Optimal path and minimal spanning trees in random weighted networks

We review results on the scaling of the optimal path length ℓ_opt in random networks with weighted links or nodes. We refer to such networks as "weighted" or "disordered" networks. The optimal path is the path with minimum sum of the weights. In strong disorder, where the maximal weight along the path dominates the sum, we find that ℓ_opt increases dramatically compared to the known small-world result for the minimum distance ℓ_min ∼ log N, where N is the number of nodes. For Erdös-Rényi (ER) networks ℓ_opt ∼ N^1/3, while for scale free (SF) networks, with degree distribution P(k) ∼ k^-λ, we find that ℓ_opt scales as N^{(λ-3)/(λ-1)} for 3 < λ < 4 and as N^1/3 for λ ≥ 4. Thus, for these networks, the small-world nature is destroyed. For 2 < λ < 3 in contrary, our numerical results suggest that ℓ_opt scales as ln ^λ-1 N, representing still a small world. We also find numerically that for weak disorder ℓ_opt ∼ ln N for ER models as well as for SF networks. We also review the transition between the strong and weak disorder regimes in the scaling properties of ℓ_opt for ER and SF networks and for a general distribution of weights τ, P(τ). For a weight distribution of the form P(τ) = 1/(aτ) with (τ_min < τ < τ_max) and a = ln τ_max/ τ_min, we find that there is a crossover network size N* = N*(a) at which the transition occurs. For N ≪ N* the scaling behavior of ℓ_opt is in the strong disorder regime, while for N ≫ N* the scaling behavior is in the weak disorder regime. The value of N* can be determined from the expression ℓ_∞ (N*) = ap_c, where ℓ_∞ is the optimal path length in the limit of strong disorder, A ≡ ap_c → ∞ and p_c is the percolation threshold of the network. We suggest that for any P(τ) the distribution of optimal path lengths has a universal form which is controlled by the scaling parameter Z = ℓ_∞/A where A ≡ p_cτ_c/ ∫₀^τc τP(τ)dτ plays the role of the disorder strength and τ_c is defined by ∫₀^τc P(τ)dτ = p_c. In case P(τ) ∼ 1/(aτ), the equation for A is reduced to A = ap_c. The relation for A is derived analytically and supported by numerical simulations for Erdös-Rényi and scale-free graphs. We also determine which form of P(τ) can lead to strong disorder A → ∞. We then study the minimum spanning tree (MST), which is the subset of links of the network connecting all nodes of the network such that it minimizes the sum of their weights. We show that the minimum spanning tree (MST) in the strong disorder limit is composed of percolation clusters, which we regard as "super-nodes", interconnected by a scale-free tree. The MST is also considered to be the skeleton of the network where the main transport occurs. We furthermore show that the MST can be partitioned into two distinct components, having significantly different transport properties, characterized by centrality - number of times a node (or link) is used by transport paths. One component the superhighways, for which the nodes (or links) with high centrality dominate, corresponds to the largest cluster at the percolation threshold (incipient infinite percolation cluster) which is a subset of the MST. The other component, roads, includes the remaining nodes, low centrality nodes dominate. We find also that the distribution of the centrality for the incipient infinite percolation cluster satisfies a power law, with an exponent smaller than that for the entire MST. We demonstrate the significance identifying the superhighways by showing that one can improve significantly the global transport by improving a very small fraction of the network, the superhighways.