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

We discuss shortest-path lengths l(r) on periodic rings of size L supplemented with an average of pL randomly located long-range links whose lengths are distributed according to Pₗ~l^-μ. Using rescaling arguments and numerical simulation on systems of up to 10⁷ sites, we show that a characteristic length ξ exists such that l(r)~r for r<ξ but l(r)~r^θₛ(μ) for rξ. For small p we find that the shortest-path length satisfies the scaling relation l(r,μ,p)/ξ=f(μ,r/ξ). Three regions with different asymptotic behaviors are found, respectively: (a) μ>2 where θₛ=1, (b) 1<μ<2 where 0<θₛ(μ)<1/2, and (c) μ<1 where l(r) behaves logarithmically, i.e., θₛ=0. The characteristic length ξ is of the form ξ~p^-ν with ν=1/(2-μ) 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.