热核PageRank计算及局部聚类算法
Computing Heat Kernel Pagerank and a Local Clustering Algorithm
摘要：
热核网页排名是个性化网页排名的一种变化，以指数形式给出。在这项工作中，我们提出了一个次线性时间算法近似的热核pagerank的图。该算法的工作原理是模拟有限长度的随机游动，并在时间上运行$O\big(\frac{\log(ε^{-1})\log n}{ε^3\log\log(ε^{-1})}\big)$，假设执行一个随机游动步骤，并从一个有界支持的分布采样需要恒定的时间。热核pagerank得到的顶点的定量排名可以用于局部聚类算法。我们提出了一个有效的本地聚类算法，发现通过执行一个热内核pagerank向量扫描削减，使用热内核pagerank近似算法作为一个子程序。具体而言，我们表明，对于一个子集$S$的Cheeger比$ϕ$，许多顶点$S$可以作为种子的热内核pagerank向量，将找到一个削减的电导$O(\sqrtϕ)$。
英文摘要：
Heat kernel pagerank is a variation of Personalized PageRank given in an exponential formulation. In this work, we present a sublinear time algorithm for approximating the heat kernel pagerank of a graph. The algorithm works by simulating random walks of bounded length and runs in time $O\big(\frac{\log(ε^{-1})\log n}{ε^3\log\log(ε^{-1})}\big)$, assuming performing a random walk step and sampling from a distribution with bounded support take constant time. The quantitative ranking of vertices obtained with heat kernel pagerank can be used for local clustering algorithms. We present an efficient local clustering algorithm that finds cuts by performing a sweep over a heat kernel pagerank vector, using the heat kernel pagerank approximation algorithm as a subroutine. Specifically, we show that for a subset $S$ of Cheeger ratio $ϕ$, many vertices in $S$ may serve as seeds for a heat kernel pagerank vector which will find a cut of conductance $O(\sqrtϕ)$.
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