  Kinetically-grown self-avoiding walks have been studied on Watts-Strogatz
small-world networks, rewired from a two-dimensional square lattice. The
maximum length L of this kind of walks is limited in regular lattices by an
attrition effect, which gives finite values for its mean value < L >. For
random networks, this mean attrition length < L > scales as a power of the
network size, and diverges in the thermodynamic limit (large system size N).
For small-world networks, we find a behavior that interpolates between those
corresponding to regular lattices and randon networks, for rewiring probability
p ranging from 0 to 1. For p < 1, the mean self-intersection and attrition
length of kinetically-grown walks are finite. For p = 1, < L > grows with
system size as N^{1/2}, diverging in the thermodynamic limit. In this limit and
close to p = 1, the mean attrition length diverges as (1-p)^{-4}. Results of
approximate probabilistic calculations agree well with those derived from
numerical simulations.
