Diagonal degree correlations vs. epidemic threshold in scale-free networks

TL;DR

This paper demonstrates that even small diagonal assortative degree correlations significantly lower the epidemic threshold in large scale-free networks, affecting epidemic spread predictions.

Contribution

It provides a mathematical proof of how diagonal degree correlations influence epidemic thresholds and explores the construction of different correlation matrices with similar properties.

Findings

Diagonal correlations lower epidemic threshold

Constructed correlation matrices with same $k_{nn}$ but different spectra

Transformations preserving $k_{nn}$ generally do not affect the threshold

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 $P(h|k)=(1-r)P^U_{hk}+r\delta_{hk}$, 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 $P(h|k)$ above, but very different elements and spectrum, and thus lead to different epidemic diffusion and threshold. Moreover, we study a subset of the admissible transformations of the form $P(h|k) \to P(h|k)+\Phi(h,k)$ with $\Phi(h,k)$ depending on a parameter which leave $k_{nn}$ 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).