Abstract Recent advances in Large Language Models have led to Large Reasoning Models, which produce step‐by‐step reasoning traces. Such traces may offer insight into how models think, improving explainability and clarifying the underlying process. These traces, however, are often verbose and complex, making them cognitively demanding to comprehend. With this in mind, we propose R e T race , an interactive system that structures and visualizes textual reasoning traces to support understanding. We use a validated reasoning taxonomy to produce structured reasoning data and investigate two types of interactive visualizations thereof. In a controlled human‐subject study, the visualizations provided more accurate comprehension of the model's reasoning and less perceived effort than a raw text baseline. The results of this study could have design implications for making long and complex machine‐generated reasoning processes more usable and transparent, an important step in AI explainability.

Abstract Edge bundling is widely used for reducing visual clutter in large 2D network and trajectory visualizations. Various edge bundling methods have been proposed, each producing qualitatively distinct outputs for the same data; however, few quantitative metrics exist for systematic evaluation. In this paper, we propose a set of quantitative metrics at the edge level (e.g., geometric properties of the resulting curves), the bundle level (e.g., thickness and number of bundles), and the global level (e.g., ambiguity and clustering). We propose a benchmark of 115 representative datasets and evaluate five representative edge bundling techniques that cover a broad range of methodological approaches. We also conduct a correlation analysis between bundling metrics and network drawing properties. To facilitate further analysis and comparison, we provide an interactive dashboard that includes all methods, metrics, and datasets, enabling side‐by‐side exploration of edge bundling effects. All supplemental materials and a link to our dashboard application are available at OSF.

In this paper, we test whether Multimodal Large Language Models (MLLMs) can match human-subject performance in tasks involving the perception of properties in network layouts. Specifically, we replicate a human-subject experiment about perceiving quality (namely stress) in network layouts using GPT-4o, Gemini-2.5 and Qwen2.5. Our experiments show that giving MLLMs the same study information as trained human participants yields performance comparable to that of human experts and exceeds that of untrained non-experts. Additionally, we show that prompt engineering that deviates from the human-subject experiment can lead to better-than-human performance in some settings. Interestingly, like human subjects, the MLLMs seem to rely on visual proxies rather than computing the actual value of stress, indicating some sense or facsimile of perception. Explanations from the models are similar to those used by the human participants (e.g., an even distribution of nodes and uniform edge lengths).

We present a novel approach to graph drawing based on reinforcement learning for minimizing the global and the local crossing number, that is, the total number of edge crossings and the maximum number of crossings on any edge, respectively. An agent learns how to move a vertex based on a given observation vector. The agent receives feedback in the form of local reward signals tied to crossing reduction. To generate an initial layout, we use a stress-based graph-drawing algorithm. We compare our method against force- and stress-based baseline algorithms as well as three established algorithms for global crossing minimization on a suite of benchmark graphs. The experiments show mixed results: our current algorithm is mainly competitive for the local crossing number.

Abstract Biofabric is a novel method for network visualization, with promising potential to highlight specific network features. Recent studies emphasize the importance of staircase motifs — equivalent to fans or stars in node‐link diagrams — within Biofabric. However, to effectively showcase these motifs, we need to formulate specialized layout algorithms. This paper introduces a method to compute optimal layouts for Biofabric, focusing on maximizing staircase formation. We present an Integer Linear Programming (ILP) model for this task and evaluate its performance in terms of scalability and output quality against a leading heuristic method, Degreecending. Our results demonstrate that the ILP approach identifies significantly more, and often longer, staircases compared to Degreecending, albeit with the trade‐off of higher computation times. Our supplemental material, including a full copy of the paper, code, and results, is available on osf.io.

Boundary labeling is a technique in computational geometry used to label sets of features in an illustration. It involves placing labels along an axis-parallel bounding box and connecting each label with its corresponding feature using non-crossing leader lines. Although boundary labeling is well-studied, semantic constraints on the labels have not been investigated thoroughly. In this paper, we introduce grouping and ordering constraints in boundary labeling: Grouping constraints enforce that all labels in a group are placed consecutively on the boundary, and ordering constraints enforce a partial order over the labels. We show that it is NP -hard to find a labeling for arbitrarily sized labels with unrestricted positions along one side of the boundary. However, we obtain polynomial-time algorithms if we restrict this problem either to uniform-height labels or to a finite set of candidate positions. Furthermore, we show that finding a labeling on two opposite sides of the boundary is NP -complete, even for uniform-height labels and finite label positions. Finally, we experimentally confirm that our approach has also practical relevance.

