We study random walk with adaptive move strategies on a class of directed
graphs with variable wiring diagram. The graphs are grown from the evolution
rules compatible with the dynamics of the world-wide Web [Tadi\'c, Physica A
{\bf 293}, 273 (2001)], and are characterized by a pair of power-law
distributions of out- and in-degree for each value of the parameter β,
which measures the degree of rewiring in the graph. The walker adapts its move
strategy according to locally available information both on out-degree of the
visited node and in-degree of target node. A standard random walk, on the other
hand, uses the out-degree only. We compute the distribution of connected
subgraphs visited by an ensemble of walkers, the average access time and
survival probability of the walks. We discuss these properties of the walk
dynamics relative to the changes in the global graph structure when the control
parameter β is varied. For β≥3, corresponding to the
world-wide Web, the access time of the walk to a given level of hierarchy on
the graph is much shorter compared to the standard random walk on the same
graph. By reducing the amount of rewiring towards rigidity limit \beta \to
\beta_c \lesss im 0.1, corresponding to the range of naturally occurring
biochemical networks, the survival probability of adaptive and standard random
walk become increasingly similar. The adaptive random walk can be used as an
efficient message-passing algorithm on this class of graphs for large degree of
rewiring.Comment: 8 pages, including 7 figures; to appear in Europ. Phys. Journal