Computing personalized PageRank quickly by exploiting graph structures

Abstract

We propose a new scalable algorithm that can compute Personalized PageRank (PPR) very quickly. The Power method is a state-of-the-art algorithm for computing exact PPR; however, it requires many iterations. Thus reducing the number of iterations is the main challenge.We achieve this by exploiting graph structures of web graphs and social networks. The convergence of our algorithm is very fast. In fact, it requires up to 7.5 times fewer iterations than the Power method and is up to five times faster in actual computation time.To the best of our knowledge, this is the first time to use graph structures explicitly to solve PPR quickly. Our contributions can be summarized as follows.1. We provide an algorithm for computing a tree decomposition, which is more efficient and scalable than any previous algorithm. 2. Using the above algorithm, we can obtain a core-tree decomposition of any web graph and social network. This allows us to decompose a web graph and a social network into (1) the core, which behaves like an expander graph, and (2) a small tree-width graph, which behaves like a tree in an algorithmic sense. 3. We apply a direct method to the small tree-width graph to construct an LU decomposition. 4. Building on the LU decomposition and using it as preconditoner, we apply GMRES method (a state-of-theart advanced iterative method) to compute PPR for whole web graphs and social networks.

TopPPR

“…In addition, considerable efforts [13,27,32,34,37,45] have been made to investigate algorithms for single-source PPR queries. The methods proposed are mostly built upon the power method [34], which is a matrix-based iterative algorithm that can answer singlesource PPR queries with any given threshold ϵ a on the absolute errors of PPR estimations.…”

“…In particular, matrix-based methods [13,15,27,32,37,45] formulate PPR as the solution to a linear system, and they apply matrix optimization approaches to reduce query costs. Local update methods [7, 8, 17-20, 26, 41], on the other hand, utilize graph traversals instead of matrix operations for PPR computation.…”

Reducing seed noise in personalized PageRank

“…Naturally, there is a trade-off between the number of partitions created for the input graph G and the accuracy: the higher the number of partitions, the faster the runtime execution (and smaller the memory requirement), but the higher the drop in accuracy. Recently, [31] proposed a fast Personalized PageRank algorithm: firstly the graph is decomposed into two parts: a core part which behaves like an expander graph and thus making the convergence of an iterative method very fast, and an almost a tree part. Authors suggest to rely on LU decomposition [18], which is a matrix factorization of the form A = LU, where L is a lower triangular matrix with unit diagonals and U is an upper triangular matrix.…”

“…Thanks to the possibility of converting (as discussed in Section 4.3.1) a given seed-set maximal PPR computation task from an expensive optimization problem into a set of linear equations of the form π i = (1−β)T G π i +βs i , the proposed RPR measures can also easily leverage other random-walks with restart approximation techniques, such as [31]: one only needs to replace the computation of π i with the selected approximate PPR technique to obtain an approximation of π i . Most importantly, though, the experimental results show that teleportation-discounting not only increases robustness of PPR scores against noise in the seed set S but also significantly improves robustness against noise introduced due to approximate computation.…”

Locality-sensitive and Re-use Promoting Personalized PageRank computations

“…Path-length based definitions, such as those used by Palmer et al (2006), Boldi et al (2011), Cohen et al (2003), Wei (2010), Xiao et al (2009), Zhou et al (2009) , are 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 useful when the relatedness can be captured solely based on the properties of the nodes and edges on the shortest path (based on some definition of path-length). Randomwalk based definitions, such as hitting distance (Chen et al, 2008;Mei et al, 2008) and personalized PageRank (PPR) score (Balmin et al, 2004;Chakrabarti, 2007;Jeh and Widom, 2002;Tong et al, 2006a;Tong et al, 2007;Liu et al, 2013;Lofgren et al, 2014;Maehara et al, 2014), of node relatedness, on the other hand, also take into account the density of the edges: intuitively, as in path-length based definitions, a node can be said to be more related to another node if there are short paths between them; however, unlike in path-based definitions, random walk-based definitions of relatedness also c...…”