  We study complex networks with weights, $w_{ij}$, associated with each link
connecting node $i$ and $j$. The weights are chosen to be correlated with the
network topology in the form found in two real world examples, (a) the
world-wide airport network, and (b) the {\it E. Coli} metabolic network. Here
$w_{ij} \sim x_{ij} (k_i k_j)^\alpha$, where $k_i$ and $k_j$ are the degrees of
nodes $i$ and $j$, $x_{ij}$ is a random number and $\alpha$ represents the
strength of the correlations. The case $\alpha > 0$ represents correlation
between weights and degree, while $\alpha < 0$ represents anti-correlation and
the case $\alpha = 0$ reduces to the case of no correlations. We study the
scaling of the lengths of the optimal paths, $\ell_{\rm opt}$, with the system
size $N$ in strong disorder for scale-free networks for different $\alpha$. We
calculate the robustness of correlated scale-free networks with different
$\alpha$, and find the networks with $\alpha < 0$ to be the most robust
networks when compared to the other values of $\alpha$. We propose an
analytical method to study percolation phenomena on networks with this kind of
correlation. We compare our simulation results with the real world-wide airport
network, and we find good agreement.
