Optimal Paths in Complex Networks with Correlated Weights: The World-wide Airport Network
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 $w_{ij} \sim x_{ij} (k_i k_j)^α$, where $k_i$ and $k_j$ are the degrees of nodes $i$ and $j$, $x_{ij}$ is a random number and $α$ represents the strength of the correlations. The case $α> 0$ represents correlation between weights and degree, while $α< 0$ represents anti-correlation and the case $α= 0$ 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 $α$. We calculate the robustness of correlated scale-free networks with different $α$, and find the networks with $α< 0$ to be the most robust networks when compared to the other values of $α$. 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.