  We investigate hide-and-seek games on complex networks using a random walk
framework. Specifically, we investigate the efficiency of various degree-biased
random walk search strategies to locate items that are randomly hidden on a
subset of vertices of a random graph. Vertices at which items are hidden in the
network are chosen at random as well, though with probabilities that may depend
on degree. We pitch various hide and seek strategies against each other, and
determine the efficiency of search strategies by computing the average number
of hidden items that a searcher will uncover in a random walk of $n$ steps. Our
analysis is based on the cavity method for finite single instances of the
problem, and generalises previous work of De Bacco et al. [1] so as to cover
degree-biased random walks. We also extend the analysis to deal with the
thermodynamic limit of infinite system size. We study a broad spectrum of
functional forms for the degree bias of both the hiding and the search strategy
and investigate the efficiency of families of search strategies for cases where
their functional form is either matched or unmatched to that of the hiding
strategy. Our results are in excellent agreement with those of numerical
simulations. We propose two simple approximations for predicting efficient
search strategies. One is based on an equilibrium analysis of the random walk
search strategy. While not exact, it produces correct orders of magnitude for
parameters characterising optimal search strategies. The second exploits the
existence of an effective drift in random walks on networks, and is expected to
be efficient in systems with low concentration of small degree nodes.
