  The dynamics of fake news and rumor spreading is investigated using a model
with three kinds of agents who are respectively the Seeds, the Agnostics and
the Others. While Seeds are the ones who start spreading the rumor being
adamantly convinced of its truth, Agnostics reject any kind of rumor and do not
believe in conspiracy theories. In between, the Others constitute the main part
of the community. While Seeds are always Believers and Agnostics are always
Indifferents, Others can switch between being Believer and Indifferent
depending on who they are discussing with. The underlying driving dynamics is
implemented via local updates of randomly formed groups of agents. In each
group, an Other turns into a Believer as soon as $m$ or more Believers are
present in the group. However, since some Believers may lose interest in the
rumor as time passes by, we add a flipping fixed rate $0<d<1$ from Believers
into Indifferents. Rigorous analysis of the associated dynamics reveals that
switching from $m=1$ to $m\ge2$ triggers a drastic qualitative change in the
spreading process. When $m=1$ even a small group of Believers may manage to
convince a large part of the community very quickly. In contrast, for $m\ge 2$,
even a substantial fraction of Believers does not prevent the rumor dying out
after a few update rounds. Our results provide an explanation on why a given
rumor spreads within a social group and not in another, and also why some
rumors will not spread in neither groups.
