Random walks are fundamental models of stochastic processes with applications in various fields, including physics, biology, and computer science. We study classical and quantum random walks under the influence of stochastic resetting on arbitrary networks. Based on the mathematical formalism of quantum stochastic walks, we provide a framework of classical and quantum walks whose evolution is determined by graph Laplacians. We study the influence of quantum effects on the stationary and long-time average probability distribution by interpolating between the classical and quantum regime. We compare our analytical results on stationary and long-time average probability distributions with numerical simulations on different networks, revealing differences in the way resets affect the sampling properties of classical and quantum walks.

中文翻译：

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从经典漫步到量子漫步，网络上的随机重置

随机游走是随机过程的基本模型，在物理，生物学和计算机科学等各个领域都有应用。我们研究随机重置对任意网络的影响下的经典和量子随机游动。基于量子随机游走的数学形式主义，我们提供了经典和量子游走的框架，其演化由图拉普拉斯算子决定。我们通过经典和量子态之间的插值研究量子效应对平稳和长期平均概率分布的影响。我们将稳态和长期平均概率分布的分析结果与不同网络上的数值模拟进行了比较，揭示了重置影响经典和量子游走的采样属性的方式上的差异。