arXiv subjects

Austin R. Benson

Publications and source records attributed to Austin R. Benson.

At least 55 records · Page 3Linked to original sources

arXiv2020

Neural Jump Stochastic Differential Equations

Many time series are effectively generated by a combination of deterministic continuous flows along with discrete jumps sparked by stochastic events. However, we usually do not have the equation of motion describing the flows, or how they are affected by jumps. To this end, we introduce Neural Jump Stochastic Differential Equations that provide a data-driven approach to learn continuous and discrete dynamic behavior, i.e., hybrid systems that both flow and jump. Our approach extends the framework of Neural Ordinary Differential Equations with a stochastic process term that models discrete events. We then model temporal point processes with a piecewise-continuous latent trajectory, where the discontinuities are caused by stochastic events whose conditional intensity depends on the latent state. We demonstrate the predictive capabilities of our model on a range of synthetic and real-world marked point process datasets, including classical point processes (such as Hawkes processes), awards on Stack Overflow, medical records, and earthquake monitoring.

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arXiv2019

Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian

Focusing on coupling between edges, we generalize the relationship between the normalized graph Laplacian and random walks on graphs by devising an appropriate normalization for the Hodge Laplacian -- the generalization of the graph Laplacian for simplicial complexes -- and relate this to a random walk on edges. Importantly, these random walks are intimately connected to the topology of the simplicial complex, just as random walks on graphs are related to the topology of the graph. This serves as a foundational step towards incorporating Laplacian-based analytics for higher-order interactions. We demonstrate how to use these dynamics for data analytics that extract information about the edge-space of a simplicial complex that complements and extends graph-based analysis. Specifically, we use our normalized Hodge Laplacian to derive spectral embeddings for examining trajectory data of ocean drifters near Madagascar and also develop a generalization of personalized PageRank for the edge-space of simplicial complexes to analyze a book co-purchasing dataset.

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arXiv2019

Retrieving Top Weighted Triangles in Graphs

Pattern counting in graphs is a fundamental primitive for many network analysis tasks, and a number of methods have been developed for scaling subgraph counting to large graphs. Many real-world networks carry a natural notion of strength of connection between nodes, which are often modeled by a weighted graph, but existing scalable graph algorithms for pattern mining are designed for unweighted graphs. Here, we develop a suite of deterministic and random sampling algorithms that enable the fast discovery of the 3-cliques (triangles) with the largest weight in a graph, where weight is measured by a generalized mean of a triangle's edges. For example, one of our proposed algorithms can find the top-1000 weighted triangles of a weighted graph with billions of edges in thirty seconds on a commodity server, which is orders of magnitude faster than existing "fast" enumeration schemes. Our methods thus open the door towards scalable pattern mining in weighted graphs.

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arXiv2019

Incrementally Updated Spectral Embeddings

Several fundamental tasks in data science rely on computing an extremal eigenspace of size $r \ll n$, where $n$ is the underlying problem dimension. For example, spectral clustering and PCA both require the computation of the leading $r$-dimensional subspace. Often, this process is repeated over time due to the possible temporal nature of the data; e.g., graphs representing relations in a social network may change over time, and feature vectors may be added, removed or updated in a dataset. Therefore, it is important to efficiently carry out the computations involved to keep up with frequent changes in the underlying data and also to dynamically determine a reasonable size for the subspace of interest. We present a complete computational pipeline for efficiently updating spectral embeddings in a variety of contexts. Our basic approach is to "seed" iterative methods for eigenproblems with the most recent subspace estimate to significantly reduce the computations involved, in contrast with a naïve approach which recomputes the subspace of interest from scratch at every step. In this setting, we provide various bounds on the number of iterations common eigensolvers need to perform in order to update the extremal eigenspace to a sufficient tolerance. We also incorporate a criterion for determining the size of the subspace based on successive eigenvalue ratios. We demonstrate the merits of our approach on the tasks of spectral clustering of temporally evolving graphs and PCA of an incrementally updated data matrix.

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arXiv2019

Pairwise Link Prediction

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.

