Spreading dynamics on small-world networks with connectivity fluctuations and correlations.

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 [Phys. Rev. Lett. 96, 038702 (2006)] I have shown that the connectivity fluctuations together with the small-world property lead to a 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.