Year: 2003
Type: article
Abstract: We study the optimal distance in networks, ${\ensuremath{\ell}}_{\mathrm{o}\mathrm{p}\mathrm{t}}$, 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 ${\ensuremath{\ell}}_{\mathrm{o}\mathrm{p}\mathrm{t}}\ensuremath{\sim}{N}^{1/3}$ in both Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi (ER) and Watts-Strogatz (WS) networks. For scale-free (SF) networks, with degree distribution $P(k)\ensuremath{\sim}{k}^{\ensuremath{-}\ensuremath{\lambda}}$, we find that ${\ensuremath{\ell}}_{\mathrm{o}\mathrm{p}\mathrm{t}}$ scales as ${N}^{(\ensuremath{\lambda}\ensuremath{-}3)/(\ensuremath{\lambda}\ensuremath{-}1)}$ for $3<\ensuremath{\lambda}<4$ and as ${N}^{1/3}$ for $\ensuremath{\lambda}\ensuremath{\ge}4$. Thus, for these networks, the small-world nature is destroyed. For $2<\ensuremath{\lambda}<3$, our numerical results suggest that ${\ensuremath{\ell}}_{\mathrm{o}\mathrm{p}\mathrm{t}}$ scales as ${ln}^{\ensuremath{\lambda}\ensuremath{-}1}N$. We also find numerically that for weak disorder ${\ensuremath{\ell}}_{\mathrm{o}\mathrm{p}\mathrm{t}}\ensuremath{\sim}\mathrm{ln}N$ for both the ER and WS models as well as for SF networks.
Source: Physical Review Letters
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2159644837