Biased-voter model: How persuasive a small group can be?

We study the voter model dynamics in the presence of confidence and bias. We assume two types of voters. Unbiased voters whose confidence is indifferent to the state of the voter and biased voters whose confidence is biased towards a common fixed preferred state. We study the problem analytically on the complete graph using mean field theory and on an Erdős-Rényi random network topology using the pair approximation, where we assume that the network of interactions topology is independent of the type of voters. We find that for the case of a random initial setup, and for sufficiently large number of voters N, the time to consensus increases proportionally to log(N)/γv, with γ the fraction of biased voters and v the parameter quantifying the bias of the voters (v = 0 no bias). We verify our analytical results through numerical simulations. We study this model on a topology of the network of interactions depending on the bias, and examine two distinct, global average-degree preserving strategies (model I and model II) to obtain such random topologies starting from the random topology independent of bias case as the initial setup. Keeping all other parameters constant, in model I, μBU, the average number of links among biased (B) and unbiased (U) voters is varied at the expense of μUU and μBB, i.e. the average number of links among only unbiased and biased voters respectively. In model II, μBU is kept constant, while μBB is varied at the expense of μUU. We find that if the agents follow the strategy described by model II, they can achieve a significant reduction in the time to reach consensus as well as an increment in the probability to reach consensus to the preferred state. Hence, persuasiveness of the biased group depends on how well its members are connected among each other, compared to how well the members of the unbiased group are connected among each other.