A Random Walk Perspective on Hide-and-Seek Games

Abstract: 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.