In various scientific and industrial domains, analyzing multivariate spatial data, i.e., vectors associated with spatial locations, is common practice. To analyze those datasets, analysts may turn to methods such as Spatial Blind Source Separation (SBSS). Designed explicitly for spatial data analysis, SBSS finds latent components in the dataset and is superior to popular non-spatial methods, like PCA. However, when analysts try different tuning parameter settings, the amount of latent components complicates analytical tasks. Based on our years-long collaboration with SBSS researchers, we propose a visualization approach to tackle this challenge. The main component is UnDRground Tubes (UT), a general-purpose idiom combining ideas from set visualization and multidimensional projections. We describe the UT visualization pipeline and integrate UT into an interactive multiple-view system. We demonstrate its effectiveness through interviews with SBSS experts, a qualitative evaluation with visualization experts, and computational experiments. SBSS experts were excited about our approach. They saw many benefits for their work and potential applications for geostatistical data analysis more generally. UT was also well received by visualization experts. Our benchmarks show that UT projections and its heuristics are appropriate.

Edge bundling algorithms can significantly improve the visualization of dense graphs by identifying and bundling together suitable groups of edges and thus reducing visual clutter. As such, bundling is often viewed as a post-processing step applied to a drawing, and the vast majority of edge bundling algorithms consider a graph and its drawing as input. A different way of thinking about edge bundling is to simultaneously optimize both the drawing and the bundling, which we investigate in this paper. We build on an earlier work where we introduced a novel algorithmic framework for bundling-aware graph drawing consisting of three main steps, namely Filter for a skeleton subgraph, Draw the skeleton, and Bundle the remaining edges against the drawing of the skeleton. We propose several alternative implementations and experimentally compare them against each other and the simple idea of first drawing the full graph and subsequently applying edge bundling to it. The experiments confirm that bundled drawings created by our Filter-Draw-Bundle framework outperform previous approaches according to metrics for edge bundling and graph drawing.

Boundary labeling is a well-known method for displaying short textual labels for a set of point features in a figure alongside the boundary of that figure. Labels and their corresponding points are connected via crossing-free leaders. We propose orbital boundary labeling as a new variant of the problem, in which (i) the figure is enclosed by a circular contour and (ii) the labels are placed as disjoint circular arcs in an annulus-shaped orbit around the contour. The algorithmic objective is to compute an orbital boundary labeling with the minimum total leader length. We identify several parameters that define the corresponding problem space: two leader types (straight or orbital-radial), label size and order, presence of candidate label positions, and constraints on where a leader attaches to its label. Our results provide polynomial-time algorithms for many variants and NP-hardness for others, using a variety of geometric and combinatorial insights.

Boundary labeling is a technique in computational geometry used to label sets of features in an illustration. It involves placing labels along an axis-parallel bounding box and connecting each label with its corresponding feature using non-crossing leader lines. Although boundary labeling is well-studied, semantic constraints on the labels have not been investigated thoroughly. In this paper, we introduce grouping and ordering constraints in boundary labeling: Grouping constraints enforce that all labels in a group are placed consecutively on the boundary, and ordering constraints enforce a partial order over the labels. We show that it is NP-hard to find a labeling for arbitrarily sized labels with unrestricted positions along one side of the boundary. However, we obtain polynomial-time algorithms if we restrict this problem either to uniform-height labels or to a finite set of candidate positions. Furthermore, we show that finding a labeling on two opposite sides of the boundary is NP-complete, even for uniform-height labels and finite label positions. Finally, we experimentally confirm that our approach has also practical relevance.

Real-world datasets often consist of quantitative and categorical variables. The analyst needs to focus on either kind separately or both jointly. We proposed a visualization technique tackling these challenges that supports visual cluster and set analysis. In this paper, we investigate how its visualization parameters affect the accuracy and speed of cluster and set analysis tasks in a controlled experiment. Our findings show that, with the proper settings, our visualization can support both task types well. However, we did not find settings suitable for the joint task, which provides opportunities for future research.

Abstract Edge‐Path bundling is a recent edge bundling approach that does not incur ambiguities caused by bundling disconnected edges together. Although the approach produces less ambiguous bundlings, it suffers from high computational cost. In this paper, we present a new Edge‐Path bundling approach that increases the computational speed of the algorithm without reducing the quality of the bundling. First, we demonstrate that biconnected components can be processed separately in an Edge‐Path bundling of a graph without changing the result. Then, we present a new edge bundling algorithm that is based on observing and exploiting a strong relationship between Edge‐Path bundling and graph spanners. Although the worst case complexity of the approach is the same as of the original Edge‐Path bundling algorithm, we conduct experiments to demonstrate that the new approach is 5–256 times faster than Edge‐Path bundling depending on the dataset, which brings its practical running time more in line with traditional edge bundling algorithms.

