Robust Naive Learning in Social Networks

We study a model of opinion exchange in social networks where a state of the world is realized and every agent receives a zero-mean noisy signal of the realized state. It is known from Golub and Jackson that under the DeGroot dynamics agents reach a consensus which is close to the state of the world when the network is large. The DeGroot dynamics, however, is highly non-robust and the presence of a single `bot' that does not adhere to the updating rule, can sway the public consensus to any other value. We introduce a variant of the DeGroot dynamics which we call \emph{ $\varepsilon$-DeGroot}. The $\varepsilon$-DeGroot dynamics approximates the standard DeGroot dynamics and like the DeGroot dynamics it is Markovian and stationary. We show that in contrast to the standard DeGroot dynamics, the $\varepsilon$-DeGroot dynamics is highly robust both to the presence of bots and to certain types of misspecifications.