Analytic treatment of consensus achievement in the single-type zealotry voter model

We introduce zealots of one opinion in the voter model on a complete graph and examine how they affect consensus achievement. Using first-step analysis for Markov chains to obtain an exact solution, we find that the mean consensus time scales with the population size N. Increasing the number of zealots, Z, will decrease the consensus time in a power law fashion for large Z. The mean magnetization was also analytically obtained and was found to contain an exponential dependence on Z. The dynamics for the complete graph are qualitatively similar to those obtained in another study for the Barabasi-Albert network. In general, the existence of zealots serves to hasten consensus, except in the case where only a few zealots oppose the vast majority.