Publications

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Network science is a powerful tool for analyzing complex systems in fields ranging from sociology to engineering to biology. This article is focused on generative models of large-scale bipartite graphs, also known as two-way graphs or two-mode networks. We propose two generative models that can be easily tuned to reproduce the characteristics of real-world networks, not just qualitatively but quantitatively. The characteristics we consider are the degree distributions and the metamorphosis coefficient. The metamorphosis coefficient, a bipartite analogue of the clustering coefficient, is the proportion of length-three paths that participate in length-four cycles. Having a high metamorphosis coefficient is a necessary condition for close-knit community structure. We define edge, node and degreewise metamorphosis coefficients, enabling a more detailed understanding of the bipartite connectivity that is not explained by degree distribution alone. Our first model, bipartite Chung-Lu, is able to reproduce real-world degree distributions, and our second model, bipartite block two-level Erdös-Rényi, reproduces both the degree distributions as well as the degreewise metamorphosis coefficients. We demonstrate the effectiveness of these models on several real-world data sets.

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The study of triangles in graphs is a standard tool in network analysis, leading to measures such as the transitivity, i.e., the fraction of paths of length two that participate in triangles. Real-world networks are often directed, and it can be difficult to meaningfully understand this network structure. We propose a collection of directed closure values for measuring triangles in directed graphs in a way that is analogous to transitivity in an undirected graph. Our study of these values reveals much information about directed triadic closure. For instance, we immediately see that reciprocal edges have a high propensity to participate in triangles. We also observe striking similarities between the triadic closure patterns of different web and social networks. We perform mathematical and empirical analysis showing that directed configuration models that preserve reciprocity cannot capture the triadic closure patterns of real networks.

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As parallel computing trends towards the exascale, scientific data produced by high-fidelity simulations are growing increasingly massive. For instance, a simulation on a three-dimensional spatial grid with 512 points per dimension that tracks 64 variables per grid point for 128 time steps yields 8 TB of data, assuming double precision. By viewing the data as a dense five-way tensor, we can compute a Tucker decomposition to find inherent low-dimensional multilinear structure, achieving compression ratios of up to 5000 on real-world data sets with negligible loss in accuracy. So that we can operate on such massive data, we present the first-ever distributed-memory parallel implementation for the Tucker decomposition, whose key computations correspond to parallel linear algebra operations, albeit with nonstandard data layouts. Our approach specifies a data distribution for tensors that avoids any tensor data redistribution, either locally or in parallel. We provide accompanying analysis of the computation and communication costs of the algorithms. To demonstrate the compression and accuracy of the method, we apply our approach to real-world data sets from combustion science simulations. We also provide detailed performance results, including parallel performance in both weak and strong scaling experiments.

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Given two sets of vectors, A = {a1→,... , am→} and B = {b1→,... , bn→}, our problem is to find the top-t dot products, i.e., the largest |ai→ · bj→| among all possible pairs. This is a fundamental mathematical problem that appears in numerous data applications involving similarity search, link prediction, and collaborative filtering. We propose a sampling-based approach that avoids direct computation of all mn dot products. We select diamonds (i.e., four-cycles) from the weighted tripartite representation of A and B. The probability of selecting a diamond corresponding to pair (i, j) is proportional to (ai→ · bj→)2, amplifying the focus on the largest-magnitude entries. Experimental results indicate that diamond sampling is orders of magnitude faster than direct computation and requires far fewer samples than any competing approach. We also apply diamond sampling to the special case of maximum inner product search, and get significantly better results than the state-of-theart hashing methods.

Network alignment is an important tool with extensive applications in comparative interactomics. Traditional approaches aim to simultaneously maximize the number of conserved edges and the underlying similarity of aligned entities. We propose a novel formulation of the network alignment problem that extends topological similarity to higher-order structures and provide a new objective function that maximizes the number of aligned substructures. This objective function corresponds to an integer programming problem, which is NP-hard. Consequently, we approximate this objective function as a surrogate function whose maximization results in a tensor eigenvalue problem. Based on this formulation, we present an algorithm called Triangular AlignMEnt (TAME), which attempts to maximize the number of aligned triangles across networks. We focus on alignment of triangles because of their enrichment in complex networks; however, our formulation and resulting algorithms can be applied to general motifs. Using a case study on the NAPABench dataset, we show that TAME is capable of producing alignments with up to 99% accuracy in terms of aligned nodes. We further evaluate our method by aligning yeast and human interactomes. Our results indicate that TAME outperforms the state-of-art alignment methods both in terms of biological and topological quality of the alignments.

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Tensor factorizations with nonnegativity constraints have found application in analysing data from cyber traffic, social networks, and other areas. We consider application data best described as being generated by a Poisson process (e.g. count data), which leads to sparse tensors that can be modelled by sparse factor matrices. In this paper, we investigate efficient techniques for computing an appropriate canonical polyadic tensor factorization based on the Kullback-Leibler divergence function. We propose novel subproblem solvers within the standard alternating block variable approach. Our new methods exploit structure and reformulate the optimization problem as small independent subproblems. We employ bound-constrained Newton and quasi-Newton methods. We compare our algorithms against other codes, demonstrating superior speed for high accuracy results and the ability to quickly find sparse solutions.

Large-scale datasets in computational chemistry typically require distributed-memory parallel methods to perform a special operation known as tensor contraction. Tensors are multidimensional arrays, and a tensor contraction is akin to matrix multiplication with special types of permutations. Creating an efficient algorithm and optimized im- plementation in this domain is complex, tedious, and error-prone. To address this, we develop a notation to express data distributions so that we can apply use automated methods to find optimized implementations for tensor contractions. We consider the spin-adapted coupled cluster singles and doubles method from computational chemistry and use our methodology to produce an efficient implementation. Experiments per- formed on the IBM Blue Gene/Q and Cray XC30 demonstrate impact both improved performance and reduced memory consumption.

We consider the problem of decomposing a real-valued symmetric tensor as the sum of outer products of real-valued vectors. Algebraic methods exist for computing complex-valued decompositions of symmetric tensors, but here we focus on real-valued decompositions, both unconstrained and nonnegative, for problems with low-rank structure. We discuss when solutions exist and how to formulate the mathematical program. Numerical results show the properties of the proposed formulations (including one that ignores symmetry) on a set of test problems and illustrate that these straightforward formulations can be effective even though the problem is nonconvex.

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