  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).
