TY - JOUR

T1 - Graph fission in an evolving voter model

AU - Durrett, Richard

AU - Gleeson, James P.

AU - Lloyd, Alun L.

AU - Mucha, Peter J.

AU - Shi, Feng

AU - Sivakoff, David

AU - Socolar, Joshua E.S.

AU - Varghese, Chris

PY - 2012/3/6

Y1 - 2012/3/6

N2 - 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 α c which does not depend on u, with ρ ≈ u for α > α c and ρ ≈ 0 for α < α c. In case (ii), the transition point α c(u) depends on the initial density u. For α > α c(u), ρ ≈ u, but for α < α c(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.

AB - 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 α c which does not depend on u, with ρ ≈ u for α > α c and ρ ≈ 0 for α < α c. In case (ii), the transition point α c(u) depends on the initial density u. For α > α c(u), ρ ≈ u, but for α < α c(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.

KW - Approximate master equation

KW - Coevolutionary network

KW - Quasi-stationary distribution

KW - Wright-Fisher diffusion

UR - http://www.scopus.com/inward/record.url?scp=84857944650&partnerID=8YFLogxK

U2 - 10.1073/pnas.1200709109

DO - 10.1073/pnas.1200709109

M3 - Article

C2 - 22355142

AN - SCOPUS:84857944650

SN - 0027-8424

VL - 109

SP - 3682

EP - 3687

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

IS - 10

ER -