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.
Graph fission in an evolving voter model
Richard Durrett et al.
PNAS March 6, 2012 vol. 109 no. 10 3682-3687