  We investigate the role of degree correlation among nodes on the stability of
complex networks, by studying spectral properties of randomly weighted matrices
constructed from directed Erd\"{o}s-R\'enyi and scale-free random graph models.
We focus on the behaviour of the largest real part of the eigenvalues,
$\lambda_\text{max}$, that governs the growth rate of perturbations about an
equilibrium (and hence, determines stability). We find that assortative mixing
by degree, where nodes with many links connect preferentially to other nodes
with many links, reduces the stability of networks. In particular, for sparse
scale-free networks with $N$ nodes, $\lambda_\text{max}$ scales as $N^\alpha$
for highly assortative networks, while for disassortative graphs,
$\lambda_\text{max}$ scales logarithmically with $N$. This difference may be a
possible reason for the prevalence of disassortative networks in nature.
