  The spread of ideas in online social networks is a crucial phenomenon to
understand nowadays the proliferation of fake news and their impact in
democracies. This makes necessary to use models that mimic the circulation of
rumors. The law of large numbers as well as the probability distribution of
contact groups allow us to construct a model with a minimum number of
hypotheses. Moreover, we can analyze with this model the presence of very
polarized groups of individuals (humans or bots) who spread a rumor as soon as
they know about it. Given only the initial number of individuals who know any
news, in a population connected by an instant messaging application, we first
deduce from our model a simple function of time to study the rumor propagation.
We then prove that the polarized groups can be detected and quantified from
empirical data. Finally, we also predict the time required by any rumor to
reach a fixed percentage of the population.
