The Offended Voter Model

TL;DR

This paper introduces a variant of the voter model on coevolving networks where interactions can lead to agreement or offense, affecting network structure and consensus, with rigorous proofs and simulations exploring different regimes.

Contribution

It provides a rigorous analysis of a new voter model variant incorporating offense and network coevolution, revealing conditions for segregation and consensus.

Findings

Segregation occurs with positive probability for all q in (0,1)

Segregation probability approaches 1 as q approaches 0

High probability of consensus when q approaches 1 fast enough

Abstract

We study a variant of the voter model on a coevolving network in which interactions of two individuals with differing opinions only lead to an agreement on one of these opinions with a fixed probability $q$. Otherwise, with probability $1-q$, both individuals become offended in the sense that they never interact again, i.e. the corresponding edge is removed from the underlying network. Eventually, these dynamics reach an absorbing state at which there is only one opinion present in each connected component of the network. If globally both opinions are present at absorption we speak of "segregation'', otherwise of "consensus''. We rigorously show that segregation and a weaker form of consensus both occur with positive probability for every $q \in (0,1)$ and that the segregation probability tends to $1$ as $q \to 0$. Furthermore, we establish that, if $q \to 1$ fast enough, with high probability the population reaches consensus while the underlying network is still densely connected. We provide results from simulations to assess the obtained bounds and to discuss further properties of the limiting state.