  We consider a model for a social network with N interacting social actors.
This model is a system of interacting marked point processes in which each
point process indicates the successive times in which a social actor expresses
a "favorable" (+1) or "contrary" (-1) opinion. The orientation and the rate at
which an actor expresses an opinion is influenced by the social pressure
exerted on this actor. The social pressure of an actor is reset to 0 when the
actor expresses an opinion, and simultaneously the social pressures on all the
other actors change by h/N in the direction of the opinion that was just
expressed.
  We prove propagation of chaos of the system, as N diverges to infinity, to a
limit nonlinear jumping stochastic differential equation. Moreover, we prove
that under certain conditions the limit system exhibits a phase transition
described as follows. If h is smaller or equal than a certain threshold, the
limit system has only the null Dirac measure as an invariant probability
measure, corresponding to a vanishing social pressure on all actors. However,
if h is greater than the threshold, the system has two additional non-trivial
invariant probability measures. One of these measures has support on the
positive real numbers and the other is obtained by symmetrization with respect
to 0, having thus support on the negative real numbers.
