Phase transitions in biased opinion dynamics with 2-choices rule

We consider a model of binary opinion dynamics where one opinion is inherently “superior” than the other, and social agents exhibit a “bias” toward the superior alternative. Specifically, it is assumed that an agent updates its choice to the superior alternative with probability α > 0 irrespective of its current opinion and opinions of other agents. With probability $1-\alpha$ , it adopts majority opinion among two randomly sampled neighbors and itself. We are interested in the time it takes for the network to converge to a consensus on the superior alternative. In a complete graph of size n, we show that irrespective of the initial configuration of the network, the average time to reach consensus scales as $\Theta(n\,\log n)$ when the bias parameter α is sufficiently high, that is, $\alpha \gt \alpha_c$ where α c is a threshold parameter that is uniquely characterized. When the bias is low, that is, when $\alpha \in (0,\alpha_c]$ , we show that the same rate of convergence can only be achieved if the initial proportion of agents with the superior opinion is above certain threshold $p_c(\alpha)$ . If this is not the case, then we show that the network takes $\Omega(\exp(\Theta(n)))$ time on average to reach consensus.