  An analytic effective medium theory is constructed to study the mean access
times for random walks on hybrid disordered structures formed by embedding
complex networks into regular lattices, considering transition rates $F$ that
are different for steps across lattice bonds from the rates $f$ across network
shortcuts. The theory is developed for structures with arbitrary shortcut
distributions and applied to a class of partially-disordered traversal enhanced
networks in which shortcuts of fixed length are distributed randomly with
finite probability. Numerical simulations are found to be in excellent
agreement with predictions of the effective medium theory on all aspects
addressed by the latter. Access times for random walks on these partially
disordered structures are compared to those on small-world networks, which on
average appear to provide the most effective means of decreasing access times
uniformly across the network.
