Long-distance characteristics of small-world networks have been studied by
means of self-avoiding walks (SAW's). We consider networks generated by
rewiring links in one- and two-dimensional regular lattices. The number of
SAW's un was obtained from numerical simulations as a function of the number
of steps n on the considered networks. The so-called connective constant,
μ=limn→∞un/un−1, which characterizes the long-distance
behavior of the walks, increases continuously with disorder strength (or
rewiring probability, p). For small p, one has a linear relation μ=μ0+ap, μ0 and a being constants dependent on the underlying
lattice. Close to p=1 one finds the behavior expected for random graphs. An
analytical approach is given to account for the results derived from numerical
simulations. Both methods yield results agreeing with each other for small p,
and differ for p close to 1, because of the different connectivity
distributions resulting in both cases.Comment: 7 pages, 5 figure