  We study opinion dynamics in a social network with stubborn agents who
influence their neighbors but who themselves always stick to their initial
opinion. We consider first the well-known DeGroot model. While it is known in
the literature that this model can lead to consensus even in the presence of a
stubborn agent, we show that the same result holds under weaker assumptions
than has been previously reported. We then consider a recent extension of the
DeGroot model in which the opinion of each agent is a random Bernoulli
distributed variable, and by leveraging on the first result we establish that
this model also leads to consensus, in the sense of convergence in probability,
in the presence of a stubborn agent. Moreover, all agents' opinions converge to
that of the stubborn agent. We also consider a variation on this model where
the stubborn agent is replaced with a drifting agent and show that herding is
achieved also in this case. Finally, we offer a detailed critique of a proof
regarding a claim about a closely related model in the recent literature.
