  We consider a system in which a group of agents represented by the vertices
of a graph synchronously update their opinion based on that of their
neighbours. If each agent adopts a positive opinion if and only if that opinion
is sufficiently popular among his neighbours, the system will eventually settle
into a fixed state or alternate between two states. If one agent acts in a
different way, other periods may arise. We show that only a small number of
periods may arise if natural restrictions are placed either on the
neighbourhood structure or on the way in which the nonconforming agent may act;
without either of these restrictions any period is possible.
