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In this paper, we consider the voter model with popularity bias. Theinfluence of each node on its neighbors depends on its degree. We find theconsensus probabilities and expected consensus times for each of the states. Wealso find the fixation probability, which is the probability that a single nodewhose state differs from every other node imposes its state on the entiresystem. In addition, we find the expected fixation time. Then two extensions tothe model are proposed and the motivations behind them are discussed. The firstone is confidence, where in addition to the states of neighbors, nodes taketheir own state into account at each update. We repeat the calculations for theaugmented model and investigate the effects of adding confidence to the model.The second proposed extension is irreversibility, where one of the states isgiven the property that once nodes adopt it, they cannot switch back. Thedynamics of densities, fixation times and consensus times are obtained.

Voter Model with Arbitrary Degree Dependence: Clout, Confidence and Irreversibility