The theory of rapid mixing random walks plays a fundamental role in the study of modern randomised algorithms. Usually, the mixing time is measured with respect to the worst initial position. It is well known that the presence of bottlenecks in a graph hampers mixing and, in particular, starting inside a small bottleneck significantly slows down the diffusion of the walk in the first steps of the process. To circumvent this problem, the average mixing time is defined to be the mixing time starting at a uniformly random vertex. In this paper we provide a general framework to show logarithmic average mixing time for random walks on graphs with small bottlenecks. The framework is especially effective on certain families of random graphs with heterogeneous properties. We demonstrate its applicability on two random models for which the mixing time was known to be of order $\log^2n$, speeding up the mixing to order $\log n$. First, in the context of smoothed analysis on connected graphs, we show logarithmic average mixing time for randomly perturbed graphs of bounded degeneracy. A particular instance is the Newman-Watts small-world model. Second, we show logarithmic average mixing time for supercritically percolated expander graphs. When the host graph is complete, this application gives an alternative proof that the average mixing time of the giant component in the supercritical Erd\H{o}s-R\'enyi graph is logarithmic.

翻译：快速混合随机游走理论在现代随机化算法研究中起着基础性作用。通常，混合时间是根据最差初始位置来衡量的。众所周知，图中的瓶颈会阻碍混合，特别是从小瓶颈内部出发会在过程的最初几步显著减缓游走的扩散。为解决这一问题，平均混合时间被定义为从均匀随机顶点出发的混合时间。本文提供了一个通用框架，以证明具有小瓶颈的图上随机游走的对数平均混合时间。该框架在具有异质特性的某些随机图族上尤为有效。我们在两个已知混合时间阶为 $\log^2n$ 的随机模型上证明了其适用性，将混合速度提升至 $\log n$ 阶。首先，在连通图的平滑分析背景下，我们展示了有界退化度随机扰动图的对数平均混合时间，其特例是纽曼-沃茨小世界模型。其次，我们证明了超临界渗流扩展图的对数平均混合时间。当宿主图为完全图时，该应用为超临界 Erdős-Rényi 图中巨分量的平均混合时间呈对数阶提供了替代性证明。