%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 }