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