Personalized PageRank on Evolving Graphs with an Incremental Index-Update Scheme

Abstract

\em Personalized PageRank (PPR) stands as a fundamental proximity measure in graph mining. Given an input graph G with the probability of decay α, a source node s and a target node t, the PPR score π(s,t) of target t with respect to source s is the probability that an α-decay random walk starting from s stops at t. A \em single-source PPR (SSPPR) query takes an input graph G with decay probability α and a source s, and then returns the PPR π(s,v) for each node v ∈ V. Since computing an exact SSPPR query answer is prohibitive, most existing solutions turn to approximate queries with guarantees. The state-of-the-art solutions for approximate SSPPR queries are index-based and mainly focus on static graphs, while real-world graphs are usually dynamically changing. However, existing index-update schemes can not achieve a sub-linear update time. Motivated by this, we present an efficient indexing scheme for single-source PPR queries on evolving graphs. Our proposed solution is based on a classic framework that combines the forward-push technique with a random walk index for approximate PPR queries. Thus, our indexing scheme is similar to existing solutions in the sense that we store pre-sampled random walks for efficient query processing. One of our main contributions is an incremental updating scheme to maintain indexed random walks in expected O(1) time after each graph update. To achieve O(1) update cost, we need to maintain auxiliary data structures for both vertices and edges. To reduce the space consumption, we further revisit the sampling methods and propose a new sampling scheme to remove the auxiliary data structure for vertices while still supporting O(1) index update cost on evolving graphs. Extensive experiments show that our update scheme achieves orders of magnitude speed-up on update performance over existing index-based dynamic schemes without sacrificing the query efficiency.