arXiv subjects

Stephen Kobourov

Publications and source records attributed to Stephen Kobourov.

At least 19 recordsLinked to original sources

arXiv2026

Visualizing High-Dimensional Graph Embeddings via Informed Multi-View Projections

Graphs are commonly visualized in 2D, where humans readily interpret spatial relationships, yet such layouts often distort higher-dimensional structure. We propose to embed graphs in high-dimensional space and search for informative 2D viewpoints that optimize aesthetic and readability metrics (e.g., edge crossings and angular resolution), enabled by a novel differentiable surrogate for edge crossings. Numerical experiments show that these viewpoints consistently outperform standard 2D layouts, and can even surpass methods explicitly designed to optimize these metrics. We further introduce DataFly, an interactive system for exploring multiple candidate viewpoints through seamless navigation. A usability study demonstrates that our approach reveals structural patterns that remain hidden in conventional 2D visualizations.

cs.LG↗

arXiv2026

Using Reinforcement Learning to Optimize the Global and Local Crossing Number

Graph drawing concerns the algorithmic visualization of graphs. A good drawing of a graph is easy to read and facilitates solving tasks on the graph. Several properties have been identified to occur in good drawings of graphs. Such properties include a low number of crossings, large angles between edges, short edges, and depicting symmetries. Many of these properties are explicitly measurable metrics. This lets us model a graph-drawing problem as a game where a single player iteratively moves vertices of a straight-line graph drawing to reduce edge crossings. We investigate whether reinforcement learning can discover effective strategies for playing this game. Our reinforcement-learning agent observes the local geometric and structural context of a vertex and selects a movement direction with the goal of reducing either the global or the local crossing number, that is, either the total number of crossings or the maximum number of crossings per edge. We compare the resulting strategies to existing methods and established crossing-minimization heuristics on standard benchmark graphs. While our approach does not out-compete state-of-the-art methods for minimizing the global crossing number, it is competitive and often superior for minimizing the local crossing number.

cs.CG↗

arXiv2026

Weighted Book Thickness

We introduce and study the weighted book thickness of graphs. A $k$-page book embedding of a graph $G=(V,E)$ is defined by a spanning cycle $C$ for $V$ (which does not need to be part of $G$) and a partition $E=\bigcup_{i=1}^{k}E_i$ such that $E\cap C\subseteq E_1$ and each graph $G_i=(V,E_i\cup C)$, for $1 \le i \le k$, is outerplane with outer cycle $C$. If $e\in E_i$, we say that $e$ appears on Page $i$. The classical book thickness of a graph $G$ is the minimum $k$ such that there exists a $k$-page book embedding of $G$, that is, the minimum (over all book embeddings of $G$) achievable maximum page an edge appears on. In contrast, the weighted book thickness is the minimum achievable average page an edge appears on. The embeddings that realize weighted book thickness can differ from those that realize (classical) book thickness. We show that, although every planar graph on at most nine vertices admits a 2-page book embedding realizing its weighted book thickness, already for ten vertices, there is a planar graph for which every realization of its weighted book thickness needs more pages than its book thickness. We prove that there even exists a 2-tree whose weighted book thickness cannot be realized on two pages. On the positive side, we show that for every graph of pathwidth at most two, the weighted book thickness can always be realized by a 2-page book embedding and such an embedding can be found in linear time. Moreover, we prove that it is NP-complete to decide if the weighted book thickness is at most $k$, for some given integer $k$.

cs.CG↗

arXiv2026

Representing Hypergraphs by Point-Line Incidences

We consider hypergraph visualizations that represent vertices as points in the plane and hyperedges as curves passing through the points of their incident vertices. Specifically, we consider several different variants of this problem by (a) restricting the curves to be lines or line segments, (b) allowing two curves to cross if they do not share an element, or not; and (c) allowing two curves to overlap or not. We show $\exists\mathbb{R}$-hardness for six of the eight resulting decision problem variants and describe polynomial-time algorithms in some restricted settings. Lastly, we briefly touch on what happens if we allow the lines of the represented hyperedges to have bends - to this we generalize a counterexample to a long-standing result that was sometimes assumed to be correct.

