## Abstract

We study the optimal distance ℓ_{opt} in random networks in the presence of disorder implemented by assigning random weights to the links. The optimal distance between two nodes is the length of the path for which the sum of weights along the path ("cost") is a minimum. We study the case of strong disorder for which the distribution of weights is so broad that its sum along any path is dominated by the largest link weight in the path. We find that in random graphs, ℓ_{opt} scales as N^{1/3}, where N is the number of nodes in the network. Thus, ℓ_{opt} increases dramatically compared to the known small-world result for the minimum distance ℓ_{min}, which scales as log N. We also study, theoretically and by simulations, scale-free networks characterized by a power law distribution for the number of links, P(k) ∼ k^{-λ}, and find that ℓ_{opt} scales as N^{1/3} for λ > 4 and as N^{(λ-3)/(λ-1)} for 3 < λ < 4. For 2 < λ < 3, our numerical results suggest that ℓ_{opt} scales logarithmically with N.

Original language | English |
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Pages (from-to) | 246-252 |

Number of pages | 7 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 330 |

Issue number | 1-2 |

DOIs | |

State | Published - 1 Dec 2003 |

Event | Randomes and Complexity - Eilat, Israel Duration: 5 Jan 2003 → 9 Jan 2003 |

### Bibliographical note

Funding Information:We thank A.-L. Barabási and S. Sreenivasan for helpful discussions, and ONR for financial support.

### Funding

We thank A.-L. Barabási and S. Sreenivasan for helpful discussions, and ONR for financial support.

Funders | Funder number |
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Office of Naval Research |

## Keywords

- Optimal path
- Percolation
- Scale-free networks
- Small-world networks
- Strong disorder