Several relaxed iteration methods for computing PageRank

https://doi.org/10.1016/j.cam.2020.113295Get rights and content

Abstract

In this paper, based on the iteration framework (Tian et al., 2019) and relaxed two-step splitting (RTSS) iteration method (Xie and Ma, 2018), we present two relaxed iteration methods for solving the PageRank problem, which are the relaxed generalized inner–outer (RGIO) and relaxed generalized two-step splitting (RGTSS) iteration methods, respectively. Next, their overall convergence properties are analyzed in detail, and choices of the parameters in these algorithms are also discussed. Finally, several numerical examples are given to illustrate the effectiveness of the proposed algorithms.

Introduction

With the development of the Internet, web search engines become more and more important in retrieving information. PageRank problem has attracted much attention in the scientific community for last decades. The PageRank problem (1.1) often arises in web ranking [1], [2], [3], [4], for example, the hyperlink structure of the web and modeling the graph by the Markov chain, etc. For the mathematical background of the PageRank problem, readers can refer to [5], [6] for more details.

In fact, the PageRank problem can be solved by computing the principal eigenvector of the Google matrix A [5] defined by Ax=xwith A=αP+(1α)veT, where α(0,1) is the damping factor to determine the weight with respect to the web link graph, PRn×n is a column-stochastic matrix and all its columns sum to one, e is a column vector of all ones, and v represents the personalization or the teleportation vector.

Since the PageRank vector x0 is a nonnegative vector and satisfies x1=1, then the PageRank problem (1.1) can be rewritten as the following linear system [6], [7], [8]: (IαP)x=(1α)v,where IRn×n is an identity matrix.

In the last decades, the iteration methods only requiring the matrix–vector products have been popular for solving the PageRank problem. The power method [5], [9] is the original method to obtain the PageRank vector by solving (1.1). However, the power method often performs poorly for the cases that the largest eigenvalue of A is not well separated from the second one and the damping factor α is close to 1. In [10], Gleich et al. proposed an inner–outer (IO) iteration method for solving (1.2), which is very efficient by solving a linear system with a lower damping factor if the damping factor α is sufficiently close to 1. Recently, by combining the two-step splitting iteration framework with the IO iteration [10], Gu et al. [11] gave the PIO iteration method. Next, Wen et al. [12] presented the MPIO iteration method including PIO iteration as its special case. Since the matrix IαP is a nonsingular Mmatrix [13], [14], a class of splitting iteration methods [15] were developed for solving (1.2). Based on the regular splittings of the matrix IαP, Tian et al. [2] gave a generalized inner–outer (GIO) iteration method. In [16], Tian et al. improved the MSI method [3] and proposed the generalized MSI method. In [17], Tian et al. proposed an MMPIO iteration method for solving the PageRank problem by combining the multi-step matrix splitting iterations instead of the power method with the IO iteration [10]. Many other algorithms are also constructed for computing the PageRank vector, such as the iteration algorithms and their references in [18], [19], [20], [21], [22], [23], [24], [25], etc.

In this paper, by using the iteration framework [26] and the GIO iteration [2], [16], two relaxed iteration methods are developed for solving (1.2). In the following sections, their convergence properties are investigated, and the choices of the parameters in these algorithms are also discussed, respectively. Finally, several numerical examples are used to show the effectiveness of our proposed algorithms.

The remainder of this paper is organized as follows. In Section 2, the RGIO iteration method and its convergence property are established. In Section 3, the RGTSS iteration method and its convergence theorem are given. In Section 4, some comparison results for the RGIO and RGTSS iteration methods are presented. Section 5 is devoted to the analyses of the choices of the parameters in these algorithms. The numerical experiments are provided in Section 6 to verify the efficiencies of our proposed algorithms. Finally, some conclusions are drawn in Section 7.

Section snippets

The GIO iteration method [2,16]

Let IαP=MN,where M is an invertible matrix. The iteration sequence derived from (2.1) for solving (1.2) has the following form: Mxk+1=Nxk+(1α)v

Based on (2.1), Tian et al. proposed the generalized inner–outer iteration method [2], [16], which can be described as follows.

The outer iteration method for solving (1.2) is (MψN)xk+1=(1ψ)Nxk+(1α)v,k=0,1,2,with 0<ψ<1.

Let f=(1ψ)Nxk+(1α)v, then the inner iteration can be defined by Myj+1=ψNyj+f,j=0,1,2,,mk1with y0=xk as the initial guess and ymk

The RGTSS iteration method

By using the following matrix splitting of the matrix IαP: IαP=M3N3,where M3=φIβP, N3=(φ1)I+(αβ)P and φ>0, Xie et al. [26] proposed the following relaxed two-step splitting (RTSS) iteration method.

The RTSS iteration method: xk+12=αPxk+(1α)v,(φIβP)xk+1=(φ1)xk+12+(αβ)Pxk+12+(1α)v.

Algorithm 3: The RTSS iteration method

Input: P,α,β,φ,m̄k,v,τ

Output: x

1: xv

2: z̃Px

3: while αz̃+(1α)vxτ

4:         xαz̃+(1α)v

5:         z̃Px

6:        f̃φ1φx+αβφz̃+1αφv

7:        for i =1: m̄k

Some comparison results for the RGIO and RGTSS iteration methods

In this section, we will give some comparison results for the RGIO and RGTSS iteration methods, respectively.

Definition 4.1

[13]

For a matrix ARn×n, A=MN is a regular splitting if M is nonsingular with M10 and N0.

Theorem 4.1

[13]

Let A=MN be a regular splitting of A. If A10, then ρ(M1N)=ρ(A1N)1+ρ(A1N)<1.

Theorem 4.2

[13]

Let A=M1N1=M2N2 be two regular splittings of A, where A10. If N2N10, then 0ρ(M11N1)ρ(M21N2)<1.

Theorem 4.3

Let IαP=MiNi(i=1,2) be two regular splittings and 0<ψ<1. If ρ(M11N1)<ρ(M21N2) and ϕ>1, then the RGIO iteration

The choices of the parameters in the RGIO and RGTSS iteration methods

In this section, we will investigate the choices of the parameters in the RGIO and RGTSS iteration methods, respectively.

For the parameters m̃k,mˆk, just as our analyses in [2], the inner iterations can achieve better convergence performances with small number of inner iterations, such as 2,3,4, since the RGIO and RGTSS iteration methods may take more computational time with larger number of inner iterations.

Next, we will pay attention to the choices of the parameter ψ,ϕ in the RGIO and RGTSS

Numerical results

In this section, three numerical examples are presented to verify the effectiveness of the RGIO and RGTSS iteration methods. The numerical experiments are implemented in Matlab R2010 on an Intel dual core processor (2.30 GHz, 8 GB RAM). Three parameters are used to test the effectiveness of these algorithms, which are the iteration step (IT), the computing time in seconds (CPU) and the relative residual (RES) defined by rk2(1α)v2with rk=(1α)v(IαP)xk. Six sparse matrices P for the

Conclusion

In this paper, we present two relaxed iteration methods for solving the PageRank problem, which are called the RGIO and RGTSS iteration methods, respectively. Next, we analyze their global convergence properties in detail. The Numerical results on several PageRank problems show the superiority of our proposed algorithms. Since these relaxed iteration methods are parameter-dependent, then how to determine the optimal parameters will be further researched in the future work.

Acknowledgments

The authors sincerely appreciate the anonymous referees for their constructive and valuable comments, which greatly improved the presentation of this paper. The first author acknowledges the partial support of Research Project Supported by Shanxi Scholarship Council of China (2020–098).

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