CLUDE: An Efficient Algorithm for LU Decomposition Over a Sequence of Evolving Graphs

Session: Matrix Factorization, Clustering and Probabilistic DataIn many applications, entities and their relationships are represented by graphs. Examples include the WWW (web pages and hyperlinks) and bibliographic networks (authors and co-authorship). A graph can be conveniently modeled by a matrix from which various quantitative measures are derived. Some example measures include PageRank and SALSA (which measure nodes’ importance), and Personalized PageRank and Random Walk with Restart (which measure proximities between nodes). To compute these measures, linear systems of the form Ax = b, where A is a matrix that captures a graph’s structure, need to be solved. To facilitate solving the linear system, the matrix A is often decomposed into two triangular matrices (L and U). In a dynamic world, the graph that models it changes with time and thus is the matrix A that represents the graph. We consider a sequence of evolving graphs and its associated sequence of evolving matrices. We study how LU-decomposition should be done over the sequence so that (1) the decomposition is efficient and (2) the resulting LU matrices best preserve the sparsity of the matrices A’s (i.e., the number of extra non-zero entries introduced in L and U are minimized.) We propose a cluster-based algorithm CLUDE for solving the problem. Through an experimental study, we show that CLUDE is about an order of magnitude faster than the traditional incremental update algorithm. The number of extra non-zero entries introduced by CLUDE is also about an order of magnitude fewer than that of the traditional algorithm. CLUDE is thus an efficient algorithm for LU decomposition that produces high-quality LU matrices over an evolving matrix sequence.published_or_final_versio

A Stochastic System Model for PageRank: Parameter Estimation and Adaptive Control

A key feature of modern web search engines is the ability to display relevant and reputable pages near the top of the list of query results. The PageRank algorithm provides one way of achieving such a useful hierarchical indexing by assigning a measure of relative importance, called the PageRank value, to each webpage. PageRank is motivated by the inherently hypertextual structure of the World Wide Web; specifically, the idea that pages with more incoming hyperlinks should be considered more popular and that popular pages should rank highly in search results, all other factors being equal. We begin by overviewing the original PageRank algorithm and discussing subsequent developments in the mathematical theory of PageRank. We focus on important contributions to improving the quality of rankings via topic-dependent or "personalized" PageRank, as well as techniques for improving the efficiency of PageRank computation based on Monte Carlo methods, extrapolation and adaptive methods, and aggregation methods We next present a model for PageRank whose dynamics are described by a controlled stochastic system that depends on an unknown parameter. The fact that the value of the parameter is unknown implies that the system is unknown. We establish strong consistency of a least squares estimator for the parameter. Furthermore, motivated by recent work on distributed randomized methods for PageRank computation, we show that the least squares estimator remains strongly consistent within a distributed framework. Finally, we consider the problem of controlling the stochastic system model for PageRank. Under various cost criteria, we use the least squares estimates of the unknown parameter to iteratively construct an adaptive control policy whose performance, according to the long-run average cost, is equivalent to the optimal stationary control that would be used if we had knowledge of the true value of the parameter. This research lays a foundation for future work in a number of areas, including testing the estimation and control procedures on real data or larger scale simulation models, considering more general parameter estimation methods such as weighted least squares, and introducing other types of control policies

Exploiting Web Matrix Permutations to Speedup PageRank Computation

Recently, the research community has devoted an increased attention to reduce the computational time needed by Web ranking algorithms. In particular, we saw many proposals to speed up the well-known PageRank algorithm used by Google. This interest is motivated by two dominant factors: (1) the Web Graph has huge dimensions and it is subject to dramatic updates in term of nodes and links - therefore PageRank assignment tends to became obsolete very soon; (2) many PageRank vectors need to be computed according to different personalization vectors chosen. In the present paper, we address this problem from a numerical point of view. First, we show how to treat dangling nodes in a way which naturally adapts to the random surfer model and preserves the sparsity of the Web Graph. This result allows to consider the PageRank computation as a sparse linear system in alternative to the commonly adopted eigenpairs interpretation. Second, we exploit the Web Matrix reducibility and compose opportunely some Web matrix permutation to speed up the PageRank computation. We tested our approaches on a Web Graphs crawled from the net. The largest one account about 24 millions nodes and more than 100 million links. Upon this Web Graph, the cost for computing the PageRank is reduced of 58% in terms of Mflops and of 89% in terms of time respect to the Power method commonly used

Graphs, Matrices, and the GraphBLAS: Seven Good Reasons

The analysis of graphs has become increasingly important to a wide range of applications. Graph analysis presents a number of unique challenges in the areas of (1) software complexity, (2) data complexity, (3) security, (4) mathematical complexity, (5) theoretical analysis, (6) serial performance, and (7) parallel performance. Implementing graph algorithms using matrix-based approaches provides a number of promising solutions to these challenges. The GraphBLAS standard (istc- bigdata.org/GraphBlas) is being developed to bring the potential of matrix based graph algorithms to the broadest possible audience. The GraphBLAS mathematically defines a core set of matrix-based graph operations that can be used to implement a wide class of graph algorithms in a wide range of programming environments. This paper provides an introduction to the GraphBLAS and describes how the GraphBLAS can be used to address many of the challenges associated with analysis of graphs.Comment: 10 pages; International Conference on Computational Science workshop on the Applications of Matrix Computational Methods in the Analysis of Modern Dat

Site-Based Partitioning and Repartitioning Techniques for Parallel PageRank Computation

Cataloged from PDF version of article.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. © 2011 IEEE