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arXiv2019

Random Spatial Network Models with Core-Periphery Structure

Core-periphery structure is a common property of complex networks, which is a composition of tightly connected groups of core vertices and sparsely connected periphery vertices. This structure frequently emerges in traffic systems, biology, and social networks via underlying spatial positioning of the vertices. While core-periphery structure is ubiquitous, there have been limited attempts at modeling network data with this structure. Here, we develop a generative, random network model with core-periphery structure that jointly accounts for topological and spatial information by "core scores" of vertices. Our model achieves substantially higher likelihood than existing generative models of core-periphery structure, and we demonstrate how the core scores can be used in downstream data mining tasks, such as predicting airline traffic and classifying fungal networks. We also develop nearly linear time algorithms for learning model parameters and network sampling by using a method akin to the fast multipole method, a technique traditional to computational physics, which allow us to scale to networks with millions of vertices with minor tradeoffs in accuracy.

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arXiv2019

Network Density of States

Spectral analysis connects graph structure to the eigenvalues and eigenvectors of associated matrices. Much of spectral graph theory descends directly from spectral geometry, the study of differentiable manifolds through the spectra of associated differential operators. But the translation from spectral geometry to spectral graph theory has largely focused on results involving only a few extreme eigenvalues and their associated eigenvalues. Unlike in geometry, the study of graphs through the overall distribution of eigenvalues - the spectral density - is largely limited to simple random graph models. The interior of the spectrum of real-world graphs remains largely unexplored, difficult to compute and to interpret. In this paper, we delve into the heart of spectral densities of real-world graphs. We borrow tools developed in condensed matter physics, and add novel adaptations to handle the spectral signatures of common graph motifs. The resulting methods are highly efficient, as we illustrate by computing spectral densities for graphs with over a billion edges on a single compute node. Beyond providing visually compelling fingerprints of graphs, we show how the estimation of spectral densities facilitates the computation of many common centrality measures, and use spectral densities to estimate meaningful information about graph structure that cannot be inferred from the extremal eigenpairs alone.

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arXiv2019

Graph-based Semi-Supervised & Active Learning for Edge Flows

We present a graph-based semi-supervised learning (SSL) method for learning edge flows defined on a graph. Specifically, given flow measurements on a subset of edges, we want to predict the flows on the remaining edges. To this end, we develop a computational framework that imposes certain constraints on the overall flows, such as (approximate) flow conservation. These constraints render our approach different from classical graph-based SSL for vertex labels, which posits that tightly connected nodes share similar labels and leverages the graph structure accordingly to extrapolate from a few vertex labels to the unlabeled vertices. We derive bounds for our method's reconstruction error and demonstrate its strong performance on synthetic and real-world flow networks from transportation, physical infrastructure, and the Web. Furthermore, we provide two active learning algorithms for selecting informative edges on which to measure flow, which has applications for optimal sensor deployment. The first strategy selects edges to minimize the reconstruction error bound and works well on flows that are approximately divergence-free. The second approach clusters the graph and selects bottleneck edges that cross cluster-boundaries, which works well on flows with global trends.

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arXiv2019

Planted Hitting Set Recovery in Hypergraphs

In various application areas, networked data is collected by measuring interactions involving some specific set of core nodes. This results in a network dataset containing the core nodes along with a potentially much larger set of fringe nodes that all have at least one interaction with a core node. In many settings, this type of data arises for structures that are richer than graphs, because they involve the interactions of larger sets; for example, the core nodes might be a set of individuals under surveillance, where we observe the attendees of meetings involving at least one of the core individuals. We model such scenarios using hypergraphs, and we study the problem of core recovery: if we observe the hypergraph but not the labels of core and fringe nodes, can we recover the "planted" set of core nodes in the hypergraph? We provide a theoretical framework for analyzing the recovery of such a set of core nodes and use our theory to develop a practical and scalable algorithm for core recovery. The crux of our analysis and algorithm is that the core nodes are a hitting set of the hypergraph, meaning that every hyperedge has at least one node in the set of core nodes. We demonstrate the efficacy of our algorithm on a number of real-world datasets, outperforming competitive baselines derived from network centrality and core-periphery measures.

