复杂网络中的随机重置随机漫步:一种离散时间方法

Thomas M. Michelitsch, Giuseppe D'Onofrio, Federico Polito, Alejandro P. Riascos
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引用次数: 0

摘要

我们考虑的是在连通的有向网络上进行重置的离散时间马尔可夫随机行走。网络中的某些节点是目标节点,我们将重点放在这些节点的首次命中统计上。在更新过程的非马尔可夫情况下,我们考虑了轻尾和胖尾的重置间分布。我们根据离散后向递推时间 PDF 推导出传播矩阵,并在光尾情况下证明了非平衡稳态的存在。为了解决非马尔可夫情况,我们推导出了一个有缺陷的传播矩阵,它描述了一种辅助行走,其特征是当行走者到达目标节点时立即杀死行走者。该传播器为目标节点提供了关于主题的首次通过统计信息。我们建立了在重置条件下行走的遍历性的充分条件。此外,我们还讨论了一般的重置机制,在这种机制下,行走是非遍历性的。最后,我们分析了具有无限均值的重置间时间分布,并将重点放在西布亚(Sibuya)情况上。我们应用这些结果研究了在瓦特-斯特罗加茨和巴拉布-阿西-阿尔伯特随机图的现实化中马尔可夫和非马尔可夫(西布亚)更新重置协议的平均首次通过时间。我们展示了平均首次通过时间与重新定位节点、目标节点和重置率的比例之间非同一般的依赖关系。事实证明,在瓦茨-斯特罗加茨图的大世界情形中,随机搜索器的效率尤其受益于重置的存在。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Random walks with stochastic resetting in complex networks: a discrete time approach
We consider a discrete-time Markovian random walk with resets on a connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent discrete-time renewal process. Some nodes of the network are target nodes, and we focus on the statistics of first hitting of these nodes. In the non-Markov case of the renewal process, we consider both light- and fat-tailed inter-reset distributions. We derive the propagator matrix in terms of discrete backward recurrence time PDFs and in the light-tailed case we show the existence of a non-equilibrium steady state. In order to tackle the non-Markov scenario, we derive a defective propagator matrix which describes an auxiliary walk characterized by killing the walker as soon as it hits target nodes. This propagator provides the information on the mean first passage statistics to the target nodes. We establish sufficient conditions for ergodicity of the walk under resetting. Furthermore, we discuss a generic resetting mechanism for which the walk is non-ergodic. Finally, we analyze inter-reset time distributions with infinite mean where we focus on the Sibuya case. We apply these results to study the mean first passage times for Markovian and non-Markovian (Sibuya) renewal resetting protocols in realizations of Watts-Strogatz and Barab\'asi-Albert random graphs. We show non trivial behavior of the dependence of the mean first passage time on the proportions of the relocation nodes, target nodes and of the resetting rates. It turns out that, in the large-world case of the Watts-Strogatz graph, the efficiency of a random searcher particularly benefits from the presence of resets.
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