Fast Approximate CoSimRanks via Random Projections

TL;DR

This paper introduces sim, a fast randomized algorithm that efficiently approximates all pairwise CoSimRank values in large graphs using random projections, significantly outperforming existing methods in speed.

Contribution

The paper presents a novel randomized algorithm for approximating CoSimRank, reducing computational complexity from cubic to near quadratic time with high accuracy guarantees.

Findings

sim is over orders of magnitude faster than existing algorithms.

On a large Twitter graph, sim computes approximate CoSimRanks within 4 hours.

sim maintains an absolute error of at most psilon with high probability.

Abstract

Given a graph $G$ with $n$ nodes and two nodes $u,v\in G$, the {\em 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 though $\epsilon$ is large. In this paper, we propose \rsim, a fast randomized algorithm for computing all pairwise CoSimRank values. The basic idea of \rsim is to approximate the $n\times n$ matrix multiplications in CoSimRank computation via random projection. Theoretically, \rsim 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 \rsim is more than orders of magnitude faster than the state of the art. In particular, on a million-edge Twitter graph, \rsim answers the $\epsilon$-approximate ($\epsilon=0.1$) all pairwise CoSimRank query within 4 hours, using a single commodity server, while existing solutions fail to terminate within a day.