We consider the Maki-Thompson model for the stochastic propagation of a
rumour within a population. We extend the original hypothesis of homogenously
mixed population by allowing for a small-world network embedding the model.
This structure is realized starting from a k-regular ring and by inserting,
in the average, c additional links in such a way that k and c are
tuneable parameter for the population architecture. We prove that this system
exhibits a transition between regimes of localization (where the final number
of stiflers is at most logarithmic in the population size) and propagation
(where the final number of stiflers grows algebraically with the population
size) at a finite value of the network parameter c. A quantitative estimate
for the critical value of c is obtained via extensive numerical simulations.Comment: 24 pages, 4 figure
