This article analyzes the meeting time between a pair of pursuer and evader performing random walks on digraphs. The existing bounds on the meeting time usually work only for certain classes of walks and cannot be used to formulate optimization problems and design robotic strategies. First, by analyzing multiple random walks on a common graph as a single random walk on the Kronecker product graph, we provide the first closed-form expression for the expected meeting time in terms of the transition matrices of the moving agents. This novel expression leads to necessary and sufficient conditions for the meeting time to be finite and to insightful graph-theoretic interpretations. Second, based on the closed-form expression, we set up and study the minimization problem for the expected capture time for a pursuer/evader pair. We report theoretical and numerical results on basic case studies to show the effectiveness of the design.

中文翻译：

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基于随机游走相遇时间的机器人监控

本文分析了一对追求者和逃避者在有向图上进行随机游走的相遇时间。会议时间的现有界限通常仅适用于某些类别的步行，不能用于制定优化问题和设计机器人策略。首先，通过将公共图上的多个随机游走分析为 Kronecker 乘积图上的单个随机游走，我们根据移动代理的转移矩阵提供了预期会面时间的第一个封闭形式表达式。这种新颖的表达导致了会议时间是有限的必要和充分条件，并导致了有见地的图论解释。其次，基于封闭形式的表达式，我们建立并研究了追求者/逃避者对预期捕获时间的最小化问题。