Incomplete ordering of the voter model on small-world networks

TL;DR

This paper studies how small-world network topology influences the voter model, revealing that long-range connections hinder complete opinion consensus in large systems but accelerate ordering in finite ones.

Contribution

It demonstrates that small-world networks prevent full order in the thermodynamic limit, contrasting with regular lattices, and shows finite systems reach consensus faster.

Findings

Long-range links inhibit complete ordering in large systems.

Finite networks reach consensus faster than regular lattices.

Stationary states with coexisting opinions are stable in large systems.

Abstract

We investigate how the topology of small-world networks affects the dynamics of the voter model for opinion formation. We show that, contrary to what occurs on regular topologies with local interactions, the voter model on small-world networks does not display the emergence of complete order in the thermodynamic limit. The system settles in a stationary state with coexisting opinions whose lifetime diverges with the system size. Hence the nontrivial connectivity pattern leads to the counterintuitive conclusion that long-range connections inhibit the ordering process. However, for networks of finite size, for which full uniformity is reached, the ordering process takes a time shorter than on a regular lattice of the same size.