  Kinetically grown self-avoiding walks on various types of generalized random
networks have been studied. Networks with short- and long-tailed degree
distributions $P(k)$ were considered ($k$, degree or connectivity), including
scale-free networks with $P(k) \sim k^{-\gamma}$. The long-range behaviour of
self-avoiding walks on random networks is found to be determined by finite-size
effects. The mean self-intersection length of non-reversal random walks, $<l>$,
scales as a power of the system size $N$: $<l > \sim N^{\beta}$, with an
exponent $\beta = 0.5$ for short-tailed degree distributions and $\beta < 0.5$
for scale-free networks with $\gamma < 3$. The mean attrition length of kinetic
growth walks, $<L>$, scales as $<L > \sim N^{\alpha}$, with an exponent
$\alpha$ which depends on the lowest degree in the network. Results of
approximate probabilistic calculations are supported by those derived from
simulations of various kinds of networks. The efficiency of kinetic growth
walks to explore networks is largely reduced by inhomogeneity in the degree
distribution, as happens for scale-free networks.