cs.CG↗

arXiv2026

Size Should Not Matter: Evaluating Network Visualizations with Stress

The normalized stress metric is widely used to assess graph drawing quality, measuring how closely distances between vertices in a layout match their graph-theoretic distances. This metric is a standard for both evaluation and optimization in many popular graph layout algorithms. However, normalized stress is highly sensitive to the scale (size) of the drawing, leading to potentially misleading comparisons between layouts produced by different algorithms. Uniformly scaling a layout can significantly alter stress values without changing the underlying structure, even to the extent that a clearly superior layout can appear to have a higher stress score than a random layout. Although this issue is recognized within the network visualization community, it is rarely addressed with sufficient detail in publications, resulting in critical calculation errors in recent studies. In this paper, we systematically examine various stress metrics used in the literature and demonstrate that commonly used metrics are affected by layout scale, undermining their reliability for comparison. We identify scale-invariant alternatives and propose scale-normalized stress for fair stress-based evaluation.

cs.CG↗

arXiv2026

3DMPE: 3D Multi-Perspective Embedding

We study 3D point cloud reconstruction from multiple partially observed 2D projections. Given two or more projections of an unknown 3D point cloud, together with cross-view point correspondences and visibility information, our goal is to recover a consistent 3D configuration when different views contain different subsets of points. We propose 3D Multi-Perspective Embedding (3DMPE), an optimization-based, training-free method that reconstructs the 3D point cloud and, in the variable-projection setting, jointly estimates the projection maps. 3DMPE extends Multi-Perspective Simultaneous Embedding to accommodate missing points and incomplete pairwise distance information across views. We consider both fixed-projection and variable-projection settings. Unlike learning-based reconstruction methods that infer shape from raw images and often depend on training data, 3DMPE operates on geometric observations with established correspondences and does not require category-specific training. Experiments on ShapeNet and Pix3D evaluate reconstruction quality using Chamfer Distance, Earth Mover Distance, and RMSE-Optimize-Align (ROA), and examine the effects of initialization, the number of views, point visibility, and several noise regimes, including noisy distances and erroneous correspondences. The results demonstrate that 3DMPE can effectively reconstruct point clouds from partial multi-view geometric observations.

cs.CV↗

arXiv2026

Rerouting Curves on Surfaces

We study the problem of reconfiguring a crossing-free embedding of a graph on a surface, with edges represented as curves, into another crossing-free embedding of the same graph on the same surface with the same fixed vertex positions. In this process, we reroute one edge at a time while maintaining crossing-free intermediate embeddings. This problem was introduced by Ito et al. [TALG 2025], who showed that even if the graph is a matching of two edges, reconfiguration is not always possible in the plane, but is always possible on the torus. For matchings of two or more edges, they gave a necessary and sufficient condition for reconfigurable embeddings in the plane, but not on the torus. Our main result is that for matchings, trees and forests, reconfiguration is always possible on the torus, and consequently, on any orientable surface of genus at least one. In addition, we provide sufficient conditions for reconfiguration on orientable surfaces of genus at least one and in the projective plane. For more general graphs, we show that reconfiguration is not always possible.

cs.CG↗

arXiv2026

Syndesmoscope: The Power of Invariant Plots Linked to Traditional Network Views

Traditional network representations, such as node-link views and adjacency matrices, can show dramatically different visual patterns, depending on the underlying layout or seriation algorithm. In contrast, invariant plots consistently surface the same visual pattern for the same input topology; yet researchers have underexplored them and have not integrated them into visualization systems. We present Syndesmoscope, an interactive system for network exploration that juxtaposes multiple views of the same network. Panes show a familiar a force-directed view alongside three panes with interpretable geometric layouts based on graph-theoretic properties: dense-sparse gradient, geodesic eccentricity, and spectral bisection. As a secondary contribution, we introduce kSnakes, a new invariant plot based on density decomposition. Syndesmoscope supports two key interactions: leapfrogging, or linked highlighting between different and interpretable visual patterns; and hopscotching, or hop-based traversal that extends data selections through the underlying topology. Through usage scenarios across a corpus of 72 diverse networks, we demonstrate how these interactions reveal network patterns inaccessible through any single view alone. Live demo available at https://syndesmoscope.vercel.app/.

