1 research outputs found
Shortest paths on systems with power-law distributed long-range connections
We discuss shortest-path lengths 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 sites, we show
that a characteristic length exists such that for
. 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 , b)
1<\xpn<2 where 0<\theta_s(\xpn)<1/2 and, c) \xpn<1 where
behaves logarithmically, i.e. . The characteristic length is
of the form 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.Comment: 10 pages, 10 figures, revtex4. Submitted to PR