  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.
