Voter model on a directed network: Role of bidirectional opinion exchanges

TL;DR

This study investigates how directed networks with bidirectional opinion exchanges influence consensus formation, showing that complex information flow patterns promote agreement on the better opinion, especially in larger networks.

Contribution

It introduces a model of the voter dynamics on directed networks with rewiring and symmetry breaking, highlighting the impact of bidirectional information flow on opinion consensus.

Findings

Larger rewiring probability enhances agreement on the better opinion.

Complex bidirectional flow slows but improves opinion consensus.

Hierarchical structure accelerates opinion agreement but is easily influenced by top voters.

Abstract

The voter model with the node update rule is numerically investigated on a directed network. We start from a directed hierarchical tree, and split and rewire each incoming arc at the probability $p$. In order to discriminate the better and worse opinions, we break the $Z_2$ symmetry ($\sigma = \pm 1$) by giving a little more preference to the opinion $\sigma = 1$. It is found that as $p$ becomes larger, introducing more complicated pattern of information flow channels, and as the network size $N$ becomes larger, the system eventually evolves to the state in which more voters agree on the better opinion, even though the voter at the top of the hierarchy keeps the worse opinion. We also find that the pure hierarchical tree makes opinion agreement very fast, while the final absorbing state can easily be influenced by voters at the higher ranks. On the other hand, although the ordering occurs much slower, the existence of complicated pattern of bidirectional information flow allows the system to agree on the better opinion.