Manycore architectures are a promising direction for single-chip performance. They typically use in-order cores and caches as building blocks and can produce good performance on regular applications. However, on irregular applications, they have low core utilization due to data-dependent control-flow and memory access.

Traditional supercomputers have focused on dense computation performance as exemplified by HPL. Graph processing applications differ with extreme irregularity ($10^{9}$imbalance in skewed, real-world graphs) that produces unpredictable work, parallelism, memory access, and communication. Together, these make scalable performance and programming difficult. We describe the UpDown system architecture, co-designed for irregular graph computations. UpDown provides efficient fine-grained thread invocations ($\sim$10 instructions), direct messaging (no network interface card) for scalable local and global messaging, and split-transaction memory operations that enable extremely high memory bandwidth. Combined with architectural support for global addressing and an aggressive network design, these UpDown features enable direct exploitation of edge and vertex parallelism, using it to deliver breakthrough graph processing performance and programmability. We evaluate the performance of the UpDown system using a challenging suite of graph applications (Pagerank, Breadth-first Search, Triangle Counting, Partial Match, etc). For a single-node, results show 100-fold performance advantage over multicore CPUs. Compared to today's fastest scalable parallel computers UpDown achieves 1000-fold performance increases. UpDown delivers these levels of performance with high-level programmability, these programs directly express vertex-edge parallelism which UpDown exploits directly in hardware.

Dense subgraph discovery is a fundamental primitive in graph and hypergraph analysis which among other applications has been used for real-time story detection on social media and improving access to data stores of social networking systems. We present several contributions for localized densest subgraph discovery, which seeks dense subgraphs located nearby given seed sets of nodes. We first introduce a generalization of a recent anchored densest subgraph problem, extending this previous objective to hypergraphs and also adding a tunable locality parameter that controls the extent to which the output set overlaps with seed nodes. Our primary technical contribution is to prove when it is possible to obtain a strongly-local algorithm for solving this problem, meaning that the runtime depends only on the size of the input set. We provide a strongly-local algorithm that applies whenever the locality parameter is not too small, and show via counterexample why strongly-local algorithms are impossible below a certain threshold. Along the way to proving our results for localized densest subgraph discovery, we also provide several advances in solving global dense subgraph discovery objectives. This includes the first strongly polynomial time algorithm for the densest supermodular set problem and a flow-based exact algorithm for a heavy and dense subgraph discovery problem in graphs with arbitrary node weights. We demonstrate our algorithms on several web-based data analysis tasks.

We study a new connection between a technical measure called $\mu$-conductance that arises in the study of Markov chains for sampling convex bodies and the network community profile that characterizes size-resolved properties of clusters and communities in social and information networks. The idea of $\mu$-conductance is similar to the traditional graph conductance, but disregards sets with small volume. We derive a sequence of optimization problems including a low-rank semi-definite program from which we can derive a lower bound on the optimal $\mu$-conductance value. These ideas give the first theoretically sound bound on the behavior of the network community profile for a wide range of cluster sizes. The algorithm scales up to graphs with hundreds of thousands of nodes and we demonstrate how our framework validates the predicted structures of real-world graphs.

We focus on the problem of aligning graphs that have a spatial basis. In such graphs, which we refer to asrigid graphs, nodes have preferred positions relative to their graph neighbors. Rigid graphs can be used to abstract objects in diverse applications such as large biomolecules, where edges corresponding to chemical bonds have preferred lengths, functional connectomes of the human brain, where edges corresponding to co-firing regions of the brain have preferred anatomical distances, and mobile device/ sensor communication logs, where edges corresponding to point-to-point communications across devices have distance constraints. Effective analysis of such graphs must account for edge lengths in addition to topological features. For instance, when identifying conserved patterns through graph alignment, it is important for matched edges to have correlated lengths, in addition to topological similarity. In this paper, we formulate the problem ofrigid graph alignmentand present a method for solving it. Our formulation of rigid graph alignment simultaneously aligns the topology of the input graphs, as well as the geometric structure represented by the edge lengths, which is solved using a block coordinate descent technique. Using detailed experiments on real and synthetic datasets, we demonstrate a number of important desirable features of our method: (i) it significantly outperforms topological and structural aligners on a wide range of problems; (ii) it scales to problems in important real-world applications; and (iii) it has excellent stability properties, in view of noise and missing data in typical applications.

Hypergraph-based machine learning methods are now widely recognized as important for modeling and using higher-order and multiway relationships between data objects. Local hypergraph clustering and semi-supervised learning specifically involve finding a well-connected set of nodes near a given set of labeled vertices. Although many methods for local graph clustering exist, there are relatively few for localized clustering in hypergraphs. Moreover, those that exist often lack flexibility to model a general class of hypergraph cut functions or cannot scale to large problems. To tackle these issues, this paper proposes a new diffusion-based hypergraph clustering algorithm that solves a quadratic hypergraph cut based objective akin to a hypergraph analog of Andersen-Chung-Lang personalized PageRank clustering for graphs. We prove that, for graphs with fixed maximum hyperedge size, this method is strongly local, meaning that its runtime only depends on the size of the output instead of the size of the hypergraph and is highly scalable. Moreover, our method enables us to compute with a wide variety of cardinality-based hypergraph cut functions. We also prove that the clusters found by solving the new objective function satisfy a Cheeger-like quality guarantee. We demonstrate that on large real-world hypergraphs our new method finds better clusters and runs much faster than existing approaches. Specifically, it runs in a few seconds for hypergraphs with a few million hyperedges compared with minutes for a flow-based technique. We furthermore show that our framework is general enough that can also be used to solve other p-norm based cut objectives on hypergraphs.

Motivated by applications in community detection and dense subgraph discovery, we consider new clustering objectives in hypergraphs and bipartite graphs. These objectives are parameterized by one or more resolution parameters in order to enable diverse knowledge discovery in complex data.

