Eur. Phys. J. B (2025) 98: 49
https://doi.org/10.1140/epjb/s10051-025-00900-x

Regular Article - Statistical and Nonlinear Physics

Competition between long-range and short-range interactions in the voter model for opinion dynamics

Department of Mathematics and Physics, University of Campania “Luigi Vanvitelli”, Caserta, Italy

^a eugenio.lippiello@unicampania.it

Received: 20 December 2024
Accepted: 11 March 2025
Published online: 21 March 2025

Abstract

The voter model is a widely used framework in sociophysics to model opinion formation based on local interactions between individuals. In this work, we investigate how the spread of consensus is affected by introducing long-range interactions. Specifically, we study a one-dimensional voter model where a fraction $\gamma$ of links connect individuals at distances r drawn from a distribution decaying as $r^{-\sigma -1}$. Our results reveal that even a small fraction of long-range interactions fundamentally alters the system’s asymptotic behavior. When long-range interactions decay rapidly $\sigma >2$, their influence is restricted to distances beyond a time-dependent threshold, $r^{\ast }(t)$. For $r<r^{\ast }(t)$, the system exhibits short-range dynamics characterized by a Gaussian-like correlation function and a diffusion-driven growth of the correlation length, $L(t)\sim t^{1/2}$. However, for $r>r^{\ast }(t)$, the correlation function transitions to a power-law decay, $r^{-\sigma -1}$, highlighting the capacity of long-range links to propagate consensus across greater distances. When long-range interactions decay more slowly ($\sigma <2$), they dominate the dynamics at all scales, leading to behavior akin to a system with only long-range interactions. Notably, in the regime $\sigma <1$ long-range links induce a stationary steady state, even for small $\gamma$.