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 \(\gamma \) 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, \(r^*(t)\). For \(r < r^*(t)\), 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 \(r > r^*(t)\), 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 \(\gamma \).