Visualizing sets of elements and their relations is an important research area in information visualization. In this paper, we present MosaicSets: a novel approach to create Euler-like diagrams from non-spatial set systems such that each element occupies one cell of a regular hexagonal or square grid. The main challenge is to find an assignment of the elements to the grid cells such that each set constitutes a contiguous region. As use case, we consider the research groups of a university faculty as elements, and the departments and joint research projects as sets. We aim at finding a suitable mapping between the research groups and the grid cells such that the department structure forms a base map layout. Our objectives are to optimize both the compactness of the entirety of all cells and of each set by itself. We show that computing the mapping is NP-hard. However, using integer linear programming we can solve real-world instances optimally within a few seconds. Moreover, we propose a relaxation of the contiguity requirement to visualize otherwise non-embeddable set systems. We present and discuss different rendering styles for the set overlays. Based on a case study with real-world data, our evaluation comprises quantitative measures as well as expert interviews.

Linear diagrams are used to visualize set systems by depicting set memberships as horizontal line segments in a matrix, where each set is represented as a row and each element as a column. Each such line segment of a set is shown in a contiguous horizontal range of cells of the matrix indicating that the corresponding elements in the columns belong to the set. As each set occupies its own row in the matrix, the total height of the resulting visualization is as large as the number of sets in the instance. Such a linear diagram can be visually sparse and intersecting sets containing the same element might be represented by distant rows. To alleviate such undesirable effects, we present LinSets.zip, a new approach that achieves a more space-efficient representation of linear diagrams. First, we minimize the total number of gaps in the horizontal segments by reordering columns, a criterion that has been shown to increase readability in linear diagrams. The main difference of LinSets.zip to linear diagrams is that multiple non-intersecting sets can be positioned in the same row of the matrix. Furthermore, we present several different rendering variations for a matrix-based representation that utilize the proposed row compression. We implemented the different steps of our approach in a visualization pipeline using integer-linear programming, and suitable heuristics aiming at sufficiently fast computations in practice. We conducted both a quantitative evaluation and a small-scale user experiment to compare the effects of compressing linear diagrams.

Edge bundling techniques cluster edges with similar attributes (i.e. similarity in direction and proximity) together to reduce the visual clutter. All edge bundling techniques to date implicitly or explicitly cluster groups of individual edges, or parts of them, together based on these attributes. These clusters can result in ambiguous connections that do not exist in the data. Confluent drawings of networks do not have these ambiguities, but require the layout to be computed as part of the bundling process. We devise a new bundling method, Edge-Path bundling, to simplify edge clutter while greatly reducing ambiguities compared to previous bundling techniques. Edge-Path bundling takes a layout as input and clusters each edge along a weighted, shortest path to limit its deviation from a straight line. Edge-Path bundling does not incur independent edge ambiguities typically seen in all edge bundling methods, and the level of bundling can be tuned through shortest path distances, Euclidean distances, and combinations of the two. Also, directed edge bundling naturally emerges from the model. Through metric evaluations, we demonstrate the advantages of Edge-Path bundling over other techniques.

We propose MetroSets, a new, flexible online tool for visualizing set systems using the metro map metaphor. We model a given set system as a hypergraph H=(V, S), consisting of a set V of vertices and a set S, which contains subsets of V called hyperedges. Our system then computes a metro map representation of H, where each hyperedge E in S corresponds to a metro line and each vertex corresponds to a metro station. Vertices that appear in two or more hyperedges are drawn as interchanges in the metro map, connecting the different sets. MetroSets is based on a modular 4-step pipeline which constructs and optimizes a path-based hypergraph support, which is then drawn and schematized using metro map layout algorithms. We propose and implement multiple algorithms for each step of the MetroSet pipeline and provide a functional prototype with easy-to-use preset configurations. Furthermore, using several real-world datasets, we perform an extensive quantitative evaluation of the impact of different pipeline stages on desirable properties of the generated maps, such as octolinearity, monotonicity, and edge uniformity.

Set systems are used to model data that naturally arises in many contexts: social networks have communities, musicians have genres, and patients have symptoms. Visualizations that accurately reflect the information in the underlying set system make it possible to identify the set elements, the sets themselves, and the relationships between the sets. In static contexts, such as print media or infographics, it is necessary to capture this information without the help of interactions. With this in mind, we consider three different systems for medium-sized set data, LineSets, EulerView, and MetroSets, and report the results of a controlled human-subjects experiment comparing their effectiveness. Specifically, we evaluate the performance, in terms of time and error, on tasks that cover the spectrum of static set-based tasks. We also collect and analyze qualitative data about the three different visualization systems. Our results include statistically significant differences, suggesting that MetroSets performs and scales better.