Winning Opinion in the Voter Model: Following Your Friends' Advice or That of Their Friends?

We investigate a variation of the classical voter model where the set of influencing agents depends on an individual's current opinion. The initial population is made up of a random sample of equally sized sub-populations for each state, and two types of interactions are considered: (i) direct neighbors and (ii) second neighbors (friends of direct neighbors, excluding the direct neighbors themselves). The neighborhood size, reflecting regular network connectivity, remains constant across all agents. Our findings show that varying the interaction range introduces asymmetries that affect the probability of consensus and convergence time. At low connectivity, direct neighbor interactions dominate, leading to consensus. As connectivity increases, the probability of either state reaching consensus becomes equal, reflecting symmetric dynamics. This asymmetric effect on the probability of consensus is shown to be independent of network topology in small-world and scale-free networks. Asymmetry also influences convergence time: while symmetric cases display decreasing times with increased connectivity, asymmetric cases show an almost linear increase. Unlike the probability of reaching consensus, the impact of asymmetry on convergence time depends on the network topology. The introduction of stubborn agents further magnifies these effects, especially when they favor the less dominant state, significantly lengthening the time to consensus. We conclude by discussing the implications of these findings for decision-making processes and political campaigns in human populations.