 1/8. At the transition, the
average component size jumps discontinuously but remains finite. In contrast, a
static random graph with the same degree distribution exhibits a second-order
phase transition at delta = 1/4, and the average component size diverges there.
These dramatic differences between grown and static random graphs stem from a
positive correlation between the degrees of connected vertices in the grown
graph--older vertices tend to have higher degree, and to link with other
high-degree vertices, merely by virtue of their age. We conclude that grown
graphs, however randomly they are constructed, are fundamentally different from
their static random graph counterparts.
@article{Callaway2001Are,
abstract = {{We analyze a minimal model of a growing network. At each time step, a new
vertex is added; then, with probability delta, two vertices are chosen
uniformly at random and joined by an undirected edge. This process is repeated
for t time steps. In the limit of large t, the resulting graph displays
surprisingly rich characteristics. In particular, a giant component emerges in
an infinite-order phase transition at delta = 1/8. At the transition, the
average component size jumps discontinuously but remains finite. In contrast, a
static random graph with the same degree distribution exhibits a second-order
phase transition at delta = 1/4, and the average component size diverges there.
These dramatic differences between grown and static random graphs stem from a
positive correlation between the degrees of connected vertices in the grown
graph--older vertices tend to have higher degree, and to link with other
high-degree vertices, merely by virtue of their age. We conclude that grown
graphs, however randomly they are constructed, are fundamentally different from
their static random graph counterparts.}},
added-at = {2019-06-10T14:53:09.000+0200},
archiveprefix = {arXiv},
author = {Callaway, Duncan S. and Hopcroft, John E. and Kleinberg, Jon M. and Newman, M. E. J. and Strogatz, Steven H.},
biburl = {https://www.bibsonomy.org/bibtex/25a0ab11c60d596847ecfe3c73364ede8/nonancourt},
citeulike-article-id = {4484147},
citeulike-linkout-0 = {http://arxiv.org/abs/cond-mat/0104546},
citeulike-linkout-1 = {http://arxiv.org/pdf/cond-mat/0104546},
citeulike-linkout-2 = {http://dx.doi.org/10.1103/physreve.64.041902},
citeulike-linkout-3 = {http://link.aps.org/abstract/PRE/v64/i4/e041902},
citeulike-linkout-4 = {http://link.aps.org/pdf/PRE/v64/i4/e041902},
day = 14,
doi = {10.1103/physreve.64.041902},
eprint = {cond-mat/0104546},
interhash = {1d038ab764a379ab36e520a05f8243c9},
intrahash = {5a0ab11c60d596847ecfe3c73364ede8},
issn = {1063-651X},
journal = {Physical Review E},
keywords = {critical-phenomena network-growth},
month = jun,
number = 4,
pages = {041902+},
posted-at = {2014-06-04 01:09:14},
priority = {2},
publisher = {American Physical Society},
timestamp = {2019-08-01T16:23:28.000+0200},
title = {{Are randomly grown graphs really random?}},
url = {http://dx.doi.org/10.1103/physreve.64.041902},
volume = 64,
year = 2001
}