Resolution parameters in graph clustering control the size and structure of clusters formed by solving a parametric objective function. Typically there is more than one meaningful way to cluster a graph, and solving the same objective function for different resolution parameters produces clusterings at different levels of granularity, each of which can be meaningful depending on the application. In this paper, we address the task of efficiently solving a parameterized graph clustering objective for all values of a resolution parameter. Specifically, we consider a new analysis-friendly objective we call LambdaPrime, involving a parameter λ ∈ (0,1). LambdaPrime is an adaptation of LambdaCC, a significant family of instances of the Correlation Clustering (minimization) problem. Indeed, LambdaPrime and LambdaCC are closely related to other parameterized clustering problems, such as parametric generalizations of modularity. They capture a number of specific clustering problems as special cases, including sparsest cut and cluster deletion. While previous work provides approximation results for a single value of the resolution parameter, we seek a set of approximately optimal clusterings for all values of λ in polynomial time. More specifically, we show that when a graph has m edges and n nodes, there exists a set of at most m clusterings such that, for every λ ∈ (0,1), the family contains an optimal solution to the LambdaPrime objective. This bound is tight on star graphs. We obtain a family of O(log n) clusterings by solving the parametric linear programming (LP) relaxation of LambdaPrime at O(log n) λ values, and rounding each LP solution using existing approximation algorithms. We prove that this is asymptotically tight: for a certain class of ring graphs, for all values of λ, Ω(log n) feasible solutions are required to provide a constant-factor approximation for the LambdaPrime LP relaxation. To minimize the size of the clustering family, we further propose an algorithm that yields a family of solutions of a size no more than twice of the minimum LP-approximating family.

Graph based semi-supervised learning is the problem of learning a labeling function for the graph nodes given a few example nodes, often called seeds, usually under the assumption that the graph’s edges indicate similarity of labels. This is closely related to the local graph clustering or community detection problem of finding a cluster or community of nodes around a given seed. For this problem, we propose a novel generalization of random walk, diffusion, or smooth function methods in the literature to a convex p-norm cut function. The need for our p-norm methods is that, in our study of existing methods, we find those principled methods based on eigenvector, spectral, random walk, or linear system often have difficulty capturing the correct boundary of a target label or target cluster. In contrast, 1-norm or maxflow-mincut based methods capture the boundary, but cannot grow from small seed set; hybrid procedures that use both have many hard to set parameters. In this paper, we propose a generalization of the objective function behind these methods involving p-norms. To solve the p-norm cut problem we give a strongly local algorithm -- one whose runtime depends on the size of the output rather than the size of the graph. Our method can be thought as a nonlinear generalization of the Anderson-Chung-Lang push procedure to approximate a personalized PageRank vector efficiently. Our procedure is general and can solve other types of nonlinear objective functions, such as p-norm variants of Huber losses. We provide a theoretical analysis of finding planted target clusters with our method and show that the p-norm cut functions improve on the standard Cheeger inequalities for random walk and spectral methods. Finally, we demonstrate the speed and accuracy of our new method in synthetic and real world datasets.

In the simplest setting, graph visualization is the problem of producing a set of two-dimensional coordinates for each node that meaningfully shows connections and latent structure in a graph. Among other uses, having a meaningful layout is often useful to help interpret the results from network science tasks such as community detection and link prediction. There are several existing graph visualization techniques in the literature that are based on spectral methods, graph embeddings, or optimizing graph distances. Despite the large number of methods, it is still often challenging or extremely time consuming to produce meaningful layouts of graphs with hundreds of thousands of vertices. Existing methods often either fail to produce a visualization in a meaningful time window, or produce a layout colorfully called a “hairball”, which does not illustrate any internal structure in the graph. Here, we show that adding higher-order information based on cliques to a classic eigenvector based graph visualization technique enables it to produce meaningful plots of large graphs. We further evaluate these visualizations along a number of graph visualization metrics and we find that it outperforms existing techniques on a metric that uses random walks to measure the local structure. Finally, we show many examples of how our algorithm successfully produces layouts of large networks. Code to reproduce our results is available.

Link prediction is a common problem in network science that transects many disciplines. The goal is to forecast the appearance of new links or to find links missing in the network. Typical methods for link prediction use the topology of the network to predict the most likely future or missing connections between a pair of nodes. However, network evolution is often mediated by higher-order structures involving more than pairs of nodes; for example, cliques on three nodes (also called triangles) are key to the structure of social networks, but the standard link prediction framework does not directly predict these structures. To address this gap, we propose a new link prediction task called "pairwise link prediction" that directly targets the prediction of new triangles, where one is tasked with finding which nodes are most likely to form a triangle with a given edge. We develop two PageRank-based methods for our pairwise link prediction problem and make natural extensions to existing link prediction methods. Our experiments on a variety of networks show that diffusion based methods are less sensitive to the type of graphs used and more consistent in their results. We also show how our pairwise link prediction framework can be used to get better predictions within the context of standard link prediction evaluation.

Flow-based methods for local graph clustering have received significant recent attention for their theoretical cut improvement and runtime guarantees. In this work we present two improvements for using flow-based methods in real-world semi-supervised clustering problems. Our first contribution is a generalized objective function that allows practitioners to place strict and soft penalties on excluding specific seed nodes from the output set. This feature allows us to avoid the tendency, often exhibited by previous flow-based methods, to contract a large seed set into a small set of nodes that does not contain all or even most of the seed nodes. Our second contribution is a fast algorithm for minimizing our generalized objective function, based on a variant of the push-relabel algorithm for computing preflows. We make our approach very fast in practice by implementing a global relabeling heuristic and employing a warm-start procedure to quickly solve related cut problems. In practice our algorithm is faster than previous related flow-based methods, and is also more robust in detecting ground truth target regions in a graph thanks to its ability to better incorporate semi-supervised information about target clusters.

