PRSim: Sublinear Time SimRank Computation on Large Power-Law Graphs Zhewei Wei, Xiaodong He, Xiaokui Xiao, Sibo Wang, Yu Liu, Xiaoyong Du, and Ji-Rong Wen Contact: zhewei@ruc.edu.cn Problems Our Results: Graph data Assume a directed graph 𝐺=(𝑉,𝐸) with 𝑛 nodes and 𝑚 edges SimRank: Two objects are similar if they are referenced by similar objects, and an object is most similar to itself. High Level ideas: 𝑠(u,v) = 1 (1− 𝑐 ) 2 𝑙=0 ∞ 𝑤∈𝑉 𝜋 𝑙 (𝑢,𝑤) 𝜋 𝑙 (𝑣,𝑤)𝜂(𝑤) Sort adj list according to in-degrees. Reversely sample backward walks Backward Search on hub nodes (with large PageRanks) to build index Applications Web Mining [Jin01] Social Network Analysis [Liben-Nowell07] Spam Detection [Spirin11] Objectives Single-source query: Given a source node u, returns SimRank s(u,v) for every v Top-k query: return v1,…,vk with highest SimRank Allow an error of predetermined ε Motivations Experiments Taxonomy Datasets and methods: Competitors: READS, SLING and TSF, the state-of-the-art index-based methods; ProbeSim and TopSim, the state-of-the-art index-free methods. 𝑐 -walk: at each step, terminates w.p. 1− 𝑐 , and move to a random in-neighbor w.p. 𝑐 SimRank s(u,v) = Pr[two 𝑐 -walks from u and v meet at the same step] Experiments Results on Real-World Graphs Outperforms competitors by at least one order of magnitudes Sampling-based algorithm Motivation 1: Linear Query Time Sublinear query time is not possible on worst-case graphs Can we achieve sub-linearity on real-world graphs? Motivation 2: SimRank v.s. Graph structure Performance of existing SimRank algorithms vary on graphs with similar number of nodes and edges How does graph structure affect SimRank algorithms? PRSim Algorithm Experiments on Synthetic Graphs All query costs reversely depend on power-law exponent 𝛾 Power-Law Graphs Faction of nodes with degree k: P 𝑘 ∼ 𝑘 −𝛾 , 𝛾>𝟏 Codes: www.weizhewei.com