Critical threshold and dynamics of a general rumor model on complex   networks
  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. 
Maziar Nekovee, Yamir Moreno, Ginestra Bianconi, Matteo Marsili