Bias Evolution in Near-Critical and Quasi-Sequential Majority Dynamics

In this paper, we study discrete time consensus among a group of nodes capable of communicating with each other and equipped with arbitrary binary valued initial decisions. At each time instant, a subset of the nodes are selected for decision update and for each node, the update is made based on the majority decisions of three further independently chosen \consultation" nodes. We consider the near-critical case where the bias towards the majority decision is very small and a function of the total number of nodes and study the evolution of bias as it progresses through each of the three regimes: small, medium and high. In each case, we obtain high probability estimates for the bias or the minority node fraction to change by a constant factor and finally combine these bounds to establish the overall consensus time. We demonstrate that the overall consensus time is of the same order as the generalized coupon collector problem with high probability, i.e., with probability converging to one as the number of nodes becomes arbitrarily large.