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

Abstract

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 $P\left(h|k\right)=\left(1-r\right)P_{hk}^U+r{\delta }_{hk}$ , where $P^U$ 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 $P\left(h|k\right)$ 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 $P\left(h|k\right)⟶P\left(h|k\right)+\mathrm{\Phi }\left(h,k\right)$ with $\mathrm{\Phi }\left(h,k\right)$ 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).