图上的多重随机游走：从少数到覆盖多数
Multiple Random Walks on Graphs: Mixing Few to Cover Many
AI总结：
研究通过改进和收紧多重随机游走的覆盖时间界限，提出部分混合时间概念，以表征最坏情况下的覆盖时间。
摘要：
图上的随机游走是许多随机算法和随机过程的基本原始操作。自然地，人们会问，通过独立并行地运行k次随机游走能获得多少收益。尽管对于许多自然网络，多重游走的覆盖时间已进行了研究，但寻找一般情况下最坏起始顶点的多重覆盖时间的特征（由Alon、Avin、Koucký、Kozma、Lotker和Tuttle于2008年提出的问题）仍然是一个开放问题。首先，我们改进并收紧了当k次随机游走从平稳分布采样的顶点出发时的平稳覆盖时间的界限。例如，我们证明了平稳覆盖时间的无条件下界为Ω((n/k) log n)，适用于任何n顶点的图G和任何1 ≤ k = o(n log n)。其次，我们建立了在几种基本网络上多重游走的平稳覆盖时间，精确到常数因子。第三，我们提出了一种框架，将最坏情况下的覆盖时间用平稳覆盖时间和一种新的、放松的多重游走混合时间概念——部分混合时间来表征。简而言之，部分混合时间仅要求特定比例的所有随机游走被混合。利用这些新概念，我们能够建立（或恢复）许多网络的最坏情况下的覆盖时间，包括扩张图、偏好附加图、网格、二叉树和超立方体。
英文摘要：
Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running $k$ multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times for worst-case start vertices (posed by Alon, Avin, Koucký, Kozma, Lotker, and Tuttle~in 2008) remains an open problem. First, we improve and tighten various bounds on the stationary cover time when $k$ random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of $Ω((n/k) \log n)$ on the stationary cover time, holding for any $n$-vertex graph $G$ and any $1 \leq k =o(n\log n )$. Secondly, we establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called the partial mixing time. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
