论文标题
部分可观测时空混沌系统的无模型预测
Cutoff for random walk on random graphs with a community structure
论文作者
论文摘要
我们考虑具有嵌入式社区结构的配置模型的变体，并研究了简单的随机步行的混合特性。每个顶点都有一个内部$ \ mathrm {deg}^{\ text {int}} \ geq 3 $和一个出发的$ \ mathrm {deg}^{\ text {out text {out}} $半边的数量。给定一个随机矩阵$ q $，我们选择了一个随机的完美匹配的半边缘，受到约束的约束，即每个顶点$ v $都有$ \ mathrm {deg}^{\ text {\ text {int}}（int}}（v）$ neighbors在其社区内部的社区中的$ $ i $ i $ i $ j $ j is j is j is j is j;假设社区的数量是恒定的，并且它们都具有可比的大小，我们证明了以下二分法：在且仅当Cheeger常数$ Q $ times $ times $ \ log n $（$ n $是$ n $是Vertices的数量）时，在结果图上显示了截断。 在[4]中，Ben-Hamou建立了一种二分法，用于在具有2个社区的类似随机图模型上进行非折叠的随机步行。我们证明了与简单随机步行的截止值相同的表征。
We consider a variant of the configuration model with an embedded community structure and study the mixing properties of a simple random walk on it. Every vertex has an internal $\mathrm{deg}^{\text{int}}\geq 3$ and an outgoing $\mathrm{deg}^{\text{out}}$ number of half-edges. Given a stochastic matrix $Q$, we pick a random perfect matching of the half-edges subject to the constraint that each vertex $v$ has $\mathrm{deg}^{\text{int}}(v)$ neighbours inside its community and the proportion of outgoing half-edges from community $i$ matched to a half-edge from community $j$ is $Q(i,j)$. Assuming the number of communities is constant and they all have comparable sizes, we prove the following dichotomy: simple random walk on the resulting graph exhibits cutoff if and only if the product of the Cheeger constant of $Q$ times $\log n$ (where $n$ is the number of vertices) diverges. In [4], Ben-Hamou established a dichotomy for cutoff for a non-backtracking random walk on a similar random graph model with 2 communities. We prove the same characterisation of cutoff holds for simple random walk.
