  The voter model is a simple agent-based model to mimic opinion dynamics in
social networks: a randomly chosen agent adopts the opinion of a randomly
chosen neighbour. This process is repeated until a consensus emerges. Although
the basic voter model is theoretically intriguing, it misses an important
feature of real opinion dynamics: it does not distinguish between an agent's
publicly expressed opinion and her inner conviction. A person may not feel
comfortable declaring her conviction if her social circle appears to hold an
opposing view. Here we introduce the Concealed Voter Model where we add a
second, concealed layer of opinions to the public layer. If an agent's public
and concealed opinions disagree, she can reconcile them by either publicly
disclosing her previously secret point of view or by accepting her public
opinion as inner conviction. We study a complete graph of agents who can choose
from two opinions. We define a martingale $M$ that determines the probability
of all agents eventually agreeing on a particular opinion. By analyzing the
evolution of $M$ in the limit of a large number of agents, we derive the
leading-order terms for the mean and standard deviation of the consensus time
(i.e. the time needed until all opinions are identical). We thereby give a
precise prediction by how much concealed opinions slow down a consensus.
