Naϊve learning in social networks with fake news : bots as a singularity

We study the impact of bots on social learning in a social network setting. Regular agents receive independent noisy signals about the true value of a variable and then communicate in a network. They na¨ıvely update beliefs by repeatedly taking weighted averages of neighbors' opinions. Bots are agents in the network that spread fake news by disseminating biased information. Our main contributions are threefold. (1) We show that the consensus of the network is a mapping of the interaction rate between the agents and bots and is discontinuous at zero mass of bots. This implies that even a comparatively "infinitesimal" small number of bots still has a sizeable impact on the consensus and hence represents an obstruction to the "wisdom of crowds". (2) We prove that the consensus gap induced by the marginal presence of bots depends neither on the agent network or bot layout nor on the assumed connection structure between agents and bots. (3) We show that before the ultimate (and bot-infected) consensus is reached, the network passes through a quasi-stationary phase which has the potential to mitigate the harmful impact of bots.