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arXiv2019

Three hypergraph eigenvector centralities

Eigenvector centrality is a standard network analysis tool for determining the importance of (or ranking of) entities in a connected system that is represented by a graph. However, many complex systems and datasets have natural multi-way interactions that are more faithfully modeled by a hypergraph. Here we extend the notion of graph eigenvector centrality to uniform hypergraphs. Traditional graph eigenvector centralities are given by a positive eigenvector of the adjacency matrix, which is guaranteed to exist by the Perron-Frobenius theorem under some mild conditions. The natural representation of a hypergraph is a hypermatrix (colloquially, a tensor). Using recently established Perron-Frobenius theory for tensors, we develop three tensor eigenvectors centralities for hypergraphs, each with different interpretations. We show that these centralities can reveal different information on real-world data by analyzing hypergraphs constructed from n-gram frequencies, co-tagging on stack exchange, and drug combinations observed in patient emergency room visits.

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arXiv2019

Computing tensor Z-eigenvectors with dynamical systems

We present a new framework for computing Z-eigenvectors of general tensors based on numerically integrating a dynamical system that can only converge to a Z-eigenvector. Our motivation comes from our recent research on spacey random walks, where the long-term dynamics of a stochastic process are governed by a dynamical system that must converge to a Z-eigenvector of a transition probability tensor. Here, we apply the ideas more broadly to general tensors and find that our method can compute Z-eigenvectors that algebraic methods like the higher-order power method cannot compute.

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arXiv2019

Link Prediction in Networks with Core-Fringe Data

Data collection often involves the partial measurement of a larger system. A common example arises in collecting network data: we often obtain network datasets by recording all of the interactions among a small set of core nodes, so that we end up with a measurement of the network consisting of these core nodes along with a potentially much larger set of fringe nodes that have links to the core. Given the ubiquity of this process for assembling network data, it is crucial to understand the role of such a `core-fringe' structure. Here we study how the inclusion of fringe nodes affects the standard task of network link prediction. One might initially think the inclusion of any additional data is useful, and hence that it should be beneficial to include all fringe nodes that are available. However, we find that this is not true; in fact, there is substantial variability in the value of the fringe nodes for prediction. Once an algorithm is selected, in some datasets, including any additional data from the fringe can actually hurt prediction performance; in other datasets, including some amount of fringe information is useful before prediction performance saturates or even declines; and in further cases, including the entire fringe leads to the best performance. While such variety might seem surprising, we show that these behaviors are exhibited by simple random graph models.

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arXiv2019

Modeling and Analysis of Tagging Networks in Stack Exchange Communities

Large Question-and-Answer (Q&A) platforms support diverse knowledge curation on the Web. While researchers have studied user behavior on the platforms in a variety of contexts, there is relatively little insight into important by-products of user behavior that also encode knowledge. Here, we analyze and model the macroscopic structure of tags applied by users to annotate and catalog questions, using a collection of 168 Stack Exchange websites. We find striking similarity in tagging structure across these Stack Exchange communities, even though each community evolves independently (albeit under similar guidelines). Using our empirical findings, we develop a simple generative model that creates random bipartite graphs of tags and questions. Our model accounts for the tag frequency distribution but does not explicitly account for co-tagging correlations. Even under these constraints, we demonstrate empirically and theoretically that our model can reproduce a number of statistical properties of the co-tagging graph that links tags appearing in the same post.

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arXiv2018

Simplicial Closure and higher-order link prediction

Networks provide a powerful formalism for modeling complex systems by using a model of pairwise interactions. But much of the structure within these systems involves interactions that take place among more than two nodes at once; for example, communication within a group rather than person-to person, collaboration among a team rather than a pair of coauthors, or biological interaction between a set of molecules rather than just two. Such higher-order interactions are ubiquitous, but their empirical study has received limited attention, and little is known about possible organizational principles of such structures. Here we study the temporal evolution of 19 datasets with explicit accounting for higher-order interactions. We show that there is a rich variety of structure in our datasets but datasets from the same system types have consistent patterns of higher-order structure. Furthermore, we find that tie strength and edge density are competing positive indicators of higher-order organization, and these trends are consistent across interactions involving differing numbers of nodes. To systematically further the study of theories for such higher-order structures, we propose higher-order link prediction as a benchmark problem to assess models and algorithms that predict higher-order structure. We find a fundamental differences from traditional pairwise link prediction, with a greater role for local rather than long-range information in predicting the appearance of new interactions.

