  In this paper, we consider the voter model with popularity bias. The influence of each node on its neighbors depends on its degree. We find the consensus probabilities and expected consensus times for each of the states. We also find the fixation probability, which is the probability that a single node whose state differs from every other node imposes its state on the entire system. In addition, we find the expected fixation time. Then two extensions to the model are proposed and the motivations behind them are discussed. The first one is confidence, where in addition to the states of neighbors, nodes take their own state into account at each update. We repeat the calculations for the augmented model and investigate the effects of adding confidence to the model. The second proposed extension is irreversibility, where one of the states is given the property that once nodes adopt it, they cannot switch back. The dynamics of densities, fixation times and consensus times are obtained. 
Voter Model with Arbitrary Degree Dependence: Clout, Confidence and   Irreversibility