H. Zhou, and R. Lipowsky. Proceedings of the National Academy of Sciences of the United States of America, 102 (29):
10052--10057(Jul 19, 2005)
DOI: 10.1073/pnas.0409296102
Abstract
A general class of dynamic models on scale-free networks is studied by analytical methods and computer simulations. Each network consists of N vertices and is characterized by its degree distribution, P(k), which represents the probability that a randomly chosen vertex is connected to k nearest neighbors. Each vertex can attain two internal states described by binary variables or Ising-like spins that evolve in time according to local majority rules. Scale-free networks, for which the degree distribution has a power law tail P(k) ∼ k -γ, are shown to exhibit qualitatively different dynamic behavior for γ < 5/2 and γ > 5/2, shedding light on the empirical observation that many real-world networks are scale-free with 2 < γ < 5/2. For 2 < γ < 5/2, strongly disordered patterns decay within a finite decay time even in the limit of infinite networks. For γ > 5/2, on the other hand, this decay time diverges as ln(N) with the network size N. An analogous distinction is found for a variety of more complex models including Hopfield models for associative memory networks. In the latter case, the storage capacity is found, within mean field theory, to be independent of N in the limit of large N for γ > 5/2 but to grow as N α with α = (5 - 2γ)/(γ - 1) for 2 < γ < 5/2.
%0 Journal Article
%1 Zhou2005Dynamic
%A Zhou, Haijun
%A Lipowsky, Reinhard
%C Max Planck Institute of Colloids and Interfaces, 14424 Potsdam, Germany.
%D 2005
%I National Academy of Sciences
%J Proceedings of the National Academy of Sciences of the United States of America
%K kinetics, opinion-models, scale-free-networks
%N 29
%P 10052--10057
%R 10.1073/pnas.0409296102
%T Dynamic pattern evolution on scale-free networks
%U http://dx.doi.org/10.1073/pnas.0409296102
%V 102
%X A general class of dynamic models on scale-free networks is studied by analytical methods and computer simulations. Each network consists of N vertices and is characterized by its degree distribution, P(k), which represents the probability that a randomly chosen vertex is connected to k nearest neighbors. Each vertex can attain two internal states described by binary variables or Ising-like spins that evolve in time according to local majority rules. Scale-free networks, for which the degree distribution has a power law tail P(k) ∼ k -γ, are shown to exhibit qualitatively different dynamic behavior for γ < 5/2 and γ > 5/2, shedding light on the empirical observation that many real-world networks are scale-free with 2 < γ < 5/2. For 2 < γ < 5/2, strongly disordered patterns decay within a finite decay time even in the limit of infinite networks. For γ > 5/2, on the other hand, this decay time diverges as ln(N) with the network size N. An analogous distinction is found for a variety of more complex models including Hopfield models for associative memory networks. In the latter case, the storage capacity is found, within mean field theory, to be independent of N in the limit of large N for γ > 5/2 but to grow as N α with α = (5 - 2γ)/(γ - 1) for 2 < γ < 5/2.
@article{Zhou2005Dynamic,
abstract = {{A general class of dynamic models on scale-free networks is studied by analytical methods and computer simulations. Each network consists of N vertices and is characterized by its degree distribution, P(k), which represents the probability that a randomly chosen vertex is connected to k nearest neighbors. Each vertex can attain two internal states described by binary variables or Ising-like spins that evolve in time according to local majority rules. Scale-free networks, for which the degree distribution has a power law tail P(k) ∼ k -γ, are shown to exhibit qualitatively different dynamic behavior for γ < 5/2 and γ > 5/2, shedding light on the empirical observation that many real-world networks are scale-free with 2 < γ < 5/2. For 2 < γ < 5/2, strongly disordered patterns decay within a finite decay time even in the limit of infinite networks. For γ > 5/2, on the other hand, this decay time diverges as ln(N) with the network size N. An analogous distinction is found for a variety of more complex models including Hopfield models for associative memory networks. In the latter case, the storage capacity is found, within mean field theory, to be independent of N in the limit of large N for γ > 5/2 but to grow as N α with α = (5 - 2γ)/(γ - 1) for 2 < γ < 5/2.}},
added-at = {2019-06-10T14:53:09.000+0200},
address = {Max Planck Institute of Colloids and Interfaces, 14424 Potsdam, Germany.},
author = {Zhou, Haijun and Lipowsky, Reinhard},
biburl = {https://www.bibsonomy.org/bibtex/23c962d6dcf9b8fab33226f43edfe5a07/nonancourt},
citeulike-article-id = {296024},
citeulike-linkout-0 = {http://dx.doi.org/10.1073/pnas.0409296102},
citeulike-linkout-1 = {http://www.pnas.org/content/102/29/10052.abstract},
citeulike-linkout-2 = {http://www.pnas.org/content/102/29/10052.full.pdf},
citeulike-linkout-3 = {http://view.ncbi.nlm.nih.gov/pubmed/16006533},
citeulike-linkout-4 = {http://www.hubmed.org/display.cgi?uids=16006533},
day = 19,
doi = {10.1073/pnas.0409296102},
interhash = {2a04add60e7b3845c486be6ebdc4061e},
intrahash = {3c962d6dcf9b8fab33226f43edfe5a07},
issn = {1091-6490},
journal = {Proceedings of the National Academy of Sciences of the United States of America},
keywords = {kinetics, opinion-models, scale-free-networks},
month = jul,
number = 29,
pages = {10052--10057},
pmid = {16006533},
posted-at = {2012-09-06 23:06:52},
priority = {2},
publisher = {National Academy of Sciences},
timestamp = {2019-08-01T16:13:01.000+0200},
title = {{Dynamic pattern evolution on scale-free networks}},
url = {http://dx.doi.org/10.1073/pnas.0409296102},
volume = 102,
year = 2005
}