Optimal Paths in Disordered Complex Networks

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We study the optimal distance in networks, ℓ opt, defined as the length of the path minimizing the total weight, in the presence of disorder. Disorder is introduced by assigning random weights to the links or nodes. For strong disorder, where the maximal weight along the path dominates the sum, we find that ℓ opt N¹/³ in both Erdős-Rényi (ER) and Watts-Strogatz (WS) networks. 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 N¹/³ for λ≥ 4. Thus, for these networks, the small-world nature is destroyed. For 2 < λ< 3, our numerical results suggest that ℓ opt scales as λ⁻¹N. We also find numerically that for weak disorder ℓ opt N for both the ER and WS models as well as for SF networks.