  We introduce a general stochastic model for the spread of rumours, and derive
mean-field equations that describe the dynamics of the model on complex social
networks (in particular those mediated by the Internet). We use analytical and
numerical solutions of these equations to examine the threshold behavior and
dynamics of the model on several models of such networks: random graphs,
uncorrelated scale-free networks and scale-free networks with assortative
degree correlations. We show that in both homogeneous networks and random
graphs the model exhibits a critical threshold in the rumour spreading rate
below which a rumour cannot propagate in the system. In the case of scale-free
networks, on the other hand, this threshold becomes vanishingly small in the
limit of infinite system size. We find that the initial rate at which a rumour
spreads is much higher in scale-free networks than in random graphs, and that
the rate at which the spreading proceeds on scale-free networks is further
increased when assortative degree correlations are introduced. The impact of
degree correlations on the final fraction of nodes that ever hears a rumour,
however, depends on the interplay between network topology and the rumour
spreading rate. Our results show that scale-free social networks are prone to
the spreading of rumours, just as they are to the spreading of infections. They
are relevant to the spreading dynamics of chain emails, viral advertising and
large-scale information dissemination algorithms on the Internet.