Diffusions, such as the heat kernel diffusion and the PageRank vector, and their relatives are widely used graph mining primitives that have been successful in a variety of contexts including community detection and semi-supervised learning. The majority of existing methods and methodology involves linear diffusions, which then yield simple algorithms involving repeated matrix-vector operations. Recent work, however, has shown that sophisticated and complicated techniques based on network embeddings and neural networks can give empirical results superior to those based on linear diffusions. In this paper, we illustrate a class of nonlinear graph diffusions that are competitive with state of the art embedding techniques and outperform classic diffusions. Our new methods enjoy much of the simplicity underlying classic diffusion methods as well. Formally, they are based on nonlinear dynamical systems that can be realized with an implementation akin to applying a nonlinear function after each matrix-vector product in a classic diffusion. This framework also enables us to easily integrate results from multiple data representations in a principled fashion. Furthermore, we have some theoretical relationships that suggest choices of the nonlinear term. We demonstrate the benefits of these techniques on a variety of synthetic and real-world data.

Finding clusters of well-connected nodes in a graph is an extensively studied problem in graph-based data analysis. Because of its many applications, a large number of distinct graph clustering objective functions and algorithms have already been proposed and analyzed. To aid practitioners in determining the best clustering approach to use in different applications, we present new techniques for automatically learning how to set clustering resolution parameters. These parameters control the size and structure of communities that are formed by optimizing a generalized objective function. We begin by formalizing the notion of a parameter fitness function, which measures how well a fixed input clustering approximately solves a generalized clustering objective for a specific resolution parameter value. Under reasonable assumptions, which suit two key graph clustering applications, such a parameter fitness function can be efficiently minimized using a bisection-like method, yielding a resolution parameter that fits well with the example clustering. We view our framework as a type of single-shot hyperparameter tuning, as we are able to learn a good resolution parameter with just a single example. Our general approach can be applied to learn resolution parameters for both local and global graph clustering objectives. We demonstrate its utility in several experiments on real-world data where it is helpful to learn resolution parameters from a given example clustering.

MOTIVATION: Precision medicine is an emerging field with hopes to improve patient treatment and reduce morbidity and mortality. To these ends, computational approaches have predicted associations among genes, chemicals and diseases. Such efforts, however, were often limited to using just some available association types. This lowers prediction coverage and, since prior evidence shows that integrating heterogeneous data is likely beneficial, it may limit accuracy. Therefore, we systematically tested whether using more association types improves prediction. RESULTS: We study multimodal networks linking diseases, genes and chemicals (drugs) by applying three diffusion algorithms and varying information content. Ten-fold cross-validation shows that these networks are internally consistent, both within and across association types. Also, diffusion methods recovered missing edges, even if all the edges from an entire mode of association were removed. This suggests that information is transferable between these association types. As a realistic validation, time-stamped experiments simulated the predictions of future associations based solely on information known prior to a given date. The results show that many future published results are predictable from current associations. Moreover, in most cases, using more association types increases prediction coverage without significantly decreasing sensitivity and specificity. In case studies, literature-supported validation shows that these predictions mimic human-formulated hypotheses. Overall, this study suggests that diffusion over a more comprehensive multimodal network will generate more useful hypotheses of associations among diseases, genes and chemicals, which may guide the development of precision therapies. AVAILABILITY AND IMPLEMENTATION: Code and data are available at https://github.com/LichtargeLab/multimodal-network-diffusion. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Traditional clustering algorithms, such as K-Means, output a clustering that is disjoint and exhaustive, i.e., every single data point is assigned to exactly one cluster. However, in many real-world datasets, clusters can overlap and there are often outliers that do not belong to any cluster. While this is a well-recognized problem, most existing algorithms address either overlap or outlier detection and do not tackle the problem in a unified way. In this paper, we propose an intuitive objective function, which we call the NEO-K-Means (Non-Exhaustive, Overlapping K-Means) objective, that captures the issues of overlap and non-exhaustiveness in a unified manner. Our objective function can be viewed as a reformulation of the traditional K-Means objective, with easy-to-understand parameters that capture the degrees of overlap and non-exhaustiveness. By considering an extension to weighted kernel K-Means, we show that we can also apply our NEO-K-Means idea to overlapping community detection, which is an important task in network analysis. To optimize the NEO-K-Means objective, we develop not only fast iterative algorithms but also more sophisticated algorithms using low-rank semidefinite programming techniques. Our experimental results show that the new objective and algorithms are effective in finding ground-truth clusterings that have varied overlap and non-exhaustiveness; for the case of graphs, we show that our method outperforms state-of-the-art overlapping community detection algorithms.

In this manuscript we present the HyperKron Graph model: an extension of the Kronecker Model, but with a distribution over hyperedges. We prove that we can efficiently generate graphs from this model in time proportional to the number of edges times a small log-factor, and find that in practice the runtime is linear with respect to the number of edges. We illustrate a number of useful features of the HyperKron model including non-trivial clustering and highly skewed degree distributions. Finally, we fit the HyperKron model to real-world networks, and demonstrate the model's flexibility with a complex application of the HyperKron model to networks with coherent feed-forward loops.

We present new results for LambdaCC and MotifCC, two recently introduced variants of the well-studied correlation clustering problem. Both variants are motivated by applications to network analysis and community detection, and have non-trivial approximation algorithms. We first show that the standard linear programming relaxation of LambdaCC has a Theta(log n) integrality gap for a certain choice of the parameter lambda. This sheds light on previous challenges encountered in obtaining parameter-independent approximation results for LambdaCC. We generalize a previous constant-factor algorithm to provide the best results, from the LP-rounding approach, for an extended range of lambda. MotifCC generalizes correlation clustering to the hypergraph setting. In the case of hyperedges of degree 3 with weights satisfying probability constraints, we improve the best approximation factor from 9 to 8. We show that in general our algorithm gives a 4(k-1) approximation when hyperedges have maximum degree k and probability weights. We additionally present approximation results for LambdaCC and MotifCC where we restrict to forming only two clusters.

