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Motivated by robotic surveillance applications, this paper studies the novelproblem of maximizing the return time entropy of a Markov chain, subject to agraph topology with travel times and stationary distribution. The return timeentropy is the weighted average, over all graph nodes, of the entropy of thefirst return times of the Markov chain; this objective function is a functionseries that does not admit in general a closed form.  The paper features theoretical and computational contributions. First, weobtain a discrete-time delayed linear system for the return time probabilitydistribution and establish its convergence properties. We show that theobjective function is continuous over a compact set and therefore admits aglobal maximum; a unique globally-optimal solution is known only for completegraphs with unitary travel times. We then establish upper and lower boundsbetween the return time entropy and the well-known entropy rate of the Markovchain. To compute the optimal Markov chain numerically, we establish theasymptotic equality between entropy, conditional entropy and truncated entropy,and propose an iteration to compute the gradient of the truncated entropy.Finally, we apply these results to the robotic surveillance problem. Ournumerical results show that, for a model of rational intruder over prototypicalgraph topologies and test cases, the maximum return time entropy chain performsbetter than several existing Markov chains.

Markov Chains with Maximum Return Time Entropy for Robotic Surveillance