Importance of small-degree nodes in assortative networks with degree-weight correlations.

It has been known that assortative network structure plays an important role in spreading dynamics for unweighted networks. Yet its influence on weighted networks is not clear, in particular when weight is strongly correlated with the degrees of the nodes as we empirically observed in Twitter. Here we use the self-consistent probability method and revised nonperturbative heterogenous mean-field theory method to investigate this influence on both susceptible-infective-recovered (SIR) and susceptible-infective-susceptible (SIS) spreading dynamics. Both our simulation and theoretical results show that while the critical threshold is not significantly influenced by the assortativity, the prevalence in the supercritical regime shows a crossover under different degree-weight correlations. In particular, unlike the case of random mixing networks, in assortative networks, the negative degree-weight correlation leads to higher prevalence in their spreading beyond the critical transmissivity than that of the positively correlated. In addition, the previously observed inhibition effect on spreading velocity by assortative structure is not apparent in negatively degree-weight correlated networks, while it is enhanced for that of the positively correlated. Detailed investigation into the degree distribution of the infected nodes reveals that small-degree nodes play essential roles in the supercritical phase of both SIR and SIS spreadings. Our results have direct implications in understanding viral information spreading over online social networks and epidemic spreading over contact networks.