Network alignment or graph matching is the classic problem of finding matching vertices between two graphs with applications in network de-anonymization and bioinformatics. There exist a wide variety of algorithms for it, but a challenging scenario for all of the algorithms is aligning two networks without any information about which nodes might be good matches. In this case, the vast majority of principled algorithms demand quadratic memory in the size of the graphs. We show that one such method---the recently proposed and theoretically grounded EigenAlign algorithm---admits a novel implementation which requires memory that is linear in the size of the graphs. The key step to this insight is identifying low-rank structure in the node-similarity matrix used by EigenAlign for determining matches. With an exact, closed-form low-rank structure, we then solve a maximum weight bipartite matching problem on that low-rank matrix to produce the matching between the graphs. For this task, we show a new, a-posteriori, approximation bound for a simple algorithm to approximate a maximum weight bipartite matching problem on a low-rank matrix. The combination of our two new methods then enables us to tackle much larger network alignment problems than previously possible and to do so quickly. Problems that take hours with existing methods take only seconds with our new algorithm. We thoroughly validate our low-rank algorithm against the original EigenAlign approach. We also compare a variety of existing algorithms on problems in bioinformatics and social networks. Our approach can also be combined with existing algorithms to improve their performance and speed.

Graph clustering, or community detection, is the task of identifying groups of closely related objects in a large network. In this paper we introduce a new community detection framework called LambdaCC that is based on a specially weighted version of correlation clustering. A key component in our methodology is a clustering resolution parameter, lambda, which implicitly controls the size and structure of clusters formed by our framework. We show that, by increasing this parameter, our objective effectively interpolates between two different strategies in graph clustering: finding a sparse cut and forming dense subgraphs. Our methodology unifies and generalizes a number of other important clustering quality functions including modularity, sparsest cut, and cluster deletion, and places them all within the context of an optimization problem that has been well studied from the perspective of approximation algorithms. Our approach to clustering is particularly relevant in the regime of finding dense clusters, as it leads to a 2-approximation for the cluster deletion problem. We use our approach to cluster several graphs, including large collaboration networks and social networks.

We present efficient coding schemes and distributed implementations of erasure coded linear system solvers. Erasure coded computations belong to the class of algorithmic fault tolerance schemes. They are based on augmenting an input dataset, executing the algorithm on the augmented dataset, and in the event of a fault, recovering the solution from the corresponding augmented solution. This process can be viewed as the computational analog of erasure coded storage schemes. The proposed technique has a number of important benefits: (i) as the hardware platform scales in size and number of faults, our scheme yields increasing improvement in resource utilization, compared to traditional schemes; (ii) the proposed scheme is easy to code - the core algorithms remain the same; and (iii) the general scheme is flexible - accommodating a range of computation and communication tradeoffs. We present new coding schemes for augmenting the input matrix that satisfy the recovery equations of erasure coding with high probability in the event of random failures. These coding schemes also minimize fill (non-zero elements introduced by the coding block), while being amenable to efficient partitioning across processing nodes. We demonstrate experimentally that our scheme adds minimal overhead for fault tolerance, yields excellent parallel efficiency and scalability, and is robust to different fault arrival models.

By studying a large number of real world graphs, we find empirical evidence that most real world graphs have a statistically significant power-law distribution with a cutoff in the singular values of the adjacency matrix and eigenvalues of the Laplacian matrix in addition to the commonly conjectured power-law in the degrees. Among these results, power-laws in the singular values appear more consistently than in the degree distribution. The exponents of the power-law distributions are much larger than previously observed. We find a surprising direct relationship between the power-law in the degree distribution and the power-law in the eigenvalues of the Laplacian that was theorized in simple models but is extremely accurate in practice. We investigate these findings in large networks by studying the cutoff value itself, which shows a scaling law for the number of elements involved in these power-laws. Using the scaling law enables us to compute only a subset of eigenvalues of large networks, up to tens of millions of vertices and billions of edges, where we find that those too show evidence of statistically significant power-laws.

Users form information trails as they checkin with a geolocation, rate items, or consume media. A common problem is to predict what a user might do next for the purposes of guidance, recommendation, or prefetching. Markov chains models have been widely used methods to study such sequences of data. First-order Markov chains are easy to estimate, but lack accuracy when history matters. Higher-order Markov chains, in contrast, have too many parameters and suffer from overfitting the training data. Fitting these parameters with regularization and smoothing only offers mild improvements. In this paper we propose the retrospective higher-order Markov process (RHOMP) as a low-parameter model for such sequences. This model is a special case of a higher-order Markov chain where the transitions depend retrospectively on a single history state instead of an arbitrary combination of history states. There are two immediate computational advantages: the number of parameters is linear in the order of the Markov chain and the model can be fit to large state spaces. Furthermore, by providing a specific structure to the higher-order chain, RHOMPs improve the model accuracy by efficiently utilizing history states without risks of overfitting the data. We demonstrate how to estimate a RHOMP from data and we demonstrate the effectiveness of our method on various real application datasets spanning geolocation data, review sequences, and business locations. The RHOMP model uniformly outperforms higher-order Markov chains, Kneser-Ney regularization, and tensor factorizations in terms of prediction accuracy.

