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 Erdos-Renyi random network topology using the pair approximation, where we assume that the network topology is independent of the type of voters. We verify our analytical results through numerical simulations. 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 $\log(N)/\gamma v$, with $\gamma$ the fraction of biased voters and $v$ a parameter measuring the bias of the voters. Finally, we study this model on a biased-dependent topology and analyze how the persuasiveness of the biased group, measured by the time needed to reach consensus in the preferred opinion, depends on how well its members are connected among each other, compared to how well the members of the unbiased group are connected among themselves.

Trabajo presentado en el Workshop on Sociophysics: Social Phenomena from a Physics Perspective, celebrado online del 18 al 22 de octubre de 2021.