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 worldwide airport network and (b) the 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 anticorrelation and the case $\alpha =0$ reduces to the case of no correlations. We study the scaling of the lengths of the optimal paths, ${\ell }_{\mathrm{opt}}$, with the system size $N$ in strong disorder for scale-free networks for different $\alpha$. We find two different universality classes for ${\ell }_{\mathrm{opt}}$ in strong disorder depending on $\alpha$: (i) if $\alpha >0$, then for $\lambda >2$ the scaling law ${\ell }_{\mathrm{opt}}\sim N^{1∕3}$, where $\lambda$ is the power-law exponent of the degree distribution of scale-free networks, and (ii) if $\alpha \le 0$, then ${\ell }_{\mathrm{opt}}\sim N^{{\nu }_{\mathrm{opt}}}$ with ${\nu }_{\mathrm{opt}}$ identical to its value for the uncorrelated case $\alpha =0$. 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, and our numerical results suggest that for scale-free networks with $\alpha <0$, the percolation threshold $p_c$ is finite for $\lambda >3$, which belongs to the same universality class as $\alpha =0$. We compare our simulation results with the real worldwide airport network, and we find good agreement.

• Received 2 August 2006

DOI:https://doi.org/10.1103/PhysRevE.74.056104