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arXiv2018

Found Graph Data and Planted Vertex Covers

A typical way in which network data is recorded is to measure all the interactions among a specified set of core nodes; this produces a graph containing this core together with a potentially larger set of fringe nodes that have links to the core. Interactions between pairs of nodes in the fringe, however, are not recorded by this process, and hence not present in the resulting graph data. For example, a phone service provider may only have records of calls in which at least one of the participants is a customer; this can include calls between a customer and a non-customer, but not between pairs of non-customers. Knowledge of which nodes belong to the core is an important piece of metadata that is crucial for interpreting the network dataset. But in many cases, this metadata is not available, either because it has been lost due to difficulties in data provenance, or because the network consists of found data obtained in settings such as counter-surveillance. This leads to a natural algorithmic problem, namely the recovery of the core set. Since the core set forms a vertex cover of the graph, we essentially have a planted vertex cover problem, but with an arbitrary underlying graph. We develop a theoretical framework for analyzing this planted vertex cover problem, based on results in the theory of fixed-parameter tractability, together with algorithms for recovering the core. Our algorithms are fast, simple to implement, and out-perform several methods based on network core-periphery structure on various real-world datasets.

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arXiv2018

Tools for higher-order network analysis

Networks are a fundamental model of complex systems throughout the sciences, and network datasets are typically analyzed through lower-order connectivity patterns described at the level of individual nodes and edges. However, higher-order connectivity patterns captured by small subgraphs, also called network motifs, describe the fundamental structures that control and mediate the behavior of many complex systems. We develop three tools for network analysis that use higher-order connectivity patterns to gain new insights into network datasets: (1) a framework to cluster nodes into modules based on joint participation in network motifs; (2) a generalization of the clustering coefficient measurement to investigate higher-order closure patterns; and (3) a definition of network motifs for temporal networks and fast algorithms for counting them. Using these tools, we analyze data from biology, ecology, economics, neuroscience, online social networks, scientific collaborations, telecommunications, transportation, and the World Wide Web.

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arXiv2018

Higher-order clustering in networks

A fundamental property of complex networks is the tendency for edges to cluster. The extent of the clustering is typically quantified by the clustering coefficient, which is the probability that a length-2 path is closed, i.e., induces a triangle in the network. However, higher-order cliques beyond triangles are crucial to understanding complex networks, and the clustering behavior with respect to such higher-order network structures is not well understood. Here we introduce higher-order clustering coefficients that measure the closure probability of higher-order network cliques and provide a more comprehensive view of how the edges of complex networks cluster. Our higher-order clustering coefficients are a natural generalization of the traditional clustering coefficient. We derive several properties about higher-order clustering coefficients and analyze them under common random graph models. Finally, we use higher-order clustering coefficients to gain new insights into the structure of real-world networks from several domains.

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arXiv2016

Motifs in Temporal Networks

Networks are a fundamental tool for modeling complex systems in a variety of domains including social and communication networks as well as biology and neuroscience. Small subgraph patterns in networks, called network motifs, are crucial to understanding the structure and function of these systems. However, the role of network motifs in temporal networks, which contain many timestamped links between the nodes, is not yet well understood. Here we develop a notion of a temporal network motif as an elementary unit of temporal networks and provide a general methodology for counting such motifs. We define temporal network motifs as induced subgraphs on sequences of temporal edges, design fast algorithms for counting temporal motifs, and prove their runtime complexity. Our fast algorithms achieve up to 56.5x speedup compared to a baseline method. Furthermore, we use our algorithms to count temporal motifs in a variety of networks. Results show that networks from different domains have significantly different motif counts, whereas networks from the same domain tend to have similar motif counts. We also find that different motifs occur at different time scales, which provides further insights into structure and function of temporal networks.

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