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

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 γ of links connect individuals at distances r drawn from a distribution decaying as r-σ-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 σ>2, their influence is restricted to distances beyond a time-dependent threshold, r∗(t). For rr∗(t), the correlation function transitions to a power-law decay, r-σ-1, highlighting the capacity of long-range links to propagate consensus across greater distances. When long-range interactions decay more slowly (σ<2), they dominate the dynamics at all scales, leading to behavior akin to a system with only long-range interactions. Notably, in the regime σ<1 long-range links induce a stationary steady state, even for small γ.