Diagonal Degree Correlations vs. Epidemic Threshold in Scale-Free Networks

We prove that the presence of a diagonal assortative degree correlation, even if small, has the effect of dramatically lowering the epidemic threshold of large scale-free networks. The correlation matrix considered is Ph|k=1-rP(hk)(U)+rd(hk), where PU is uncorrelated and r (the Newman assortativity coefficient) can be very small. The effect is uniform in the scale exponent gamma if the network size is measured by the largest degree n. We also prove that it is possible to construct, via the Porto-Weber method, correlation matrices which have the same k(nn) as the Ph|k above, but very different elements and spectra, and thus lead to different epidemic diffusion and threshold. Moreover, we study a subset of the admissible transformations of the form P(h|k)-> P(h|k)+Phi(h,k) with Phi(h,k) depending on a parameter which leaves k(nn) invariant. Such transformations affect in general the epidemic threshold. We find, however, that this does not happen when they act between networks with constant k(nn), i.e., networks in which the average neighbor degree is independent from the degree itself (a wider class than that of strictly uncorrelated networks).