As a measure of vertex importance according to the graph structure, PageRank has been widely applied in various fields. While many PageRank algorithms have been proposed in the past decades, few of them take into account whether the graph under investigation is directed or not. Thus, some important properties of undirected graph\textemdash symmetry on edges, for example\textemdash is ignored. In this paper, we propose a parallel PageRank algorithm specifically designed for undirected graphs that can fully leverage their symmetry. Formally, our algorithm extends the Chebyshev Polynomial approximation from the field of real function to the field of matrix function. Essentially, it reflects the symmetry on edges of undirected graph and the density of diagonalizable matrix. Theoretical analysis indicates that our algorithm has a higher convergence rate and requires less computation than the Power method, with the convergence rate being up to 50\% higher with a damping factor of $c=0.85$. Experiments on six datasets illustrate that our algorithm with 38 parallelism can be up to 39 times faster than the Power method.

翻译：作为衡量图结构中顶点重要性的指标，PageRank已广泛应用于多个领域。尽管过去几十年中提出了许多PageRank算法，但鲜有算法考虑所研究图是否为有向图，因此无向图的重要特性（如边的对称性）往往被忽略。本文提出一种专门针对无向图的并行PageRank算法，该算法能够充分利用无向图的对称性。形式上，本算法将实函数域的切比雪夫多项式逼近扩展至矩阵函数域，其本质体现了无向图边对称性与可对角化矩阵的稠密性。理论分析表明，与幂法相比，本算法具有更高的收敛速度且计算量更小——当阻尼因子c=0.85时，收敛速度可提升高达50%。基于六个数据集的实验表明，本算法在38路并行度下运行速度可比幂法快39倍。