A multi-opinion evolving voter model with infinitely many phase transitions

TL;DR

This paper introduces a model where individuals' opinions and social networks coevolve, revealing infinitely many phase transitions and simple formulas for end states based on a key parameter.

Contribution

It extends previous two-opinion models to multiple opinions, demonstrating the existence of infinitely many phase transitions and deriving simple formulas for final states.

Findings

Model exhibits infinitely many phase transitions.

End states are described by simple formulas in terms of a parameter.

The model generalizes previous two-opinion coevolution models.

Abstract

We consider an idealized model in which individuals' changing opinions and their social network coevolve, with disagreements between neighbors in the network resolved either through one imitating the opinion of the other or by reassignment of the discordant edge. Specifically, an interaction between $x$ and one of its neighbors $y$ leads to $x$ imitating $y$ with probability $(1-\alpha)$ and otherwise (i.e., with probability $\alpha$) $x$ cutting its tie to $y$ in order to instead connect to a randomly chosen individual. Building on previous work about the two-opinion case, we study the multiple-opinion situation, finding that the model has infinitely many phase transitions. Moreover, the formulas describing the end states of these processes are remarkably simple when expressed as a function of $\beta = \alpha/(1-\alpha)$.