Local graph clustering methods aim to find a cluster of nodes by exploring a small region of the graph. These methods are attractive because they enable targeted clustering around a given seed node and are faster than traditional global graph clustering methods because their runtime does not depend on the size of the input graph. However, current local graph partitioning methods are not designed to account for the higher-order structures crucial to the network, nor can they effectively handle directed networks. Here we introduce a new class of local graph clustering methods that address these issues by incorporating higher-order network information captured by small subgraphs, also called network motifs. We develop the Motif-based Approximate Personalized PageRank (MAPPR) algorithm that finds clusters containing a seed node with minimal \emph{motif conductance}, a generalization of the conductance metric for network motifs. We generalize existing theory to prove the fast running time (independent of the size of the graph) and obtain theoretical guarantees on the cluster quality (in terms of motif conductance). We also develop a theory of node neighborhoods for finding sets that have small motif conductance, and apply these results to the case of finding good seed nodes to use as input to the MAPPR algorithm. Experimental validation on community detection tasks in both synthetic and real-world networks, shows that our new framework MAPPR outperforms the current edge-based personalized PageRank methodology.

A multimodal network encodes relationships between the same set of nodes in multiple settings, and network alignment is a powerful tool for transferring information and insight between a pair of networks. We propose a method for multimodal network alignment that computes a matrix which indicates the alignment, but produces the result as a low-rank factorization directly. We then propose new methods to compute approximate maximum weight matchings of low-rank matrices to produce an alignment. We evaluate our approach by applying it on synthetic networks and use it to de-anonymize a multimodal transportation network.

Correlation clustering is a technique for aggregating data based on qualitative information about which pairs of objects are labeled `similar' or `dissimilar.' Because the optimization problem is NP-hard, much of the previous literature focuses on finding approximation algorithms. In this paper we explore how to solve the correlation clustering objective exactly when the data to be clustered can be represented by a low-rank matrix. We prove in particular that correlation clustering can be solved in polynomial time when the underlying matrix is positive semidefinite with small constant rank, but that the task remains NP-hard in the presence of even one negative eigenvalue. Based on our theoretical results, we develop an algorithm for efficiently ``solving'' low-rank positive semidefinite correlation clustering by employing a procedure for zonotope vertex enumeration. We demonstrate the effectiveness and speed of our algorithm by using it to solve several clustering problems on both synthetic and real-world data.

MOTIVATION: Diffusion-based network models are widely used for protein function prediction using protein network data and have been shown to outperform neighborhood-based and module-based methods. Recent studies have shown that integrating the hierarchical structure of the Gene Ontology (GO) data dramatically improves prediction accuracy. However, previous methods usually either used the GO hierarchy to refine the prediction results of multiple classifiers, or flattened the hierarchy into a function-function similarity kernel. No study has taken the GO hierarchy into account together with the protein network as a two-layer network model. RESULTS: We first construct a Bi-relational graph (Birg) model comprised of both protein-protein association and function-function hierarchical networks. We then propose two diffusion-based methods, BirgRank and AptRank, both of which use PageRank to diffuse information on this two-layer graph model. BirgRank is a direct application of traditional PageRank with fixed decay parameters. In contrast, AptRank utilizes an adaptive diffusion mechanism to improve the performance of BirgRank. We evaluate the ability of both methods to predict protein function on yeast, fly and human protein datasets, and compare with four previous methods: GeneMANIA, TMC, ProteinRank and clusDCA. We design four different validation strategies: missing function prediction, de novo function prediction, guided function prediction and newly discovered function prediction to comprehensively evaluate predictability of all six methods. We find that both BirgRank and AptRank outperform the previous methods, especially in missing function prediction when using only 10% of the data for training. AVAILABILITY AND IMPLEMENTATION: The MATLAB code is available at https://github.rcac.purdue.edu/mgribsko/aptrank . CONTACT: [email protected]. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Locally-biased graph algorithms are algorithms that attempt to find local or small-scale structure in a large data graph. In some cases, this can be accomplished by adding some sort of locality constraint and calling a traditional graph algorithm; but more interesting are locally-biased graph algorithms that compute answers by running a procedure that does not even look at most of the input graph. This corresponds more closely to what practitioners from various data science domains do, but it does not correspond well with the way that algorithmic and statistical theory is typically formulated. Recent work from several research communities has focused on developing locally-biased graph algorithms that come with strong complementary algorithmic and statistical theory and that are useful in practice in downstream data science applications. We provide a review and overview of this work, highlighting commonalities between seemingly different approaches, and highlighting promising directions for future work.

Network alignment has 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 provides 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 identify a closely related surrogate function whose maximization results in a tensor eigenvector problem. Based on this formulation, we present an algorithm called Triangular AlignMEnt (TAME), which attempts to maximize the number of aligned triangles across networks. Using a case study on the NAPAbench dataset, we show that triangular alignment is capable of producing mappings with high node correctness. We further evaluate our method by aligning yeast and human interactomes. Our results indicate that TAME outperforms the state-of-art alignment methods in terms of conserved triangles. In addition, we show that the number of conserved triangles is more significantly correlated, compared to the conserved edge, with node correctness and co-expression of edges. Our formulation and resulting algorithms can be easily extended to arbitrary motifs.

Many recently proposed graph-processing frameworks utilize powerful computer clusters with dozens of cores to process massive graphs. Their usability and flexibility often come at a cost. We demonstrate that custom software written for “nanocomputers,” including a credit-card-sized Raspberry Pi, a low-cost ARM server, and an Intel Atom computer, can process the same graphs. Our implementations of PageRank and connected components stream graphs from external storage while performing computation in the limited main memory on these nanocomputers. The results show that a $100 computer with an Intel Atom core can compute PageRank and connected components on a 1.5-billion-edge Twitter graph as quickly as graph-processing systems running on machines with up to 48 cores. As people continue to apply graph computations to large datasets, this research suggests that there may be cost and energy advantages to using nanocomputers.

