动态图上的可逆随机游动
Reversible Random Walks on Dynamic Graphs
摘要：
最近，动态图上的随机游动由于其对现实世界网络的时变结构的适应性而得到了研究。一般来说，对于惰性简单随机游走，静态图和动态图设置之间存在巨大的差距：尽管$O(n^3)$覆盖时间对于任何$n$顶点的静态图都有显示，但存在具有指数命中时间的边变化动态图。另一方面，以前的工作表明，随机游走的动态图上的时间齐次平稳分布的行为几乎相同的静态图。在本文中，我们加强了这种洞察力，获得一般和改进的界限。具体地说，我们考虑一个随机游动根据序列$(P_t)_{t\geq 1}$的不可约和可逆的过渡矩阵，使所有的$P_t$有相同的平稳分布。我们根据最坏固定$P_t$，根据随机游动的击中时间和松弛时间来约束混合时间、击中时间和覆盖时间。此外，我们还得到了动态图上多重随机游动的击中时间、覆盖时间和合并时间的第一界。这些界可以看作是著名的静态图上的随机游动的界的扩展。我们的结果推广了以前的上界为特定的随机游动动态图，例如，懒惰简单随机游动和$d_{\max}$ -懒惰游动，并在各种情况下给出了改进的和紧的上界.作为一个有趣的结果，我们的推广，我们得到严格的限制懒惰大都会步行[Nonaka，Ono，Sadakane，和Yamashita，TCS 10]的任何动态图：$O(n^2)$混合时间，$O(n^2)$命中时间，和$O(n^2\log n)$覆盖时间。此外，我们的合并时间界限意味着在动态图上拉投票的共识时间界限。
英文摘要：
Recently, random walks on dynamic graphs have been studied because of their adaptivity to the time-varying structure of real-world networks. In general, there is a tremendous gap between static and dynamic graph settings for the lazy simple random walk: Although $O(n^3)$ cover time was shown for any static graphs of $n$ vertices, there is an edge-changing dynamic graph with an exponential hitting time. On the other hand, previous works indicate that the random walk on a dynamic graph with a time-homogeneous stationary distribution behaves almost identically to that on a static graph. In this paper, we strengthen this insight by obtaining general and improved bounds. Specifically, we consider a random walk according to a sequence $(P_t)_{t\geq 1}$ of irreducible and reversible transition matrices such that all $P_t$ have the same stationary distribution. We bound the mixing, hitting, and cover times in terms of the hitting and relaxation times of the random walk according to the worst fixed $P_t$. Moreover, we obtain the first bounds of the hitting and cover times of multiple random walks and the coalescing time on dynamic graphs. These bounds can be seen as an extension of the well-known bounds of random walks on static graphs. Our results generalize the previous upper bounds for specific random walks on dynamic graphs, e.g., lazy simple random walks and $d_{\max}$-lazy walks, and give improved and tight upper bounds in various cases. As an interesting consequence of our generalization, we obtain tight bounds for the lazy Metropolis walk [Nonaka, Ono, Sadakane, and Yamashita, TCS10] on any dynamic graph: $O(n^2)$ mixing time, $O(n^2)$ hitting time, and $O(n^2\log n)$ cover time. Additionally, our coalescing time bound implies the consensus time bound of the pull voting on a dynamic graph.
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。
