We present an exact solution of percolation in a generalized class of Watts-Strogatz graphs defined on a one-dimensional underlying lattice. We find a nonclassical critical point in the limit of the number of long-range bonds in the system going to zero, with a discontinuity in the percolation probability and a divergence in the mean finite-cluster size. We show that the critical behavior falls into one of three regimes depending on the proportion of occupied long-range to unoccupied nearest-neighbor bonds, with each regime being characterized by different critical exponents. The three regimes can be united by a single scaling function around the critical point. These results can be used to identify the number of long-range links necessary to secure connectivity in a communication or transportation chain. As an example, we can resolve the communication problem in a game of “telephone.”
%0 Journal Article
%1 Cohen2009Unusual
%A Cohen, Reuven
%A Dawid, Daryush J.
%A Kardar, Mehran
%A Yam, Yaneer B.
%D 2009
%I APS
%J Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)
%K small-world percolation critical-phenomena networks explosive-percolation critical-points
%N 6
%P 066112+
%R 10.1103/physreve.79.066112
%T Unusual percolation in simple small-world networks
%U http://dx.doi.org/10.1103/physreve.79.066112
%V 79
%X We present an exact solution of percolation in a generalized class of Watts-Strogatz graphs defined on a one-dimensional underlying lattice. We find a nonclassical critical point in the limit of the number of long-range bonds in the system going to zero, with a discontinuity in the percolation probability and a divergence in the mean finite-cluster size. We show that the critical behavior falls into one of three regimes depending on the proportion of occupied long-range to unoccupied nearest-neighbor bonds, with each regime being characterized by different critical exponents. The three regimes can be united by a single scaling function around the critical point. These results can be used to identify the number of long-range links necessary to secure connectivity in a communication or transportation chain. As an example, we can resolve the communication problem in a game of “telephone.”
@article{Cohen2009Unusual,
abstract = {{We present an exact solution of percolation in a generalized class of Watts-Strogatz graphs defined on a one-dimensional underlying lattice. We find a nonclassical critical point in the limit of the number of long-range bonds in the system going to zero, with a discontinuity in the percolation probability and a divergence in the mean finite-cluster size. We show that the critical behavior falls into one of three regimes depending on the proportion of occupied long-range to unoccupied nearest-neighbor bonds, with each regime being characterized by different critical exponents. The three regimes can be united by a single scaling function around the critical point. These results can be used to identify the number of long-range links necessary to secure connectivity in a communication or transportation chain. As an example, we can resolve the communication problem in a game of \“telephone.\”}},
added-at = {2019-06-10T14:53:09.000+0200},
author = {Cohen, Reuven and Dawid, Daryush J. and Kardar, Mehran and Yam, Yaneer B.},
biburl = {https://www.bibsonomy.org/bibtex/2db180f38a76d89fb81acf1ddc4237877/nonancourt},
citeulike-article-id = {4971588},
citeulike-linkout-0 = {http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal\&id=PLEEE8000079000006066112000001\&idtype=cvips\&gifs=yes},
citeulike-linkout-1 = {http://link.aps.org/abstract/PRE/v79/e066112},
citeulike-linkout-2 = {http://dx.doi.org/10.1103/physreve.79.066112},
doi = {10.1103/physreve.79.066112},
interhash = {7e35680b4fa4b11fe430fa904be3d5a8},
intrahash = {db180f38a76d89fb81acf1ddc4237877},
issn = {1539-3755},
journal = {Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)},
keywords = {small-world percolation critical-phenomena networks explosive-percolation critical-points},
month = jun,
number = 6,
pages = {066112+},
posted-at = {2009-10-21 14:45:19},
priority = {2},
publisher = {APS},
timestamp = {2019-08-23T10:59:34.000+0200},
title = {{Unusual percolation in simple small-world networks}},
url = {http://dx.doi.org/10.1103/physreve.79.066112},
volume = 79,
year = 2009
}