This paper introduces a decentralized state-dependent Markov chain synthesis (DSMC) algorithm for finite-state Markov chains. We present a state-dependent consensus protocol that achieves exponential convergence under mild technical conditions, without relying on any connectivity assumptions regarding the dynamic network topology. Utilizing the proposed consensus protocol, we develop the DSMC algorithm, updating the Markov matrix based on the current state while ensuring the convergence conditions of the consensus protocol. This result establishes the desired steady-state distribution for the resulting Markov chain, ensuring exponential convergence from all initial distributions while adhering to transition constraints and minimizing state transitions. The DSMC's performance is demonstrated through a probabilistic swarm guidance example, which interprets the spatial distribution of a swarm comprising a large number of mobile agents as a probability distribution and utilizes the Markov chain to compute transition probabilities between states. Simulation results demonstrate faster convergence for the DSMC based algorithm when compared to the previous Markov chain based swarm guidance algorithms.

翻译：本文提出了一种适用于有限状态马尔可夫链的去中心化状态依赖马尔可夫链合成算法。我们提出了一种去中心化状态依赖共识协议，该协议在温和的技术条件下实现指数收敛，且无需依赖动态网络拓扑的任何连通性假设。利用所提出的共识协议，我们开发了DSMC算法，在确保共识协议收敛条件的同时，基于当前状态更新马尔可夫矩阵。该结果为所得马尔可夫链建立了所需的稳态分布，确保从所有初始分布出发实现指数收敛，同时满足转移约束并最小化状态转移。通过一个概率蜂群引导示例展示了DSMC的性能，该示例将包含大量移动代理的蜂群空间分布解释为概率分布，并利用马尔可夫链计算状态间的转移概率。仿真结果表明，与以往的基于马尔可夫链的蜂群引导算法相比，基于DSMC的算法具有更快的收敛速度。