  Infectious diseases and computer malwares spread among humans and computers
through the network of contacts among them. These networks are characterized by
wide connectivity fluctuations, connectivity correlations and the small-world
property. In a previous work [A. Vazquez, Phys. Rev. Lett. 96, 038702 (2006)] I
have shown that the connectivity fluctuations together with the small-world
property lead to a novel spreading law, characterized by an initial power law
growth with an exponent determined by the average node distance on the network.
Here I extend these results to consider the influence of connectivity
correlations which are generally observed in real networks. I show that
assortative and disassortative connectivity correlations enhance and diminish,
respectively, the range of validity of this spreading law. As a corollary I
obtain the region of connectivity fluctuations and degree correlations
characterized by the absence of an epidemic threshold. These results are
relevant for the spreading of infectious diseases, rumors, and information
among humans and the spreading of computer viruses, email worms and hoaxes
among computer users.
