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Documents authored by Miller, Jacob

Document
Poster Abstract
DOI: 10.4230/LIPIcs.GD.2026.41
Size Should Not Matter: Evaluating Network Visualizations with Stress (Poster Abstract)

Authors: Reyan Ahmed, Cesim Erten, Stephen Kobourov, Jonah Lotz, Jacob Miller, and Hamlet Taraz

Published in: LIPIcs, Volume 396, 34th International Symposium on Graph Drawing and Network Visualization (GD 2026)

Abstract
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 a scale-invariant alternative and propose scale-normalized stress for fair stress-based evaluation.
Cite as

Reyan Ahmed, Cesim Erten, Stephen Kobourov, Jonah Lotz, Jacob Miller, and Hamlet Taraz. Size Should Not Matter: Evaluating Network Visualizations with Stress (Poster Abstract). In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 41:1-41:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ahmed_et_al:LIPIcs.GD.2026.41,
  author =	{Ahmed, Reyan and Erten, Cesim and Kobourov, Stephen and Lotz, Jonah and Miller, Jacob and Taraz, Hamlet},
  title =	{{Size Should Not Matter: Evaluating Network Visualizations with Stress}},
  booktitle =	{34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
  pages =	{41:1--41:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-452-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{396},
  editor =	{L\"{o}ffler, Maarten and Miksch, Silvia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.41},
  URN =		{urn:nbn:de:0030-drops-279363},
  doi =		{10.4230/LIPIcs.GD.2026.41},
  annote =	{Keywords: Normalized stress, layout evaluation, scale-invariant metrics}
}
Document
Poster Abstract
DOI: 10.4230/LIPIcs.GD.2026.44
Toroidal Graph Drawing by Multidimensional Scaling (Poster Abstract)

Authors: Timo Brand, Sebastian Damrich, Stephen Kobourov, and Jacob Miller

Published in: LIPIcs, Volume 396, 34th International Symposium on Graph Drawing and Network Visualization (GD 2026)

Abstract
We present a framework for multidimensional scaling of graph data to the flat torus. Compared to prior approaches, we contribute a closed-form geodesic distance computation, and learnable parameters for the torus size and aspect ratio, with an extension to rhombic tori. On synthetic benchmarks, our layouts achieve lower normalized stress than Euclidean baselines, and on real toroidal data our method recovers the underlying structure.
Cite as

Timo Brand, Sebastian Damrich, Stephen Kobourov, and Jacob Miller. Toroidal Graph Drawing by Multidimensional Scaling (Poster Abstract). In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 44:1-44:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{brand_et_al:LIPIcs.GD.2026.44,
  author =	{Brand, Timo and Damrich, Sebastian and Kobourov, Stephen and Miller, Jacob},
  title =	{{Toroidal Graph Drawing by Multidimensional Scaling}},
  booktitle =	{34th International Symposium on Graph Drawing and Network Visualization (GD 2026)},
  pages =	{44:1--44:5},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-452-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{396},
  editor =	{L\"{o}ffler, Maarten and Miksch, Silvia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2026.44},
  URN =		{urn:nbn:de:0030-drops-279393},
  doi =		{10.4230/LIPIcs.GD.2026.44},
  annote =	{Keywords: Graph layout, non-Euclidean geometry, multidimensional scaling}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.38
Stress in Graph Drawings: Perception, Preference, and Performance

Authors: Gavin J. Mooney, Jacob Miller, Michael Wybrow, Stephen Kobourov, and Helen C. Purchase

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
Stress in a graph drawing has been a popular layout principle for more than two decades. Low stress drawings exhibit the property that the geometric distances between all pairs of nodes correlate with the shortest paths between them. The assumption has always been that low stress drawings are "nicer" and better support human perception and comprehension than high stress drawings. In this paper, we put these assumptions to the test. We use a normalised scale-independent and rotation-independent metric for stress; this is necessary to ensure strict controls on our experimental stimuli. We report on three experiments, exploring human perception of stress, preference for stress, and the effect of stress on a graph performance task. We conclude that people can see stress in a graph drawing, that they prefer low stress drawings, and that their performance in a shortest path task improves as stress decreases - thus empirically confirming long-standing assumptions.
Cite as

