PageRank is a famous measure of graph centrality that has numerous applications in practice. The problem of computing a single node's PageRank has been the subject of extensive research over a decade. However, existing methods still incur large time complexities despite years of efforts. Even on undirected graphs where several valuable properties held by PageRank scores, the problem of locally approximating the PageRank score of a target node remains a challenging task. Two commonly adopted techniques, Monte-Carlo based random walks and backward push, both cost $O(n)$ time in the worst-case scenario, which hinders existing methods from achieving a sublinear time complexity like $O(\sqrt{m})$ on an undirected graph with $n$ nodes and $m$ edges. In this paper, we focus on the problem of single-node PageRank computation on undirected graphs. We propose a novel algorithm, SetPush, for estimating single-node PageRank specifically on undirected graphs. With non-trival analysis, we prove that our SetPush achieves the $\tilde{O}\left(\min\left\{d_t, \sqrt{m}\right\}\right)$ time complexity for estimating the target node $t$'s PageRank with constant relative error and constant failure probability on undirected graphs. We conduct comprehensive experiments to demonstrate the effectiveness of SetPush.

翻译：摘要：PageRank是一种著名的图中心性度量方法，在实际中具有广泛应用。过去十年来，计算单个节点PageRank值的问题一直是研究的重点。然而，尽管经过多年努力，现有方法仍存在高时间复杂度的困境。即使是在无向图中（PageRank值具有若干有价值的性质），局部近似目标节点PageRank值仍是一项具有挑战性的任务。两种常用技术——基于蒙特卡洛的随机游走和后向推算法——在最坏情况下均需 $O(n)$ 时间，这阻碍了现有方法在具有 $n$ 个节点和 $m$ 条边的无向图上实现 $O(\sqrt{m})$ 级别的次线性时间复杂度。本文聚焦无向图上单节点PageRank计算问题，提出了一种新颖算法SetPush，专门针对无向图进行单节点PageRank估计。通过非平凡分析，我们证明在无向图上，SetPush能以恒定相对误差和恒定失败概率估计目标节点 $t$ 的PageRank值，并实现 $\tilde{O}\left(\min\left\{d_t, \sqrt{m}\right\}\right)$ 时间复杂度。我们通过全面的实验验证了SetPush算法的有效性。