cs.HC↗

arXiv2026

Class Angular Distortion Index for Dimensionality Reduction

Dimensionality reduction (DR) techniques are often characterized by whether they preserve global, high-level structures in the data or local, neighborhood structures. This distinction matters in visualization: global methods can obscure clusters while local methods can over-emphasize them. Yet, even when clusters appear distinct, their relative arrangement in the projection may be arbitrary or misleading, a common issue in techniques such as t-SNE and UMAP. Existing cluster quality metrics either only measure cluster separability or assume spherical, globular clusters in the original space. We introduce the Class Angular Distortion Index (CADI), a metric that uses internal angles among point triples to determine the faithfulness of cluster organization in a projection. We show cases on both real and synthetic data where existing cluster metrics fail, but CADI provides an interpretable result. Since it relies on computing angles, CADI is also differentiable, enabling optimization. We demonstrate this with a CADI-based DR technique.

cs.LG↗

arXiv2026

Bridging Graph Drawing and Dimensionality Reduction with Stochastic Stress Optimization

Both Dimensionality Reduction (DR) and Graph Drawing (GD) aim to visualize abstract, non-linear structures, yet rely on different optimization paradigms. This contrast is evident in Multidimensional Scaling (MDS), which typically depends on the SMACOF algorithm despite graph drawing results showing that simpler stochastic optimization schemes can be more effective for the same objective. We bridge these domains by adapting Stochastic Gradient Descent (SGD) techniques from graph drawing to vector data embedding. We present a scikit-learn compatible estimator that minimizes global stress through local pairwise updates, improving upon the existing implementation. Experiments on standard high-dimensional benchmarks show that our stochastic solver converges substantially faster than SMACOF while achieving comparable or lower stress.

cs.LG↗

arXiv2026

Exploring MLLMs Perception of Network Visualization Principles

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).

cs.HC↗

arXiv2026

Visualization Tasks for Unlabeled Graphs

We investigate tasks that can be accomplished with unlabeled graphs, which are graphs with nodes that do not have persistent or semantically meaningful labels attached. New visualization techniques to represent unlabeled graphs have been proposed, but more understanding of unlabeled graph tasks is required before these techniques can be adequately evaluated. Some network visualization tasks apply to both labeled and unlabeled graphs, but many do not translate between these contexts. We propose a data abstraction model that distinguishes the Unlabeled context from the increasingly semantically rich Labeled, Attributed, and Augmented contexts. We filter tasks collected and gleaned from the literature according to our data abstraction and analyze the surfaced tasks, leading to a taxonomy of abstract tasks for unlabeled graphs. Our task taxonomy is organized according to the Target data under consideration, the Action intended by the user, and the Scope of the data at play. We show the descriptive power of this task abstraction by connecting to concrete examples from previous frameworks, and connecting these abstractions to real-world problems. To showcase the evaluative power of the taxonomy, we perform a preliminary assessment across 6 different network visualization idioms for each task. For each combination of task and visual encoding, we consider the effort required from viewers, the likelihood of task success, and how both factors vary between small-scale and large-scale graphs. Supplemental materials are available at osf.io/e23mr.

cs.HC↗

arXiv2026

Simultaneous Embedding of Two Paths on the Grid

We study the problem of simultaneous geometric embedding of two paths without self-intersections on an integer grid. We show that minimizing the length of the longest edge of such an embedding is NP-hard. We also show that we can minimize in $O(n^{3/2})$ time the perimeter of an integer grid containing such an embedding if one path is $x$-monotone and the other is $y$-monotone.

cs.CG↗

arXiv2025

Hypergraphs as Metro Maps: Drawing Paths with Few Bends in Trees, Cacti, and Plane 4-Graphs

