Given an undirected graph $G=(V, E)$, the Personalized PageRank (PPR) of $t\in V$ with respect to $s\in V$, denoted $π(s,t)$, is the probability that an $α$-discounted random walk starting at $s$ terminates at $t$. We study the time complexity of estimating $π(s,t)$ with constant relative error and constant failure probability, whenever $π(s,t)$ is above a given threshold parameter $δ\in(0,1)$. We consider common graph-access models and furthermore study the single source, single target, and single node (PageRank centrality) variants of the problem. We provide a complete characterization of PPR estimation in undirected graphs by giving tight bounds (up to logarithmic factors) for all problems and model variants in both the worst-case and average-case setting. This includes both new upper and lower bounds. Tight bounds were recently obtained by Bertram, Jensen, Thorup, Wang, and Yan for directed graphs. However, their lower bound constructions rely on asymmetry and therefore do not carry over to undirected graphs. At the same time, undirected graphs exhibit additional structure that can be exploited algorithmically. Our results resolve the undirected case by developing new techniques that capture both aspects, yielding tight bounds.

翻译：给定一个无向图$G=(V, E)$，节点$t\in V$相对于节点$s\in V$的个性化PageRank（PPR），记作$π(s,t)$，是指一个从$s$出发、以$α$为折扣因子的随机游走终止于$t$的概率。我们研究在$π(s,t)$高于给定阈值参数$δ\in(0,1)$时，以恒定相对误差和恒定失败概率估计$π(s,t)$的时间复杂度。我们考虑了常见的图访问模型，并进一步研究了问题的单源、单目标及单节点（PageRank中心性）变体。通过为所有问题及模型变体在最坏情况和平均情况下给出（对数因子内的）紧确界，我们提供了无向图中PPR估计的完整刻画。这包括新的上界和下界结果。Bertram、Jensen、Thorup、Wang和Yan最近在有向图中获得了紧确界。然而，他们的下界构造依赖于图的有向性，因此无法直接推广到无向图。同时，无向图展现出额外的结构特性，可在算法层面加以利用。我们的研究通过发展能够同时捕捉这两方面的新技术，解决了无向图情形，从而得到了紧确界。