Bi-layer voter model: modeling intolerant/tolerant positions and bots in opinion dynamics

Abstract

The diffusion of opinions in social networks is a relevant process for adopting positions and attracting potential voters in political campaigns. Opinion polarization, bias, targeted diffusion, and the radicalization of postures are key elements for understanding the voting dynamics’ challenges. In particular, social bots are currently a new element that can have a pronounced effect on the formation of opinions during electoral processes by, for instance, creating fake accounts in social networks to manipulate elections. Here, we propose a voter model incorporating bots and radical or intolerant individuals in the decision-making process. The dynamics of the system occur in a multiplex network of interacting agents composed of two layers, one for the dynamics of opinions where agents choose between two possible alternatives, and the other for the tolerance dynamics, in which agents adopt one of the two tolerance levels. The tolerance accounts for the likelihood to change opinion in an interaction, with tolerant (intolerant) agents switching opinion with probability 1.0 ($\gamma \le 1$). We find that intolerance leads to a consensus of tolerant agents during an initial stage that scales as $\tau ^+ \sim \gamma ^{-1} \ln N$, who then reach an opinion consensus during the second stage in a time that scales as $\tau \sim N$, where N is the number of agents. Therefore, very intolerant agents ($\gamma \ll 1$) could considerably slow down dynamics towards the final consensus state. We also find that the inclusion of a fraction $\sigma _{{\mathbb {B}}}^-$ of bots breaks the symmetry between both opinions, driving the system to a consensus of intolerant agents with the bots’ opinion. Thus, bots eventually impose their opinion to the entire population, in a time that scales as $\tau _B^- \sim \gamma ^{-1}$ for $\gamma \ll \sigma _{{\mathbb {B}}}^-$ and $\tau _B^- \sim 1/\sigma _{{\mathbb {B}}}^-$ for $\sigma _{{\mathbb {B}}}^- \ll \gamma$.

Appendix A: complement of the explicit transitions rules

In this section, we explicitly write all transitions between opinion and tolerance states of agents in the model without bots (Table 1) and with bots (Table 2). The notations $A^+, A^-, B^+$ and $B^-$ correspond to states of agents with opinion and tolerance A and $+$, A and −, B and $+$, and B and −, respectively. In a single time step $\Delta t = 1/N$ of the dynamics, one node is chosen at random. Then, this node copies the tolerance of a random neighbor in the ±-layer, and the opinion of a random neighbor in the AB-layer. In Tables 1 and 2, the states on the left and right of a given pair correspond, respectively, to the focal agent—who changes state—and the random neighbor on the corresponding layer. Only situations that lead to a state change are included in the tables.