Many graph-based learning problems can be cast as finding a good set of vertices nearby a seed set, and a powerful methodology for these problems is based on minimum cuts and maximum flows. We introduce and analyze a new method for locally-biased graph-based learning called SimpleLocal, which finds good conductance cuts near a set of seed vertices. An important feature of our algorithm is that it is strongly-local, meaning it does not need to explore the entire graph to find cuts that are locally optimal. This method is related to other strongly-local flow-based methods, but it enables a simple implementation. We also show how it achieves localization through an implicit l1-norm penalty term. As a flow-based method, our algorithm exhibits several advantages in terms of cut optimality and accurate identification of target regions in a graph. We demonstrate the power of SimpleLocal solving segmentation problems on a 467 million edge graph based on an MRI scan.

Collaborative filtering is a popular technique to infer users' preferences on new content based on the collective information of all users preferences. Recommender systems then use this information to make personalized suggestions to users. When users accept these recommendations it creates a feedback loop in the recommender system, and these loops iteratively influence the collaborative filtering algorithm's predictions over time. We investigate whether it is possible to identify items affected by these feedback loops. We state sufficient assumptions to deconvolve the feedback loops while keeping the inverse solution tractable. We furthermore develop a metric to unravel the recommender system's influence on the entire user-item rating matrix. We use this metric on synthetic and real-world datasets to (1) identify the extent to which the recommender system affects the final rating matrix, (2) rank frequently recommended items, and (3) distinguish whether a user's rated item was recommended or an intrinsic preference. Our results indicate that it is possible to recover the ratings matrix of intrinsic user preferences using a single snapshot of the ratings matrix without any temporal information.

Spectral clustering and co-clustering are well-known techniques in data analysis, and recent work has extended spectral clustering to square, symmetric tensors and hypermatrices derived from a network. We develop a new tensor spectral co-clustering method that simultaneously clusters the rows, columns, and slices of a nonnegative three-mode tensor and generalizes to tensors with any number of modes. The algorithm is based on a new random walk model which we call the super-spacey random surfer. We show that our method out-performs state-of-the-art co-clustering methods on several synthetic datasets with ground truth clusters and then use the algorithm to analyze several real-world datasets.

Clustering is one of the most fundamental and important tasks in data mining. Traditional clustering algorithms, such as K-means, assign every data point to exactly one cluster. However, in real-world datasets, the clusters may overlap with each other. Furthermore, often, there are outliers that should not belong to any cluster. We recently proposed the NEO-K-Means (Non-Exhaustive, Overlapping K-Means) objective as a way to address both issues in an integrated fashion. Optimizing this discrete objective is NP-hard, and even though there is a convex relaxation of the objective, straightforward convex optimization approaches are too expensive for large datasets. A practical alternative is to use a low-rank factorization of the solution matrix in the convex formulation. The resulting optimization problem is non-convex, and we can locally optimize the objective function using an augmented Lagrangian method. In this paper, we consider two fast multiplier methods to accelerate the convergence of an augmented Lagrangian scheme: a proximal method of multipliers and an alternating direction method of multipliers (ADMM). For the proximal augmented Lagrangian or proximal method of multipliers, we show a convergence result for the non-convex case with bound-constrained subproblems. These methods are up to 13 times faster—with no change in quality—compared with a standard augmented Lagrangian method on problems with over 10,000 variables and bring runtimes down from over an hour to around 5 minutes.

Community detection is an important task in network analysis. A community (also referred to as a cluster) is a set of cohesive vertices that have more connections inside the set than outside. In many social and information networks, these communities naturally overlap. For instance, in a social network, each vertex in a graph corresponds to an individual who usually participates in multiple communities. In this paper, we propose an efficient overlapping community detection algorithm using a seed expansion approach. The key idea of our algorithm is to find good seeds, and then greedily expand these seeds based on a community metric. Within this seed expansion method, we investigate the problem of how to determine good seed nodes in a graph. In particular, we develop new seeding strategies for a personalized PageRank clustering scheme that optimizes the conductance community score. An important step in our method is the neighborhood inflation step where seeds are modified to represent their entire vertex neighborhood. Experimental results show that our seed expansion algorithm outperforms other state-of-the-art overlapping community detection methods in terms of producing cohesive clusters and identifying ground-truth communities. We also show that our new seeding strategies are better than existing strategies, and are thus effective in finding good overlapping communities in real-world networks.

Graph-based learning methods have a variety of names including semi-supervised and transductive learning. They typically use a diffusion to propagate labels from a small set of nodes with known class labels to the remaining nodes of the graph. While popular, these algorithms, when implemented in a straightforward fashion, are extremely sensitive to the details of the graph construction. Here, we provide four procedures to help make them more robust: recognizing implicit regularization in the diffusion, using a scalable push method to evaluate the diffusion, using rank-based rounding, and densifying the graph through a matrix polynomial. We study robustness with respect to the details of graph constructions, errors in node labeling, degree variability, and a variety of other real-world heterogeneities, studying these methods through a precise relationship with mincut problems. For instance, the densification strategy explicitly adds new weighted edges to a sparse graph. We find that this simple densification creates a graph where multiple diffusion methods are robust to several types of errors. This is demonstrated by a study with predicting product categories from an Amazon co-purchasing network.

Clustering is one of the most fundamental tasks in data mining. To analyze complex real-world data emerging in many data-centric applications, the problem of non-exhaustive, overlapping clustering has been studied where the goal is to find overlapping clusters and also detect outliers simultaneously. We propose a novel convex semidefinite program (SDP) as a relaxation of the non-exhaustive, overlapping clustering problem. Although the SDP formulation enjoys attractive theoretical properties with respect to global optimization, it is computationally intractable for large problem sizes. As an alternative, we optimize a low-rank factorization of the solution. The resulting problem is non-convex, but has a smaller number of solution variables. We construct an optimization solver using an augmented Lagrangian methodology that enables us to deal with problems with tens of thousands of data points. The new solver provides more accurate and reliable answers than other approaches. By exploiting the connection between graph clustering objective functions and a kernel k-means objective, our new low-rank solver can also compute overlapping communities of social networks with state-of-the-art accuracy.

