Graph fission in an evolving voter model

TLDR

The study examines a simplified coevolving voter model on a social network where each node holds one of two opinions and both opinions and links evolve together. The authors aim to determine how the final minority opinion fraction ρ depends on the rewiring probability α and the initial opinion density u. At each step a random edge is selected; if its endpoints disagree, with probability 1–α one adopts the other's opinion, otherwise with probability α the edge is severed and one endpoint rewires to a random node either sharing its opinion or chosen uniformly from the network, and the process ends when no discordant edges remain. They find that in the rewiring-to-same-opinion case a universal critical αc separates regimes where ρ≈u or ρ≈0, whereas in the random-rewiring case αc depends on u, yielding ρ≈u for α>αc and ρ(α,u)=ρ(α,½) for α<αc, with simulations and analytic arguments explaining the markedly different phase transitions.

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 &gt; α c and ρ ≈ 0 for α &lt; α c . In case ( ii ), the transition point α c ( u ) depends on the initial density u . For α &gt; α c ( u ), ρ ≈ u , but for α &lt; 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.