Multilevel Aggregation Methods for Small-World Graphs with Application to Random-Walk Ranking

We describe multilevel aggregation in the specific context of using Markov chains to rank the nodes of graphs. More generally, aggregation is a graph coarsening technique that has a wide range of possible uses regarding information retrieval applications. Aggregation successfully generates efficient multilevel methods for solving nonsingular linear systems and various eigenproblems from discretized partial differential equations, which tend to involve mesh-like graphs. Our primary goal is to extend the applicability of aggregation to similar problems on small-world graphs, with a secondary goal of developing these methods for eventual applicability towards many other tasks such as using the information in the hierarchies for node clustering or pattern recognition. The nature of small-world graphs makes it difficult for many coarsening approaches to obtain useful hierarchies that have complexity on the order of the number of edges in the original graph while retaining the relevant properties of the original graph. Here, for a set of synthetic graphs with the small-world property, we show how multilevel hierarchies formed with non-overlapping strength-based aggregation have optimal or near optimal complexity. We also provide an example of how these hierarchies are employed to accelerate convergence of methods that calculate the stationary probability vector of large, sparse, irreducible, slowly-mixing Markov chains on such small-world graphs. The stationary probability vector of a Markov chain allows one to rank the nodes in a graph based on the likelihood that a long random walk visits each node. These ranking approaches have a wide range of applications including information retrieval and web ranking, performance modeling of computer and communication systems, analysis of social networks, dependability and security analysis, and analysis of biological systems

An Extension of Two Conjugate Direction Methods to Markov Chain Problems

Motivated by the recent applications of the conjugate residual method to nonsymmetric linear systems by Sogabe, Sugihara and Zhang [An extension of the conjugate residual method to nonsymmetric linear systems. J. Comput. Appl. Math., Vol. 266, 2009, pp. 103--113], this paper describes two conjugate direction methods, BiCR and BiCG, and attempts to extend their applications to compute the stationary probability distribution for an irreducible Markov chain with the aim of finding an alternative basic solver. Numerical experiments show the feasibility of the BiCR and BiCG to some extent, with applications to several practical Markov chain problems

An Efficient Visual Analysis Method for Cluster Tendency Evaluation, Data Partitioning and Internal Cluster Validation

Visual methods have been extensively studied and performed in cluster data analysis. Given a pairwise dissimilarity matrix D of a set of n objects, visual methods such as Enhanced-Visual Assessment Tendency (E-VAT) algorithm generally represent D as an n times n image I( overlineD) where the objects are reordered to expose the hidden cluster structure as dark blocks along the diagonal of the image. A major constraint of such methods is their lack of ability to highlight cluster structure when D contains composite shaped datasets. This paper addresses this limitation by proposing an enhanced visual analysis method for cluster tendency assessment, where D is mapped to D' by graph based analysis and then reordered to overlineD' using E-VAT resulting graph based Enhanced Visual Assessment Tendency (GE-VAT). An Enhanced Dark Block Extraction (E-DBE) for automatic determination of the number of clusters in I( overlineD') is then proposed as well as a visual data partitioning method for cluster formation from I( overlineD') based on the disparity between diagonal and off-diagonal blocks using permuted indices of GE-VAT. Cluster validation measures are also performed to evaluate the cluster formation. Extensive experimental results on several complex synthetic, UCI and large real-world data sets are analyzed to validate our algorithm

A Self-learning Algebraic Multigrid Method for Extremal Singular Triplets and Eigenpairs

A self-learning algebraic multigrid method for dominant and minimal singular triplets and eigenpairs is described. The method consists of two multilevel phases. In the first, multiplicative phase (setup phase), tentative singular triplets are calculated along with a multigrid hierarchy of interpolation operators that approximately fit the tentative singular vectors in a collective and self-learning manner, using multiplicative update formulas. In the second, additive phase (solve phase), the tentative singular triplets are improved up to the desired accuracy by using an additive correction scheme with fixed interpolation operators, combined with a Ritz update. A suitable generalization of the singular value decomposition is formulated that applies to the coarse levels of the multilevel cycles. The proposed algorithm combines and extends two existing multigrid approaches for symmetric positive definite eigenvalue problems to the case of dominant and minimal singular triplets. Numerical tests on model problems from different areas show that the algorithm converges to high accuracy in a modest number of iterations, and is flexible enough to deal with a variety of problems due to its self-learning properties.Comment: 29 page

