Optimal paths in complex networks with correlated weights: The worldwide airport network

We study complex networks with weights wᵢⱼ 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 wᵢⱼ~xᵢⱼ(kᵢkⱼ)^α, where kᵢ and kⱼ are the degrees of nodes i and j, xᵢⱼ 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 anticorrelation and the case α=0 reduces to the case of no correlations. We study the scaling of the lengths of the optimal paths, ₒₚₜ, with the system size N in strong disorder for scale-free networks for different α. We find two different universality classes for ₒₚₜ in strong disorder depending on α: (i) if α>0, then for λ>2 the scaling law ₒₚₜ~N1∕3, where λ is the power-law exponent of the degree distribution of scale-free networks, and (ii) if α≤0, then ₒₚₜ~N^νₒₚₜ with νₒₚₜ identical to its value for the uncorrelated case α=0. 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, and our numerical results suggest that for scale-free networks with α<0, the percolation threshold pc is finite for λ>3, which belongs to the same universality class as α=0. We compare our simulation results with the real worldwide airport network, and we find good agreement.