Personalized PageRank Vectors are widely used as fundamental graph-learning tools for detecting anomalous spammers, learning graph embeddings, and training graph neural networks. The well-known local FwdPush algorithm approximates PPVs and has a sublinear rate of $O\big(\frac{1}{\alpha\epsilon}\big)$. A recent study found that when high precision is required, FwdPush is similar to the power iteration method, and its run time is pessimistically bounded by $O\big(\frac{m}{\alpha} \log\frac{1}{\epsilon}\big)$. This paper looks closely at calculating PPVs for both directed and undirected graphs. By leveraging the linear invariant property, we show that FwdPush is a variant of Gauss-Seidel and propose a Successive Over-Relaxation based method, FwdPushSOR to speed it up by slightly modifying FwdPush. Additionally, we prove FwdPush has local linear convergence rate $O\big(\tfrac{\text{vol}(S)}{\alpha} \log\tfrac{1}{\epsilon}\big)$ enjoying advantages of two existing bounds. We also design a new local heuristic push method that reduces the number of operations by 10-50 percent compared to FwdPush. For undirected graphs, we propose two momentum-based acceleration methods that can be expressed as one-line updates and speed up non-acceleration methods by$\mathcal{O}\big(\tfrac{1}{\sqrt{\alpha}}\big)$. Our experiments on six real-world graph datasets confirm the efficiency of FwdPushSOR and the acceleration methods for directed and undirected graphs, respectively.

翻译：个性化PageRank向量被广泛用作基础图学习工具，用于检测异常垃圾邮件发送者、学习图嵌入以及训练图神经网络。著名的局部FwdPush算法能够近似计算PPV，其收敛速度达到亚线性$O\big(\frac{1}{\alpha\epsilon}\big)$。最近一项研究发现，当需要高精度时，FwdPush与幂迭代方法类似，其运行时间悲观情况下有界为$O\big(\frac{m}{\alpha} \log\frac{1}{\epsilon}\big)$。本文仔细研究了有向图和无向图的PPV计算方法。通过利用线性不变性，我们证明FwdPush是高斯-赛德尔方法的一种变体，并提出了一种基于连续超松弛的方法FwdPushSOR，通过轻微修改FwdPush来加速计算。此外，我们证明了FwdPush具有局部线性收敛速度$O\big(\tfrac{\text{vol}(S)}{\alpha} \log\tfrac{1}{\epsilon}\big)$，这同时具备了现有两种界值的优势。我们还设计了一种新的局部启发式推送方法，与FwdPush相比，将操作次数减少了10-50%。对于无向图，我们提出了两种基于动量的加速方法，这些方法可以表达为单行更新，并且将非加速方法的速度提升$\mathcal{O}\big(\tfrac{1}{\sqrt{\alpha}}\big)$。我们在六个真实世界图数据集上的实验分别验证了FwdPushSOR以及针对有向图和无向图的加速方法的有效性。