We consider a simplified model of a social network in which individuals have one of two opinions (called 0 and 1) and their opinions and the network connections coevolve. Edges are picked at random. If the two connected individuals hold different opinions then, with probability 1 - α, one imitates the opinion of the other; otherwise (i.e., with probability α), the link between them is broken and one of them makes a new connection to an individual chosen at random (i) from those with the same opinion or (ii) from the network as a whole. The evolution of the system stops when there are no longer any discordant edges connecting individuals with different opinions. Letting ρ be the fraction of voters holding the minority opinion after the evolution stops, we are interested in how ρ depends on α and the initial fraction u of voters with opinion 1. In case (i), there is a critical value αc which does not depend on u, with ρ ≈ u for α > αc and ρ ≈ 0 for α < αc. In case (ii), the transition point αc(u) depends on the initial density u. For α > αc(u), ρ ≈ u, but for α < αc(u), we have ρ(α,u) = ρ(α,1/2). Using simulations and approximate calculations, we explain why these two nearly identical models have such dramatically different phase transitions.
%0 Journal Article
%1 Durrett2012
%A Durrett, Richard
%A Gleeson, James P.
%A Lloyd, Alun L.
%A Mucha, Peter J.
%A Shi, Feng
%A Sivakoff, David
%A Socolar, Joshua E. S.
%A Varghese, Chris
%D 2012
%J Proceedings of the National Academy of Sciences
%K adaptive-networks fragmentation link-update networks opinion-formation phase-transition voter-model
%N 10
%P 3682--3687
%R 10.1073/pnas.1200709109
%T Graph fission in an evolving voter model
%V 109
%X We consider a simplified model of a social network in which individuals have one of two opinions (called 0 and 1) and their opinions and the network connections coevolve. Edges are picked at random. If the two connected individuals hold different opinions then, with probability 1 - α, one imitates the opinion of the other; otherwise (i.e., with probability α), the link between them is broken and one of them makes a new connection to an individual chosen at random (i) from those with the same opinion or (ii) from the network as a whole. The evolution of the system stops when there are no longer any discordant edges connecting individuals with different opinions. Letting ρ be the fraction of voters holding the minority opinion after the evolution stops, we are interested in how ρ depends on α and the initial fraction u of voters with opinion 1. In case (i), there is a critical value αc which does not depend on u, with ρ ≈ u for α > αc and ρ ≈ 0 for α < αc. In case (ii), the transition point αc(u) depends on the initial density u. For α > αc(u), ρ ≈ u, but for α < αc(u), we have ρ(α,u) = ρ(α,1/2). Using simulations and approximate calculations, we explain why these two nearly identical models have such dramatically different phase transitions.
@article{Durrett2012,
abstract = {We consider a simplified model of a social network in which individuals have one of two opinions (called 0 and 1) and their opinions and the network connections coevolve. Edges are picked at random. If the two connected individuals hold different opinions then, with probability 1 - α, one imitates the opinion of the other; otherwise (i.e., with probability α), the link between them is broken and one of them makes a new connection to an individual chosen at random (i) from those with the same opinion or (ii) from the network as a whole. The evolution of the system stops when there are no longer any discordant edges connecting individuals with different opinions. Letting ρ be the fraction of voters holding the minority opinion after the evolution stops, we are interested in how ρ depends on α and the initial fraction u of voters with opinion 1. In case (i), there is a critical value αc which does not depend on u, with ρ ≈ u for α > αc and ρ ≈ 0 for α < αc. In case (ii), the transition point αc(u) depends on the initial density u. For α > αc(u), ρ ≈ u, but for α < αc(u), we have ρ(α,u) = ρ(α,1/2). Using simulations and approximate calculations, we explain why these two nearly identical models have such dramatically different phase transitions.},
added-at = {2012-07-05T11:05:19.000+0200},
author = {Durrett, Richard and Gleeson, James P. and Lloyd, Alun L. and Mucha, Peter J. and Shi, Feng and Sivakoff, David and Socolar, Joshua E. S. and Varghese, Chris},
biburl = {https://www.bibsonomy.org/bibtex/2a5bbf803377a0c219b1f4859cf60f668/rincedd},
doi = {10.1073/pnas.1200709109},
interhash = {66d12362250d51e4f3757a3ea1c3fe4b},
intrahash = {a5bbf803377a0c219b1f4859cf60f668},
journal = {Proceedings of the National Academy of Sciences},
keywords = {adaptive-networks fragmentation link-update networks opinion-formation phase-transition voter-model},
number = 10,
pages = {3682--3687},
timestamp = {2012-07-05T11:05:19.000+0200},
title = {Graph fission in an evolving voter model},
volume = 109,
year = 2012
}