A hypergraph consists of a set of vertices and a set of subsets of vertices, called hyperedges. In the metro map metaphor, each hyperedge is represented by a path (the metro line) and the union of all these paths is the support graph (metro network) of the hypergraph. Formally speaking, a path-based support is a graph together with a set of paths. We consider the problem of constructing drawings of path-based supports that (i) minimize the sum of the number of bends on all paths, (ii) minimize the maximum number of bends on any path, or (iii) maximize the number of 0-bend paths, then the number of 1-bend paths, etc. We concentrate on straight-line drawings of path-based tree and cactus supports as well as orthogonal drawings of path-based plane supports with maximum degree 4.

cs.CG↗

arXiv2025

"Normalized Stress" is Not Normalized: How to Interpret Stress Correctly

Stress is among the most commonly employed quality metrics and optimization criteria for dimension reduction projections of high dimensional data. Complex, high dimensional data is ubiquitous across many scientific disciplines, including machine learning, biology, and the social sciences. One of the primary methods of visualizing these datasets is with two dimensional scatter plots that visually capture some properties of the data. Because visually determining the accuracy of these plots is challenging, researchers often use quality metrics to measure projection accuracy or faithfulness to the full data. One of the most commonly employed metrics, normalized stress, is sensitive to uniform scaling of the projection, despite this act not meaningfully changing anything about the projection. We investigate the effect of scaling on stress and other distance based quality metrics analytically and empirically by showing just how much the values change and how this affects dimension reduction technique evaluations. We introduce a simple technique to make normalized stress scale invariant and show that it accurately captures expected behavior on a small benchmark.

cs.LG↗

arXiv2025

ReTrace: Interactive Visualizations for Reasoning Traces of Large Reasoning Models

Recent advances in Large Language Models have led to Large Reasoning Models, which produce step-by-step reasoning traces. These traces offer insight into how models think and their goals, improving explainability and helping users follow the logic, learn the process, and even debug errors. These traces, however, are often verbose and complex, making them cognitively demanding to comprehend. We address this challenge with ReTrace, 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 user study, both visualizations enabled users to comprehend the model's reasoning more accurately and with 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.

cs.HC↗

arXiv2025

How Scale Breaks "Normalized Stress" and KL Divergence: Rethinking Quality Metrics

Complex, high-dimensional data is ubiquitous across many scientific disciplines, including machine learning, biology, and the social sciences. One of the primary methods of visualizing these datasets is with two-dimensional scatter plots that visually capture some properties of the data. Because visually determining the accuracy of these plots is challenging, researchers often use quality metrics to measure the projection's accuracy and faithfulness to the original data. One of the most commonly employed metrics, normalized stress, is sensitive to uniform scaling (stretching, shrinking) of the projection, despite this act not meaningfully changing anything about the projection. Another quality metric, the Kullback--Leibler (KL) divergence used in the popular t-Distributed Stochastic Neighbor Embedding (t-SNE) technique, is also susceptible to this scale sensitivity. We investigate the effect of scaling on stress and KL divergence analytically and empirically by showing just how much the values change and how this affects dimension reduction technique evaluations. We introduce a simple technique to make both metrics scale-invariant and show that it accurately captures expected behavior on a small benchmark.

cs.LG↗

arXiv2025

Drawing Trees and Cacti with Integer Edge Lengths on a Polynomial-Size Grid

A strengthened version of Harborth's well-known conjecture -- known as Kleber's conjecture -- states that every planar graph admits a planar straight-line drawing where every edge has integer length and each vertex is restricted to the integer grid. Positive results for Kleber's conjecture are known for planar 3-regular graphs, for planar graphs that have maximum degree 4, and for planar 3-trees. However, all but one of the existing results are existential and do not provide bounds on the required grid size. In this paper, we provide polynomial-time algorithms for computing crossing-free straight-line drawings of trees and cactus graphs with integer edge lengths and integer vertex position on polynomial-size integer grids.

cs.CG↗