  We derive mean-field equations that describe the dynamics of a general model
of rumor spreading on complex networks, and use analytical and numerical
solutions of these equations to examine the threshold behavior and dynamics of
the model on 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 rumor
spreading rate below which a rumor 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 rumor 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 rumor, however, depends on the interplay between network
topology and the rumor spreading rate. Our results show that scale-free
networks are prone to the spreading of rumors, just as they are to the
spreading of infections. They are relevant to the spreading dynamics of chain
emails, viral advertising and large-scale data dissemination algorithms on the
Internet, the so-called gossip protocols.
