Abstract
We discuss shortest-path lengths on periodic rings of size L supplemented with an average of randomly located long-range links whose lengths are distributed according to Using rescaling arguments and numerical simulation on systems of up to sites, we show that a characteristic length exists such that for but for For small p we find that the shortest-path length satisfies the scaling relation Three regions with different asymptotic behaviors are found, respectively: (a) where (b) where and (c) where behaves logarithmically, i.e., The characteristic length is of the form with 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.
- Received 9 January 2002
DOI:https://doi.org/10.1103/PhysRevE.65.056709
©2002 American Physical Society