Off-diagonal low-rank preconditioner for difficult PageRank problems

PageRank problem is the cornerstone of Google search engine and is usually stated as solving a huge linear system. Moreover, when the damping factor approaches 1, the spectrum properties of this system deteriorate rapidly and this system becomes difficult to solve. In this paper, we demonstrate that the coefficient matrix of this system can be transferred into a block form by partitioning its rows into special sets. In particular, the off-diagonal part of the block coefficient matrix can be compressed by a simple low-rank factorization, which can be beneficial for solving the PageRank problem. Hence, a matrix partition method is proposed to discover the special sets of rows for supporting the low rank factorization. Then a preconditioner based on the low-rank factorization is proposed for solving difficult PageRank problems. Numerical experiments are presented to support the discussions and to illustrate the effectiveness of the proposed methods. (C) 2018 Elsevier B.V. All rights reserved

Multi-Step Low-Rank Decomposition of Large PageRank Matrices

The PageRank model, initially proposed by Google for search engine rankings, provides a useful network centrality measure to identify the most important nodes within large graphs arising in several applications. However, its computation is often very difficult due to the huge sizes of the networks and the unfavourable spectral properties of the associated matrices. We present a novel multi-step low-rank factorization that can be used to reduce the huge memory cost demanded for realistic PageRank calculations. Finally, we present some directions of future research

Convergence of iterative aggregation/disaggregation methods based on splittings with cyclic iteration matrices

Iterative aggregation/disaggregation methods (IAD) belong to competitive tools for computation the characteristics of Markov chains as shown in some publications devoted to testing and comparing various methods designed to this purpose. According to Dayar T., Stewart W.J., ``Comparison of partitioning techniques for two-level iterative solvers on large, sparse Markov chains,\u27\u27 SIAM J. Sci. Comput., Vol.21, No. 5, 1691-1705 (2000), the IAD methods are effective in particular when applied to large ill posed problems. One of the purposes of this paper is to contribute to a possible explanation of this fact. The novelty may consist of the fact that the IAD algorithms do converge independently of whether the iteration matrix of the corresponding process is primitive or not. Some numerical tests are presented and possible applications mentioned; e.g. computing the PageRank

Solving Directed Laplacian Systems in Nearly-Linear Time through Sparse LU Factorizations

We show how to solve directed Laplacian systems in nearly-linear time. Given a linear system in an $n \times n$ Eulerian directed Laplacian with $m$ nonzero entries, we show how to compute an $\epsilon$-approximate solution in time $O(m \log^{O(1)} (n) \log (1/\epsilon))$. Through reductions from [Cohen et al. FOCS'16] , this gives the first nearly-linear time algorithms for computing $\epsilon$-approximate solutions to row or column diagonally dominant linear systems (including arbitrary directed Laplacians) and computing $\epsilon$-approximations to various properties of random walks on directed graphs, including stationary distributions, personalized PageRank vectors, hitting times, and escape probabilities. These bounds improve upon the recent almost-linear algorithms of [Cohen et al. STOC'17], which gave an algorithm to solve Eulerian Laplacian systems in time $O((m+n2^{O(\sqrt{\log n \log \log n})})\log^{O(1)}(n \epsilon^{-1}))$. To achieve our results, we provide a structural result that we believe is of independent interest. We show that Laplacians of all strongly connected directed graphs have sparse approximate LU-factorizations. That is, for every such directed Laplacian ${\mathbf{L}}$, there is a lower triangular matrix $\boldsymbol{\mathit{{\mathfrak{L}}}}$ and an upper triangular matrix $\boldsymbol{\mathit{{\mathfrak{U}}}}$, each with at most $\tilde{O}(n)$ nonzero entries, such that their product $\boldsymbol{\mathit{{\mathfrak{L}}}} \boldsymbol{\mathit{{\mathfrak{U}}}}$ spectrally approximates ${\mathbf{L}}$ in an appropriate norm. This claim can be viewed as an analogue of recent work on sparse Cholesky factorizations of Laplacians of undirected graphs. We show how to construct such factorizations in nearly-linear time and prove that, once constructed, they yield nearly-linear time algorithms for solving directed Laplacian systems.Comment: Appeared in FOCS 201

Parametric controllability of the personalized PageRank: Classic model vs biplex approach

[EN] Measures of centrality in networks defined by means of matrix algebra, like PageRank-type centralities, have been used for over 70 years. Recently, new extensions of PageRank have been formulated and may include a personalization (or teleportation) vector. It is accepted that one of the key issues for any centrality measure formulation is to what extent someone can control its variability. In this paper, we compare the limits of variability of two centrality measures for complex networks that we call classic PageRank (PR) and biplex approach PageRank (BPR). Both centrality measures depend on the so-called damping parameter alpha that controls the quantity of teleportation. Our first result is that the intersection of the intervals of variation of both centrality measures is always a nonempty set. Our second result is that when alpha is lower that 0.48 (and, therefore, the ranking is highly affected by teleportation effects) then the upper limits of PR are more controllable than the upper limits of BPR; on the contrary, when alpha is greater than 0.5 (and we recall that the usual PageRank algorithm uses the value 0.85), then the upper limits of PR are less controllable than the upper limits of BPR, provided certain mild assumptions on the local structure of the graph. Regarding the lower limits of variability, we give a result for small values of alpha. We illustrate the results with some analytical networks and also with a real Facebook network.This work has been partially supported by the Spanish Ministry of Science, Innovation and Universities under Project Nos. PGC2018-101625-B-I00, MTM2016-76808-P, and MTM2017-84194-P (AEI/FEDER, UE).Flores, J.; García, E.; Pedroche Sánchez, F.; Romance, M. (2020). 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Sparsified Block Elimination for Directed Laplacians

We show that the sparsified block elimination algorithm for solving undirected Laplacian linear systems from [Kyng-Lee-Peng-Sachdeva-Spielman STOC'16] directly works for directed Laplacians. Given access to a sparsification algorithm that, on graphs with $n$ vertices and $m$ edges, takes time $\mathcal{T}_{\rm S}(m)$ to output a sparsifier with $\mathcal{N}_{\rm S}(n)$ edges, our algorithm solves a directed Eulerian system on $n$ vertices and $m$ edges to $\epsilon$ relative accuracy in time $$O(\mathcal{T}_{\rm S}(m) + {\mathcal{N}_{\rm S}(n)\log {n}\log(n/\epsilon)}) + \tilde{O}(\mathcal{T}_{\rm S}(\mathcal{N}_{\rm S}(n)) \log n),$$ where the $\tilde{O}(\cdot)$ notation hides $\log\log(n)$ factors. By previous results, this implies improved runtimes for linear systems in strongly connected directed graphs, PageRank matrices, and asymmetric M-matrices. When combined with slower constructions of smaller Eulerian sparsifiers based on short cycle decompositions, it also gives a solver that runs in $O(n \log^{5}n \log(n / \epsilon))$ time after $O(n^2 \log^{O(1)} n)$ pre-processing. At the core of our analyses are constructions of augmented matrices whose Schur complements encode error matrices