Optimal Paths in Disordered Complex Networks

TL;DR

This paper investigates how disorder affects the optimal path length in various complex networks, revealing different scaling behaviors depending on network type and disorder strength, with implications for network robustness and efficiency.

Contribution

It provides a comprehensive analysis of optimal path scaling in disordered networks, including new results for scale-free networks across different degree exponents.

Findings

Strong disorder leads to $ ext{opt} ext{-}path ext{ length} o N^{1/3}$ in ER and WS networks.

In scale-free networks, $ ext{opt} ext{-}path$ scales as $N^{( ext{lambda}-3)/( ext{lambda}-1)}$ for $3< ext{lambda}<4$.

Weak disorder results in $ ext{opt} ext{-}path o ext{log} N$ in all studied network models.

Abstract

We study the optimal distance in networks, $\ell_{\scriptsize opt}$, 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 $\ell_{\scriptsize opt}\sim N^{1/3}$ in both Erd\H{o}s-R\'enyi (ER) and Watts-Strogatz (WS) networks. For scale free (SF) networks, with degree distribution $P(k) \sim k^{-\lambda}$, we find that $\ell_{\scriptsize opt}$ scales as $N^{(\lambda - 3)/(\lambda - 1)}$ for $3<\lambda<4$ and as $N^{1/3}$ for $\lambda\geq 4$. Thus, for these networks, the small-world nature is destroyed. For $2 < \lambda < 3$, our numerical results suggest that $\ell_{\scriptsize opt}$ scales as $\ln^{\lambda-1}N$. We also find numerically that for weak disorder $\ell_{\scriptsize opt}\sim\ln N$ for both the ER and WS models as well as for SF networks.