  We consider a discrete opinion formation problem in a setting where agents
are influenced by both information diffused by their social relations and from
recommendations received directly from the social media manager. We study how
the "strength" of the influence of the social media and the homophily ratio
affect the probability of the agents of reaching a consensus and how these
factors can determine the type of consensus reached. In a simple 2-symmetric
block model we prove that agents converge either to a consensus or to a
persistent disagreement. In particular, we show that when the homophily ratio
is large, the social media has a very low capacity of determining the outcome
of the opinion dynamics. On the other hand, when the homophily ratio is low,
the social media influence can have an important role on the dynamics, either
by making harder to reach a consensus or inducing it on extreme opinions.
Finally, in order to extend our analysis to more general and realistic settings
we give some experimental evidences that our results still hold on general
networks.