Distributed Spectral Graph Methods for Analyzing Large-Scale Unstructured Biomedical Data

There is an ever-expanding body of biological data, growing in size and complexity, out- stripping the capabilities of standard database tools or traditional analysis techniques. Such examples include molecular dynamics simulations, drug-target interactions, gene regulatory networks, and high-throughput imaging. Large-scale acquisition and curation biological data has already yielded results in the form of lower costs for genome sequencing and greater cov- erage in databases such as GenBank, and is viewed as the future of biocuration. The “big data” philosophy and its associated paradigms and frameworks have the potential to uncover solutions to problems otherwise intractable with more traditional investigative techniques. Here, we focus on two biological systems whose data form large, undirected graphs. First, we develop a quantitative model of ciliary motion phenotypes, using spectral graph methods for unsupervised latent pattern discovery. Second, we apply similar techniques to identify a mapping between physiochemical structure and odor percept in human olfaction. In both cases, we experienced computational bottlenecks in our statistical machinery, necessitating the creation of a new analysis framework. At the core of this framework is a distributed hierarchical eigensolver, which we compare directly to other popular solvers. We demon- strate its essential role in enabling the discovery of novel ciliary motion phenotypes and in identifying physiochemical-perceptual associations

Laboratory Directed Research and Development FY2010 Annual Report

A premier applied-science laboratory, Lawrence Livermore National Laboratory (LLNL) has at its core a primary national security mission - to ensure the safety, security, and reliability of the nation's nuclear weapons stockpile without nuclear testing, and to prevent and counter the spread and use of weapons of mass destruction: nuclear, chemical, and biological. The Laboratory uses the scientific and engineering expertise and facilities developed for its primary mission to pursue advanced technologies to meet other important national security needs - homeland defense, military operations, and missile defense, for example - that evolve in response to emerging threats. For broader national needs, LLNL executes programs in energy security, climate change and long-term energy needs, environmental assessment and management, bioscience and technology to improve human health, and for breakthroughs in fundamental science and technology. With this multidisciplinary expertise, the Laboratory serves as a science and technology resource to the U.S. government and as a partner with industry and academia. This annual report discusses the following topics: (1) Advanced Sensors and Instrumentation; (2) Biological Sciences; (3) Chemistry; (4) Earth and Space Sciences; (5) Energy Supply and Use; (6) Engineering and Manufacturing Processes; (7) Materials Science and Technology; Mathematics and Computing Science; (8) Nuclear Science and Engineering; and (9) Physics

Laboratory Directed Research and Development FY2010 Annual Report

A premier applied-science laboratory, Lawrence Livermore National Laboratory (LLNL) has at its core a primary national security mission - to ensure the safety, security, and reliability of the nation's nuclear weapons stockpile without nuclear testing, and to prevent and counter the spread and use of weapons of mass destruction: nuclear, chemical, and biological. The Laboratory uses the scientific and engineering expertise and facilities developed for its primary mission to pursue advanced technologies to meet other important national security needs - homeland defense, military operations, and missile defense, for example - that evolve in response to emerging threats. For broader national needs, LLNL executes programs in energy security, climate change and long-term energy needs, environmental assessment and management, bioscience and technology to improve human health, and for breakthroughs in fundamental science and technology. With this multidisciplinary expertise, the Laboratory serves as a science and technology resource to the U.S. government and as a partner with industry and academia. This annual report discusses the following topics: (1) Advanced Sensors and Instrumentation; (2) Biological Sciences; (3) Chemistry; (4) Earth and Space Sciences; (5) Energy Supply and Use; (6) Engineering and Manufacturing Processes; (7) Materials Science and Technology; Mathematics and Computing Science; (8) Nuclear Science and Engineering; and (9) Physics