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) ∼ k^-γ. 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, , scales as a power of the system size N: ∼ N^β, with an

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... xponent β = 0.5 for short-tailed degree distributions and β < 0.5 for scale-free networks with γ < 3. The mean attrition length of kinetic growth walks, , scales as ∼ N^α, with an exponent α 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.