title: Graph fission in an evolving voter model.
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 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.
authors.affiliation: Department of Mathematics, Duke University, Box 90320, Durham, NC 27708, USA. rtd@math.duke.edu
authors.forename: Richard
authors.initials: R
authors.lastname: Durrett
authors.forename: James P
authors.initials: JP
authors.lastname: Gleeson
authors.forename: Alun L
authors.initials: AL
authors.lastname: Lloyd
authors.forename: Peter J
authors.initials: PJ
authors.lastname: Mucha
authors.forename: Feng
authors.initials: F
authors.lastname: Shi
authors.forename: David
authors.initials: D
authors.lastname: Sivakoff
authors.forename: Joshua E S
authors.initials: JE
authors.lastname: Socolar
authors.forename: Chris
authors.initials: C
authors.lastname: Varghese
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D004058:Diffusion; D006801:Humans; D015233:Models, Statistical; D008962:Models, Theoretical; D011057:Politics; D011336:Probability; D011639:Public Opinion; D012944:Social Support