稀疏随机图上混合时间的比较
Comparing mixing times on sparse random graphs
摘要：
人们很自然地认为，非回溯随机游动会比简单随机游动混合得更快，但到目前为止，这只在正则图中得到了证明。为了分析典型的不规则图，设$G$是一个随机图，其顶点数为$n$，最小度为3，度分布具有指数尾。我们确定了$G$上简单随机游动的最坏情况下的精确混合时间，并表明，以很高的概率，它表现出截止时间$\mathbf{h}^{-1} \log n$，其中$\mathbf{h}$是渐近熵的高尔顿-沃森树上的简单随机游动近似$G$本地。（以前，这只是典型的起点。此外，我们表明，这个渐近混合时间是严格大于混合时间的nonbacktracking步行，通过一个微妙的比较上的Galton-Watson树的熵。
英文摘要：
It is natural to expect that nonbacktracking random walk will mix faster than simple random walks, but so far this has only been proved in regular graphs. To analyze typical irregular graphs, let $G$ be a random graph on $n$ vertices with minimum degree 3 and a degree distribution that has exponential tails. We determine the precise worst-case mixing time for simple random walk on $G$, and show that, with high probability, it exhibits cutoff at time $\mathbf{h}^{-1} \log n$, where $\mathbf{h}$ is the asymptotic entropy for simple random walk on a Galton--Watson tree that approximates $G$ locally. (Previously this was only known for typical starting points.) Furthermore, we show that this asymptotic mixing time is strictly larger than the mixing time of nonbacktracking walk, via a delicate comparison of entropies on the Galton-Watson tree.
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