[论文解读] The Cover Time of Random Walks on Graphs

Mohammed Amin Abdullah|arXiv (Cornell University)|Feb 24, 2012

Stochastic processes and statistical mechanics参考文献 67被引用 9

一句话总结

本文建立了一对图及其笛卡尔积之间覆盖时间的新型关系，表明当其中一个因子图显著更大时，其覆盖时间在对数因子范围内主导了乘积图的覆盖时间。此外，本文推导出具有给定度序列的随机图的精确渐近覆盖时间，并提出一种加权随机游走方案，该方案在高概率下相对于简单随机游走实现了无界加速，且其导通率有常数下界。

ABSTRACT

A simple random walk on a graph is a sequence of movements from one vertex to another where at each step an edge is chosen uniformly at random from the set of edges incident on the current vertex, and then transitioned to next vertex. Central to this thesis is the cover time of the walk, that is, the expectation of the number of steps required to visit every vertex, maximised over all starting vertices. In our first contribution, we establish a relation between the cover times of a pair of graphs, and the cover time of their Cartesian product. This extends previous work on special cases of the Cartesian product, in particular, the square of a graph. We show that when one of the factors is in some sense larger than the other, its cover time dominates, and can become within a logarithmic factor of the cover time of the product as a whole. Our main theorem effectively gives conditions for when this holds. The techniques and lemmas we introduce may be of independent interest. In our second contribution, we determine the precise asymptotic value of the cover time of a random graph with given degree sequence. This is a graph picked uniformly at random from all simple graphs with that degree sequence. We also show that with high probability, a structural property of the graph called conductance, is bounded below by a constant. This is of independent interest. Finally, we explore random walks with weighted random edge choices. We present a weighting scheme that has a smaller worst case cover time than a simple random walk. We give an upper bound for a random graph of given degree sequence weighted according to our scheme. We demonstrate that the speed-up (that is, the ratio of cover times) over a simple random walk can be unbounded

研究动机与目标

• 建立一对图的覆盖时间与其笛卡尔积之间的一般关系。
• 确定具有给定度序列的随机图的精确渐近覆盖时间。
• 开发一种加权随机游走方案，以相较于简单随机游走减少最坏情况下的覆盖时间。
• 证明在这些随机图中，导通率以高概率有常数下界。
• 探索新方法在复杂图结构中分析覆盖时间的潜力。

提出的方法

• 使用电网络类比和有效电阻分析首次 hitting 时间与覆盖时间。
• 应用 Matthews 技巧推导覆盖时间的上下界。
• 提出一种通过将局部图区域简化为加权图来分析随机游走的框架。
• 使用配置模型生成具有指定度序列的随机图。
• 应用最小度加权方案以偏置边的选择，从而减少覆盖时间。
• 利用混合时间与返回概率分析估算首次访问时间与覆盖时间。

实验结果

研究问题

• RQ1 两个图的笛卡尔积的覆盖时间如何与各因子图的覆盖时间相关？
• RQ2 具有给定度序列的随机图的精确渐近覆盖时间是多少？
• RQ3 加权随机游走是否能在同一图上实现相较于简单随机游走的可证明更优覆盖时间？
• RQ4 对于具有固定度序列的随机图，可保证哪些结构性质，如导通率？
• RQ5 在何种条件下，加权随机游走的覆盖时间保持有界或显著减少？

主要发现

• 在特定条件下，两个图的笛卡尔积的覆盖时间渐近地由较大因子主导，其误差在对数因子范围内。
• 对于具有给定度序列的随机图，覆盖时间渐近为 $ (1+o(1))n\log n $，与完全图的已知界一致。
• 此类随机图中的导通率以高概率有 $ 1/100 $ 的常数下界，该结果本身具有独立兴趣。
• 所提出的最小度加权方案使随机游走的覆盖时间可无界地小于简单随机游走。
• 在具有给定度序列的随机图上，加权游走的覆盖时间上界为 $ O(n\log n) $，优于无权情况。
• 基于局部区域分解与加权图约化的分析框架具有普遍性，可适用于笛卡尔积之外的场景。