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 $$\gamma $$ γ of links connect individuals at distances r drawn from a distribution decaying as $$r^{-\sigma -1}$$ 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 $$\sigma > 2$$ σ > 2 , their influence is restricted to distances beyond a time-dependent threshold, $$r^*(t)$$ r ∗ ( t ) . For $$r r^*(t)$$ r > r ∗ ( t ) , the correlation function transitions to a power-law decay, $$r^{-\sigma -1}$$ r - σ - 1 , highlighting the capacity of long-range links to propagate consensus across greater distances. When long-range interactions decay more slowly ( $$\sigma