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Enhanced preprocessed multi-step splitting iterations for computing PageRank

Guangcong Meng, Yuehua Feng, Yongxin Dong

TL;DR

The paper tackles slow convergence in PageRank as the damping factor $\alpha$ nears 1 by introducing a novel multi-step splitting framework (MIIO) and two accelerated variants that fuse this framework with Krylov subspace methods: Arnoldi-MIIO using thick restarted Arnoldi and GArnoldi-MIIO using adaptive Generalized Arnoldi. The authors provide theoretical convergence bounds showing contraction factors that guarantee rapid progression, and they validate the approach with extensive experiments on large, sparse Google matrices, demonstrating substantial speedups in CPU time and reduced matrix-vector product counts compared to state-of-the-art baselines. The work advances practical PageRank computation by delivering faster, scalable algorithms with solid convergence guarantees, enabling efficient ranking on very large graphs. The proposed methods hold potential for broader impact in large-scale Markov chain computations and search engine ranking pipelines where damping is close to 1.

Abstract

In recent years, the PageRank algorithm has garnered significant attention due to its crucial role in search engine technologies and its applications across various scientific fields. It is well-known that the power method is a classical method for computing PageRank. However, there is a pressing demand for alternative approaches that can address its limitations and enhance its efficiency. Specifically, the power method converges very slowly when the damping factor is close to 1. To address this challenge, this paper introduces a new multi-step splitting iteration approach for accelerating PageRank computations. Furthermore, we present two new approaches for computating PageRank, which are modifications of the new multi-step splitting iteration approach, specifically utilizing the thick restarted Arnoldi and generalized Arnoldi methods. We provide detailed discussions on the construction and theoretical convergence results of these two approaches. Extensive experiments using large test matrices demonstrate the significant performance improvements achieved by our proposed algorithms.

Enhanced preprocessed multi-step splitting iterations for computing PageRank

TL;DR

The paper tackles slow convergence in PageRank as the damping factor nears 1 by introducing a novel multi-step splitting framework (MIIO) and two accelerated variants that fuse this framework with Krylov subspace methods: Arnoldi-MIIO using thick restarted Arnoldi and GArnoldi-MIIO using adaptive Generalized Arnoldi. The authors provide theoretical convergence bounds showing contraction factors that guarantee rapid progression, and they validate the approach with extensive experiments on large, sparse Google matrices, demonstrating substantial speedups in CPU time and reduced matrix-vector product counts compared to state-of-the-art baselines. The work advances practical PageRank computation by delivering faster, scalable algorithms with solid convergence guarantees, enabling efficient ranking on very large graphs. The proposed methods hold potential for broader impact in large-scale Markov chain computations and search engine ranking pipelines where damping is close to 1.

Abstract

In recent years, the PageRank algorithm has garnered significant attention due to its crucial role in search engine technologies and its applications across various scientific fields. It is well-known that the power method is a classical method for computing PageRank. However, there is a pressing demand for alternative approaches that can address its limitations and enhance its efficiency. Specifically, the power method converges very slowly when the damping factor is close to 1. To address this challenge, this paper introduces a new multi-step splitting iteration approach for accelerating PageRank computations. Furthermore, we present two new approaches for computating PageRank, which are modifications of the new multi-step splitting iteration approach, specifically utilizing the thick restarted Arnoldi and generalized Arnoldi methods. We provide detailed discussions on the construction and theoretical convergence results of these two approaches. Extensive experiments using large test matrices demonstrate the significant performance improvements achieved by our proposed algorithms.
Paper Structure (11 sections, 8 theorems, 62 equations, 3 figures, 4 tables)

This paper contains 11 sections, 8 theorems, 62 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The IIO iteration for PageRank
  4. The thick restarted Arnoldi algorithm
  5. The Adaptively Accelerated Arnoldi method for computing PageRank.
  6. Proposed Approaches
  7. A new iteration for PageRank
  8. An Arnoldi-MIIO algorithm for computing PageRank
  9. A GArnoldi-MIIO algorithm for computing PageRank
  10. Numerical Experiments
  11. Conclusions

Key Result

Theorem 1

The iteration matrix $M(\alpha, \beta)$ of the MIIO iteration is given by and the modulus of its eigenvalues is bounded by where $\alpha \in (0, 1)$, $\beta \in(0, \alpha)$, $m_1$ and $m_2$ are two multiple iteration parameters. Therefore, it holds that $\rho\left(M\left(\alpha, \beta\right)\right)<1$. In other words, the MIIO iteration converges to the unique solution $x^* \in \mathbb{C}^n$ of

Figures (3)

  • Figure 1: Convergence of the computation for the web-Stanford matrix when $m$ = 8, $p$ = 4, maxit = 10.
  • Figure 2: Convergence of the computation for the Stanford-Berkeley matrix when $m$ = 8, $p$ = 4, maxit = 10.
  • Figure 3: Convergence of the computation for the web-Google matrix when $m$ = 8, $p$ = 4, maxit = 10.

Theorems & Definitions (15)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Theorem 2: Wei
  • Theorem 3: Sylvester inequality HornJohnson
  • Theorem 4
  • proof
  • Remark 3
  • Theorem 5: JIA19971
  • ...and 5 more