  We study binary opinion dynamics in a fully connected network of interacting
agents. The agents are assumed to interact according to one of the following
rules: (1) Voter rule: An updating agent simply copies the opinion of another
randomly sampled agent; (2) Majority rule: An updating agent samples multiple
agents and adopts the majority opinion in the selected group. We focus on the
scenario where the agents are biased towards one of the opinions called the
{\em preferred opinion}. Using suitably constructed branching processes, we
show that under both rules the mean time to reach consensus is $\Theta(\log
N)$, where $N$ is the number of agents in the network. Furthermore, under the
majority rule model, we show that consensus can be achieved on the preferred
opinion with high probability even if it is initially the opinion of the
minority. We also study the majority rule model when stubborn agents with fixed
opinions are present. We find that the stationary distribution of opinions in
the network in the large system limit using mean field techniques.
