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

Abstract

We investigate a variation of the classical voter model in which the set of influencing agents depends on an individual's current opinion. The initial population consists 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, is kept constant across all agents. Our findings reveal that varying the interaction range introduces asymmetries that influence the probability of consensus and convergence time. At low connectivity, direct neighbor interactions dominate, driving consensus. As connectivity increases, the probability of consensus for either state becomes equal, mirroring symmetric dynamics. It is shown that this asymmetric effect on the probability of consensus is independent of the network topology in small-world and scale-free networks. Asymmetry is also reflected in convergence time: while symmetric cases show decreasing times with increased connectivity, asymmetric cases exhibit an almost linear increase. Contrary to the probability of achieving consensus, the effect of asymmetry on the consensus time does depend on the network topology. The introduction of stubborn agents further accentuates these effects, particularly when they favor the less dominant state, significantly increasing consensus time. We conclude by discussing the implications of these findings for decision-making processes and political campaigns in human populations.