Shortest paths on systems with power-law distributed long-range connections

We discuss shortest-path lengths $\mathcal{l}\mathrm{}\left(r\right)\mathrm{}$ on periodic rings of size L supplemented with an average of $\mathrm{pL}$ randomly located long-range links whose lengths are distributed according to $P_l\sim l^{-\mu }.$ Using rescaling arguments and numerical simulation on systems of up to ${10}^7$ sites, we show that a characteristic length $\xi$ exists such that $\mathcal{l}\mathrm{}\left(r\right)\mathrm{}\sim r$ for $r<\mathrm{}\xi$ but $\mathcal{l}\mathrm{}\left(r\right)\mathrm{}\sim r^{{\theta }_s\left(\mu \right)}$ for $r\gg \xi .$ For small p we find that the shortest-path length satisfies the scaling relation $\mathcal{l}(r,\mu ,p)/\xi =f(\mu ,r/\xi ).\mathrm{}$ Three regions with different asymptotic behaviors are found, respectively: (a) $\mu >2\mathrm{}$ where ${\theta }_s=1,$ (b) $1<\mathrm{}\mu <2\mathrm{}$ where $0<\mathrm{}{\theta }_s\left(\mu \right)<\mathrm{}1/2,$ and (c) $\mu <1\mathrm{}$ where $\mathcal{l}\mathrm{}\left(r\right)\mathrm{}$ behaves logarithmically, i.e., ${\theta }_s=0.\mathrm{}$ The characteristic length $\xi$ is of the form $\xi \sim p^{-\nu }$ with $\nu =1/\left(2\mathrm{}-\mu \right)$ in region (b), but depends on L as well in region (c). A directed model of shortest paths is solved and compared with numerical results.