Fast Approximate All Pairwise CoSimRanks via Random Projection

Abstract

Given a graph G with n nodes, and two nodes $u,v\in G$, the CoSimRank value s(u, v) quantifies the similarity between u and v based on graph topology. Compared to SimRank, CoSimRank is shown to be more accurate and effective in many real-world applications including synonym expansion, lexicon extraction, and entity relatedness in knowledge graphs. The computation of all pairwise CoSimRanks in G is highly expensive and challenging. Existing solutions all focus on devising approximate algorithms for the computation of all pairwise CoSimRanks. To attain a desired absolute accuracy guarantee $\epsilon$, the state-of-the-art approximate algorithm for computing all pairwise CoSimRanks requires $O(n^3\log _2(\ln (\frac{1}{\epsilon })))$ time, which is prohibitively expensive even $\epsilon$ is large. In this paper, we propose $\mathtt {RPCS}$, a fast randomized algorithm for computing all pairwise CoSimRank values. The basic idea of $\mathtt {RPCS}$ is to approximate the $n\times n$ matrix multiplications in CoSimRank computation via random projection. Theoretically, $\mathtt {RPCS}$ runs in $O(\frac{n^2\ln (n)}{\epsilon ^2}\ln (\frac{1}{\epsilon }))$ time and meanwhile ensures an absolute error of at most $\epsilon$ in each CoSimRank value in G with a high probability. Extensive experiments using six real graphs demonstrate that $\mathtt {RPCS}$ is more than up to orders of magnitude faster than the state of the art. In particular, on a million-edge Twitter graph, $\mathtt {RPCS}$ answers the $\epsilon$-approximate ($\epsilon =0.1$) all pairwise CoSimRank query within 4 h, using a single commodity server, while existing solutions fail to terminate within a day.