Unifying the Global and Local Approaches: An Efficient Power Iteration with Forward Push

Personalized PageRank (PPR) is a critical measure on the importance of a node $t$ to a source node $s$ in a graph. A Single-Source PPR (SSPPR) query computes the PPR's of all the nodes with respect to $s$ on a directed graph $G$ with $n$ nodes and $m$ edges, and it is an essential operation widely used in graph applications. In this paper, we propose novel algorithms for solving two variants of SSPPR: (i) high-precision queries and (ii) approximate queries. For the high-precision queries, Power Iteration (PowItr) and Forward Push (FwdPush) are two fundamental approaches. Given an absolute error threshold $\lamda$, the only known bound of FwdPush is $O(\frac{m}{\lamda})$, much worse than the $O(m \log \frac{1}{\lamda})$-bound of PowItr. Whether FwdPush can achieve the same running time bound as PowItr does still remains an open question in the research community. We give a positive answer to this question by showing that the running time of a common implementation of FwdPush is actually bounded by $O(m \cdot \log \frac{1}{\lamda})$. Based on this finding, we propose a new algorithm, called Power Iteration with Forward Push (PowForPush), which incorporates both strengths of PowItr and FwdPush. For approximate queries (with a relative error $\eps$), we propose a new algorithm, called SpeedPPR, with overall expected time bounded by $O(n \cdot \log n \cdot \log \frac{1}{\eps})$ on scale-free graphs. This bound greatly improves the $O(\frac{n \cdot \log n}{\eps})$ bound of a state-of-the-art algorithm FORA. We conduct extensive experiments on six real datasets. The experimental results show that PowForPush outperforms the state-of-the-art algorithm BePi by up to an order of magnitude in both efficiency and accuracy. Furthermore, our SpeedPPR also outperforms FORA by up to an order of magnitude in all aspects includes query time, accuracy, pre-processing time as well as index size.