Comparison of Krylov subspace methods on the PageRank problem

Corso, Gianna M. Del; Gullì, Antonio; Romani, Francesco

doi:10.1016/j.cam.2006.10.080

Cited by 18 publications

(28 citation statements)

References 14 publications

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“…We can see that the error curves of the Power iteration and GMRES are completely overlapped, because, as there are too many larger eigenvalues, there is no good polynomial p that shows the rapid convergence of GMRES. Since the computational cost of a single iteration of GMRES is more expensive than that of the Power iteration [18,26], we can conclude that GMRES cannot outperform the Power iteration for non-expander graphs. …”

Section: Iterative Methods Is Fast On Expander Graphsmentioning

confidence: 99%

“…Krylov subspace methods are regarded by numerical analysts as state-of-the-art for solving large sparse linear equations. However, with respect to computing PPR, it has been reported that Krylov methods do not outperform simple Power iteration [18,26] especially for web graphs. In comparison of Krylov subspace methods and Power iterations to compute PPR, Gleich, Zhukov, and Berkhin [26] have stated the following:…”

Section: Basic Algorithmsmentioning

confidence: 99%

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Computing personalized PageRank quickly by exploiting graph structures

et al. 2014

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We propose a new scalable algorithm that can compute Personalized PageRank (PPR) very quickly. The Power method is a state-of-the-art algorithm for computing exact PPR; however, it requires many iterations. Thus reducing the number of iterations is the main challenge.We achieve this by exploiting graph structures of web graphs and social networks. The convergence of our algorithm is very fast. In fact, it requires up to 7.5 times fewer iterations than the Power method and is up to five times faster in actual computation time.To the best of our knowledge, this is the first time to use graph structures explicitly to solve PPR quickly. Our contributions can be summarized as follows.1. We provide an algorithm for computing a tree decomposition, which is more efficient and scalable than any previous algorithm. 2. Using the above algorithm, we can obtain a core-tree decomposition of any web graph and social network. This allows us to decompose a web graph and a social network into (1) the core, which behaves like an expander graph, and (2) a small tree-width graph, which behaves like a tree in an algorithmic sense. 3. We apply a direct method to the small tree-width graph to construct an LU decomposition. 4. Building on the LU decomposition and using it as preconditoner, we apply GMRES method (a state-of-theart advanced iterative method) to compute PPR for whole web graphs and social networks.

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Section: Iterative Methods Is Fast On Expander Graphsmentioning

confidence: 99%

Section: Basic Algorithmsmentioning

confidence: 99%

Computing personalized PageRank quickly by exploiting graph structures

et al. 2014

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“…Gleich, Zhukov, and Berkhin [18] and Del Corso, Gullí, and Romani [13] explored the performance of preconditioned BiCG-STAB on the PageRank system. We have modified the MATLAB implementation of BiCG-STAB to use the 1-norm of the residual as the stopping criterion.…”

Section: Parallel Speedupmentioning

confidence: 99%

“…Gleich, Zhukov, and Berkhin [18] and Del Corso, Gullí, and Romani [13] examined the behavior of Krylov subspace methods on the system…”

Section: Inner-outer Gauss-seidel Iterationsmentioning

confidence: 99%

An Inner-Outer Iteration for Computing PageRank

Gleich¹,

Gray²,

Greif³

et al. 2010

SIAM J. Sci. Comput.

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We present a new iterative scheme for PageRank computation. The algorithm is applied to the linear system formulation of the problem, using inner-outer stationary iterations. It is simple, can be easily implemented and parallelized, and requires minimal storage overhead. Our convergence analysis shows that the algorithm is effective for a crude inner tolerance and is not sensitive to the choice of the parameters involved. The same idea can be used as a preconditioning technique for nonstationary schemes. Numerical examples featuring matrices of dimensions exceeding 100,000,000 in sequential and parallel environments demonstrate the merits of our technique. Our code is available online for viewing and testing, along with several large scale examples.

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“…These techniques mainly aim to increase the convergence rate of the power method [30], which is the de-facto method for PageRank computation with low memory requirement. Among the recently proposed linear system approaches, there are Krylov subspace methods [22], [25], which are applied together with various preconditioners. These methods decrease the number of iterations for convergence at the expense of increased computation per iteration and increased space consumption.…”

mentioning

confidence: 99%

Site-Based Partitioning and Repartitioning Techniques for Parallel PageRank Computation

Cevahir

Aykanat

Türk

et al. 2011

IEEE Trans. Parallel Distrib. Syst.

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Abstract-The PageRank algorithm is an important component in effective web search. At the core of this algorithm are repeated sparse matrix-vector multiplications where the involved web matrices grow in parallel with the growth of the web and are stored in a distributed manner due to space limitations. Hence, the PageRank computation, which is frequently repeated, must be performed in parallel with high-efficiency and low-preprocessing overhead while considering the initial distributed nature of the web matrices. Our contributions in this work are twofold. We first investigate the application of state-of-the-art sparse matrix partitioning models in order to attain high efficiency in parallel PageRank computations with a particular focus on reducing the preprocessing overhead they introduce. For this purpose, we evaluate two different compression schemes on the web matrix using the site information inherently available in links. Second, we consider the more realistic scenario of starting with an initially distributed data and extend our algorithms to cover the repartitioning of such data for efficient PageRank computation. We report performance results using our parallelization of a state-of-the-art PageRank algorithm on two different PC clusters with 40 and 64 processors. Experiments show that the proposed techniques achieve considerably high speedups while incurring a preprocessing overhead of several iterations (for some instances even less than a single iteration) of the underlying sequential PageRank algorithm.

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Comparison of Krylov subspace methods on the PageRank problem

Cited by 18 publications

References 14 publications

Computing personalized PageRank quickly by exploiting graph structures

Computing personalized PageRank quickly by exploiting graph structures

An Inner-Outer Iteration for Computing PageRank

Site-Based Partitioning and Repartitioning Techniques for Parallel PageRank Computation

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