  The ability of a society to make the right decisions on relevant matters
relies on its capability to properly aggregate the noisy information spread
across the individuals it is made of. In this paper we study the information
aggregation performance of a stylized model of a society whose most influential
individuals - the leaders - are highly connected among themselves and
uninformed. Agents update their state of knowledge in a Bayesian manner by
listening to their neighbors. We find analytical and numerical evidence of a
transition, as a function of the noise level in the information initially
available to agents, from a regime where information is correctly aggregated to
one where the population reaches consensus on the wrong outcome with finite
probability. Furthermore, information aggregation depends in a non-trivial
manner on the relative size of the clique of leaders, with the limit of a
vanishingly small clique being singular.
