Graph fission in an evolving voter model

• Published in: Proceedings of the National Academy of Sciences - PNAS Vol. 109; no. 10; pp. 3682 - 3687
• Main Authors: Durrett, Richard, Gleeson, James P, Lloyd, Alun L, Mucha, Peter J, Shi, Feng, Sivakoff, David, Socolar, Joshua E. S, Varghese, Chris
• Format: Journal Article
• Language: English
• Published: United States National Academy of Sciences 06.03.2012; National Acad Sciences
• Subjects: Algorithms; Approximation; attitudes and opinions; Coevolution; Computer Simulation; Density; Diffusion; Epidemics; Humans; Modeling; Models, Statistical; Models, Theoretical; Opinions; phase transition; Phase transitions; Physical Sciences; Politics; Probability; Public Opinion; Simulation; Simulations; Social networking; Social networks; Social Support; Statistical graphs; Vertices; Voters; Zero
• ISSN: 0027-8424; 1091-6490
• DOI: 10.1073/pnas.1200709109

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.