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

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