Spectral graph theory-based methods represent an important class of tools for studying the structure of networks. Spectral methods are based on a first-order Markov chain derived from a random walk on the graph and thus they cannot take advantage of important higher-order network substructures such as triangles, cycles, and feed-forward loops. Here we propose a Tensor Spectral Clustering (TSC) algorithm that allows for modeling higher-order network structures in a graph partitioning framework. Our TSC algorithm allows the user to specify which higher-order network structures (cycles, feed-forward loops, etc.) should be preserved by the network clustering. Higher-order network structures of interest are represented using a tensor, which we then partition by developing a multilinear spectral method. Our framework can be applied to discovering layered flows in networks as well as graph anomaly detection, which we illustrate on synthetic networks. In directed networks, a higher-order structure of particular interest is the directed 3-cycle, which captures feedback loops in networks. We demonstrate that our TSC algorithm produces large partitions that cut fewer directed 3-cycles than standard spectral clustering algorithms.

Traditional clustering algorithms, such as k-means, output a clustering that is disjoint and exhaustive, that is, every single data point is assigned to exactly one cluster. However, in real datasets, clusters can overlap and there are often outliers that do not belong to any cluster. This is a well recognized problem that has received much attention in the past, and several algorithms, such as fuzzy k-means have been proposed for overlapping clustering. However, most existing algorithms address either overlap or outlier detection and do not tackle the problem in a unified way. In this paper, we propose a simple and intuitive objective function that captures the issues of overlap and non-exhaustiveness in a unified manner. Our objective function can be viewed as a reformulation of the traditional k-means objective, with easy-to-understand parameters that capture the degrees of overlap and non-exhaustiveness. By studying the objective, we are able to obtain a simple iterative algorithm which we call NEO-K-Means (Non-Exhaustive Overlapping K-Means). Furthermore, by considering an extension to weighted kernel k-means, we can tackle the case of non-exhaustive and overlapping graph clustering. This extension allows us to apply our NEO-K-Means algorithm to the community detection problem, which is an important task in network analysis. Our experimental results show that the new objective and algorithm are effective in finding ground-truth clusterings that have varied overlap and non-exhaustiveness; for the case of graphs, we show that our algorithm outperforms state-of-the-art overlapping community detection methods.

We formalize and illustrate the general concept of algorithmic anti-differentiation: given an algorithmic procedure, e.g., an approximation algorithm for which worst-case approximation guarantees are available or a heuristic that has been engineered to be practically-useful but for which a precise theoretical understanding is lacking, an algorithmic anti-derivative is a precise statement of an optimization problem that is exactly solved by that procedure. We explore this concept with a case study of approximation algorithms for finding locally-biased partitions in data graphs, demonstrating connections between min-cut objectives, a personalized version of the popular PageRank vector, and the highly effective "push" procedure for computing an approximation to personalized PageRank. We show, for example, that this latter algorithm solves (exactly, but implicitly) an l1-regularized l2-regression problem, a fact that helps to explain its excellent performance in practice. We expect that, when available, these implicit optimization problems will be critical for rationalizing and predicting the performance of many approximation algorithms on realistic data.

The heat kernel is a type of graph diffusion that, like the much-used personalized PageRank diffusion, is useful in identifying a community nearby a starting seed node. We present the first deterministic, local algorithm to compute this diffusion and use that algorithm to study the communities that it produces. Our algorithm is formally a relaxation method for solving a linear system to estimate the matrix exponential in a degree-weighted norm. We prove that this algorithm stays localized in a large graph and has a worst-case constant runtime that depends only on the parameters of the diffusion, not the size of the graph. On large graphs, our experiments indicate that the communities produced by this method have better conductance than those produced by PageRank, although they take slightly longer to compute. On a real-world community identification task, the heat kernel communities perform better than those from the PageRank diffusion.

The QR factorization and the SVD are two fundamental matrix decompositions with applications throughout scientific computing and data analysis. For matrices with many more rows than columns, so-called “tall-and-skinny matrices,” there is a numerically stable, efficient, communication-avoiding algorithm for computing the QR factorization. It has been used in traditional high performance computing and grid computing environments. For MapReduce environments, existing methods to compute the QR decomposition use a numerically unstable approach that relies on indirectly computing the Q factor. In the best case, these methods require only two passes over the data. In this paper, we describe how to compute a stable tall-and-skinny QR factorization on a MapReduce architecture in only slightly more than 2 passes over the data. We can compute the SVD with only a small change and no difference in performance. We present a performance comparison between our new direct TSQR method, indirect TSQR methods that use the communication-avoiding TSQR algorithm, and a standard unstable implementation for MapReduce (Cholesky QR). We find that our new stable method is competitive with unstable methods for matrices with a modest number of columns. This holds both in a theoretical performance model as well as in an actual implementation.

Community detection is an important task in network analysis. A community (also referred to as a cluster) is a set of cohesive vertices that have more connections inside the set than outside. In many social and information networks, these communities naturally overlap. For instance, in a social network, each vertex in a graph corresponds to an individual who usually participates in multiple communities. One of the most successful techniques for finding overlapping communities is based on local optimization and expansion of a community metric around a seed set of vertices. In this paper, we propose an efficient overlapping community detection algorithm using a seed set expansion approach. In particular, we develop new seeding strategies for a personalized PageRank scheme that optimizes the conductance community score. The key idea of our algorithm is to find good seeds, and then expand these seed sets using the personalized PageRank clustering procedure. Experimental results show that this seed set expansion approach outperforms other state-of-the-art overlapping community detection methods. We also show that our new seeding strategies are better than previous strategies, and are thus effective in finding good overlapping clusters in a graph.

