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. (C) 2020 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.