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.