Graphs are commonly used to encode relationships among entities, yet their abstractness makes them difficult to analyze. Node-link diagrams are popular for drawing graphs, and force-directed layouts provide a flexible method for node arrangements that use local relationships in an attempt to reveal the global shape of the graph. However, clutter and overlap of unrelated structures can lead to confusing graph visualizations. This paper leverages the persistent homology features of an undirected graph as derived information for interactive manipulation of force-directed layouts. We first discuss how to efficiently extract 0-dimensional persistent homology features from both weighted and unweighted undirected graphs. We then introduce the interactive persistence barcode used to manipulate the force-directed graph layout. In particular, the user adds and removes contracting and repulsing forces generated by the persistent homology features, eventually selecting the set of persistent homology features that most improve the layout. Finally, we demonstrate the utility of our approach across a variety of synthetic and real datasets.

Projection techniques are often used to visualize high-dimensional data, allowing users to better understand the overall structure of multi-dimensional spaces on a 2D screen. Although many such methods exist, comparably little work has been done on generalizable methods of inverse-projection -the process of mapping the projected points, or more generally, the projection space back to the original high-dimensional space. In this article we present NNInv, a deep learning technique with the ability to approximate the inverse of any projection or mapping. NNInv learns to reconstruct high-dimensional data from any arbitrary point on a 2D projection space, giving users the ability to interact with the learned high-dimensional representation in a visual analytics system. We provide an analysis of the parameter space of NNInv, and offer guidance in selecting these parameters. We extend validation of the effectiveness of NNInv through a series of quantitative and qualitative analyses. We then demonstrate the method's utility by applying it to three visualization tasks: interactive instance interpolation, classifier agreement, and gradient visualization.

A hallmark of visual analytics is its ability to support users in translating broad, open-ended analytic questions (e.g., "is our company succeeding?") into verifiable hypotheses that can be tested from the available data (e.g., "our total revenue increased this past quarter"). However, the process of converting open-ended analytic questions into testable hypotheses is complex and often ill-defined. Beyond high-level descriptions, the visual analytics literature lacks a formalization that can be operationalized for analysis and automation. In this paper, we propose a novel grammar to express hypothesis-based analytic questions for visual analysis. Drawing from prior work in science and education, our grammar defines a formal way to express sets of verifiable hypotheses as a "hypothesis space". Our proposed use of hypothesis spaces contributes a new lens to unify concepts of user goals, the capabilities of a dataset, visual analysis, and testable hypotheses. As a result, we can reformulate abstract classes of visual analysis goals, such as analytic and data-related tasks, in a way that is suitable for analysis and automation. We demonstrate use cases of our grammar in real-world analytic applications including VAST challenges, Kaggle competitions, and pre-existing task taxonomies. Finally, we provide design opportunities in which our grammar can be operationalized to articulate analysis tasks, evaluate visualization systems, and support hypothesis-based reasoning in visual analytic tools.

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