Edge-based Local Push for Personalized PageRank

Abstract

Personalized PageRank (PPR) is a popular node proximity metric in graph mining and network research. Given a graph 𝐺 = (𝑉 , 𝐸) and a source node 𝑠 ∈ 𝑉 , a single-source PPR (SSPPR) query asks for the PPR value 𝝅 (𝑢) with respect to 𝑠, which represents the relative importance of node 𝑢 in the context of the source node 𝑠. SSPPR queries are widely used in recommendation, web mining, and graph representation learning, and their efficient processing has been intensively investigated over the last several decades. Among existing algorithms for SSPPR queries, LocalPush is a fundamental method which serves as a cornerstone for subsequent algorithms. In LocalPush, a push operation is a crucial primitive operation, which distributes the probability at a node 𝑢 to ALL 𝑢's neighbors via the corresponding edges. Although this push operation works well on unweighted graphs, unfortunately, it can be rather inefficient on weighted graphs. In particular, on unbalanced weighted graphs where only a few of these edges take the majority of the total weight among them, the push operation would have to distribute "insignificant" probabilities along those edges which just take the minor weights, resulting in expensive overhead.To resolve this issue, in this paper, we propose the EdgePush algorithm, a novel method for computing SSPPR queries on weighted graphs. EdgePush decomposes the aforementioned push operations in edge-based push, allowing the algorithm to operate at the edge level granularity. Hence, it can flexibly distribute the probabilities according to edge weights. Furthermore, our EdgePush allows a finegrained termination threshold for each individual edge, leading to a superior complexity over LocalPush. Notably, we prove that Edge-Push improves the theoretical query cost of LocalPush by an order of up to 𝑂 (𝑛) when the graph's weights are unbalanced, both in terms of ℓ 1 -error and normalized additive error. Our experimental results demonstrate that EdgePush significantly outperforms state-of-theart baselines in terms of query efficiency on large motif-based and real-world weighted graphs. Additionally, we apply EdgePush to the local clustering problem, where EdgePush is an order of magnitude faster than state-of-the-art baselines while maintaining the same level of clustering quality.