  We study the dynamics of an epidemic-like model for the spread of a rumor on
a small-world network. It has been shown that this model exhibits a transition
between regimes of localization and propagation at a finite value of the
network randomness. Here, by numerical means, we perform a quantitative
characterization of the evolution in the two regimes. The variant of dynamic
small worlds, where the quenched disorder of small-world networks is replaced
by randomly changing connections between individuals, is also analyzed in
detail and compared with a mean-field approximation.
