  We show that a simple model for the propagation of a rumor on a small-world
network exhibits critical behavior at a finite randomness of the underlying
graph. The transition occurs between a regime where the rumor "dies" in a small
neighborhood of its origin, and a regime where it spreads over a finite
fraction of the whole population. Critical exponents are evaluated through
finite-size scaling analysis, and the dependence of the critical randomness
with the network connectivity is studied. The behavior of this system as a
function of the network randomness bears noticeable similarities with an
epidemiological model reported recently [M. Kuperman and G. Abramson, Phys.
Rev. Lett. 86, 2909 (2001)], in spite of substantial differences in the
respective dynamical rules.
