Optimal Paths in Disordered Complex Networks
Abstract
We study the optimal distance in networks, ℓopt, 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 ℓopt∼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 ℓopt 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 ℓopt scales as ln(λ-1N. We also find numerically that for weak disorder ℓopt∼ln(N for both the ER and WS models as well as for SF networks.
- Publication:
-
Physical Review Letters
- Pub Date:
- October 2003
- DOI:
- 10.1103/PhysRevLett.91.168701
- arXiv:
- arXiv:cond-mat/0305051
- Bibcode:
- 2003PhRvL..91p8701B
- Keywords:
-
- 89.75.Hc;
- Networks and genealogical trees;
- Condensed Matter - Soft Condensed Matter
- E-Print:
- 5 pages, 4 figures, accepted for publication in Physical Review Letters