Parallel multisplitting iteration methods based on M-splitting for the PageRank problem

A general multi-splitting iteration method for computing PageRank

“…where A =P T = α P + (1 − α)ve T is the Google matrix with P =P T , and x ≥ 0 is a nonnegative vector satisfying x 1 = 1. Since x 1 = 1, then the PageRank problem (1.1) can be reformulated as the following linear system (Gu et al 2015;Huang and Ma 2015;Gleich et al 2010;Gu and Wang 2013):…”

“…Next, Wen et al (2017) gave the MPIO iteration method, which is a variant of the PIO iteration. By making use of the regular splittings of the matrix I − α P, a class of splitting iteration methods (Huang and Ma 2015) was established for solving (1.2). More numerical algorithms for PageRank problem can be found in Wu et al (2013), Lin et al (2009), Langville and Meyer (2005), Lee et al (2003) and Kamvar et al (2004) and references therein.…”

Parallel two-stage algorithms for solving the PageRank problem

“…Using this formulation, researchers have recently experimented with new PageRank techniques [12,14,15,16] such as inner-outer or multisplitting iterations for solving this linear system. However, the matrix I − αP ′ can be pretty dense, whenever the number of dangling nodes is large because these completely sparse columns are replaced with completely dense columns.…”

“…However, the normalized eigenvector problem can be solved using its linear system formulation opening new research lines [10]. Recently, some approaches based on linear system formulations have been considered, see e.g., [11,12,13,14,15,16] and the references cited therein.…”

The modified matrix splitting iteration method for computing PageRank problem

“…Based on the matrix splitting I − αP = (I − βP) − (α − β)P, where 0 < β < α, an inner-outer iteration method [24] was proposed for solving (1.2). Since the matrix I − αP is a nonsingular M−matrix [2,20], a class of splitting iteration methods were presented in [7], which are obtained from the M-splittings of the matrix I − αP.…”