Network alignment generalizes and unifies several approaches for forming a matching or alignment between the vertices of two graphs. We study a mathematical programming framework for network alignment problem and a sparse variation of it where only a small number of matches between the vertices of the two graphs are possible. We propose a new message passing algorithm that allows us to compute, very efficiently, approximate solutions to the sparse network alignment problems with graph sizes as large as hundreds of thousands of vertices. We also provide extensive simulations comparing our algorithms with two of the best solvers for network alignment problems on two synthetic matching problems, two bioinformatics problems, and three large ontology alignment problems including a multilingual problem with a known labeled alignment.

Our goal is to predict the output of a parameterized computer simulation code given a database of outputs at different parameter values. To do so, we investigate a particular model reduction technique that interpolates the right singular vectors in the singular value decomposition of the matrix of outputs. A common observation about these singular vectors is that they become more oscillatory as the index of the singular vectors increases. We use this property to split the singular vectors into “signal” and “noise” regions. The model reduction then interpolates the “signal” and uses the “noise” to estimate the uncertainty in the result. This methodology requires a big-data approach because the simulations we study produce snapshots with hundreds or thousands of timesteps on thousands to millions of nodal values. Each simulation output is then a vector with millions to billions of values. We utilize a MapReduce-based SVD routine to compute the SVD of the snapshot matrix.

The communities of a social network are sets of vertices with more connections inside the set than outside. We theoretically demonstrate that two commonly observed properties of social networks, heavy-tailed degree distributions and large clustering coefficients, imply the existence of vertex neighborhoods (also known as egonets) that are themselves good communities. We evaluate these neighborhood communities on a range of graphs. What we find is that the neighborhood communities can exhibit conductance scores that are as good as the Fiedler cut. Also, the conductance of neighborhood communities shows similar behavior as the network community profile computed with a personalized PageRank community detection method. Neighborhood communities give us a simple and powerful heuristic for speeding up local partitioning methods. Since finding good seeds for the PageRank clustering method is difficult, most approaches involve an expensive sweep over a great many starting vertices. We show how to use neighborhood communities to quickly generate a small set of seeds.

Network alignment is an optimization problem to find the best one-to-one map between the vertices of a pair of graphs that overlaps as many edges as possible. It is a relaxation of the graph isomorphism problem and is closely related to the subgraph isomorphism problem. The best current approaches are entirely heuristic and iterative in nature. They generate real-valued heuristic weights that must be rounded to find integer solutions. This rounding requires solving a bipartite maximum weight matching problem at each iteration in order to avoid missing high quality solutions. We investigate substituting a parallel, half-approximation for maximum weight matching instead of an exact computation. Our experiments show that the resulting difference in solution quality is negligible. We demonstrate almost a 20-fold speedup using 40 threads on an 8 processor Intel Xeon E7-8870 system and now solve real-world problems in 36 seconds instead of 10 minutes.

Most graph decomposition procedures seek to partition a graph into disjoint sets of vertices. Motivated by applications of clustering in distributed computation, we describe a graph decomposition algorithm for the paradigm where the partitions intersect. This algorithm covers the vertex set with a collection of overlapping clusters. Each vertex in the graph is well-contained within some cluster in the collection. We then describe a framework for distributed computation across a collection of overlapping clusters and describe how this framework can be used in various algorithms based on the graph diffusion process. In particular, we focus on two illustrative examples: (i) the simulation of a randomly walking particle and (ii) the solution of a linear system, e.g. PageRank. Our simulation results for these two cases show a significant reduction in swapping between clusters in a random walk, a significant decrease in communication volume during a linear system solve in a geometric mesh, and some ability to reduce the communication volume during a linear system solve in an information network.

The process of rank aggregation is intimately intertwined with the structure of skew symmetric matrices. We apply recent advances in the theory and algorithms of matrix completion to skew-symmetric matrices. This combination of ideas produces a new method for ranking a set of items. The essence of our idea is that a rank aggregation describes a partially filled skew-symmetric matrix. We extend an algorithm for matrix completion to handle skew-symmetric data and use that to extract ranks for each item.

PageRank computes the importance of each node in a directed graph under a random surfer model governed by a teleportation parameter. Commonly denoted alpha, this parameter models the probability of following an edge inside the graph or, when the graph comes from a network of web pages and links, clicking a link on a web page. We empirically measure the teleportation parameter based on browser toolbar logs and a click trail analysis. For a particular user or machine, such analysis produces a value of alpha. We find that these values nicely fit a Beta distribution with mean edge-following probability between 0.3 and 0.7, depending on the site. Using these distributions, we compute PageRank scores where PageRank is computed with respect to a distribution as the teleportation parameter, rather than a constant teleportation parameter. These new metrics are evaluated on the graph of pages in Wikipedia.

We propose a new distributed algorithm for sparse variants of the network alignment problem, which occurs in a variety of data mining areas including systems biology, database matching, and computer vision. Our algorithm uses a belief propagation heuristic and provides near optimal solutions for this NP-hard combinatorial optimization problem. We show that our algorithm is faster and outperforms or ties existing algorithms on synthetic problems, a problem in bioinformatics, and a problem in ontology matching. We also provide a unified framework for studying and comparing all network alignment solvers.

In this paper, we consider the application of the singular value decomposition (SVD) to a search term suggestion system in a pay-for-performance search market. We propose a positive and negative refinement method based on orthogonal subspace projections. We demonstrate that SVD subspace-based methods: 1) expand coverage by reordering the results, and 2) enhance the clustered structure of the data. The numerical experiments reported in this paper were performed on Overture's pay-per-performance search market data.