Gavin J. Mooney, Jacob Miller, Michael Wybrow, Stephen Kobourov, and Helen C. Purchase. Stress in Graph Drawings: Perception, Preference, and Performance. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 38:1-38:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mooney_et_al:LIPIcs.GD.2025.38,
  author =	{Mooney, Gavin J. and Miller, Jacob and Wybrow, Michael and Kobourov, Stephen and Purchase, Helen C.},
  title =	{{Stress in Graph Drawings: Perception, Preference, and Performance}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{38:1--38:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.38},
  URN =		{urn:nbn:de:0030-drops-250240},
  doi =		{10.4230/LIPIcs.GD.2025.38},
  annote =	{Keywords: Graph Drawing, Graph Drawing Metrics, Stress, Visual Perception, User Study}
}
Document
Poster Abstract
DOI: 10.4230/LIPIcs.GD.2025.48
Drawing Trees and Cacti with Integer Edge Lengths on a Polynomial-Size Grid (Poster Abstract)

Authors: Henry Förster, Stephen Kobourov, Jacob Miller, and Johannes Zink

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
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. 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. We also give an historic overview of planar straight-line graph drawing results.
Cite as

Henry Förster, Stephen Kobourov, Jacob Miller, and Johannes Zink. Drawing Trees and Cacti with Integer Edge Lengths on a Polynomial-Size Grid (Poster Abstract). In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 48:1-48:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{forster_et_al:LIPIcs.GD.2025.48,
  author =	{F\"{o}rster, Henry and Kobourov, Stephen and Miller, Jacob and Zink, Johannes},
  title =	{{Drawing Trees and Cacti with Integer Edge Lengths on a Polynomial-Size Grid}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{48:1--48:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.48},
  URN =		{urn:nbn:de:0030-drops-250349},
  doi =		{10.4230/LIPIcs.GD.2025.48},
  annote =	{Keywords: Harborth’s conjecture, tree drawings, cactus drawings, grid drawings}
}
Document
DOI: 10.4230/LIPIcs.GD.2024.21
The Perception of Stress in Graph Drawings

Authors: Gavin J. Mooney, Helen C. Purchase, Michael Wybrow, Stephen G. Kobourov, and Jacob Miller

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Abstract
Most of the common graph layout principles (a.k.a. "aesthetics") on which many graph drawing algorithms are based are easy to define and to perceive. For example, the number of pairs of edges that cross each other, how symmetric a drawing looks, the aspect ratio of the bounding box, or the angular resolution at the nodes. The extent to which a graph drawing conforms to these principles can be determined by looking at how it is drawn - that is, by looking at the marks on the page - without consideration for the underlying structure of the graph. A key layout principle is that of optimising "stress", the basis for many algorithms such as the popular Kamada & Kawai algorithm and several force-directed algorithms. The stress of a graph drawing is, loosely speaking, the extent to which the geometric distance between each pair of nodes is proportional to the shortest path between them - over the whole graph drawing. The definition of stress therefore relies on the underlying structure of the graph (the "paths") in a way that other layout principles do not, making stress difficult to describe to novices unfamiliar with graph drawing principles, and, we believe, difficult to perceive. We conducted an experiment to see whether people (novices as well as experts) can see stress in graph drawings, and found that it is possible to train novices to "see" stress - even if their perception strategies are not based on the definitional concepts.
Cite as

Gavin J. Mooney, Helen C. Purchase, Michael Wybrow, Stephen G. Kobourov, and Jacob Miller. The Perception of Stress in Graph Drawings. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mooney_et_al:LIPIcs.GD.2024.21,
  author =	{Mooney, Gavin J. and Purchase, Helen C. and Wybrow, Michael and Kobourov, Stephen G. and Miller, Jacob},
  title =	{{The Perception of Stress in Graph Drawings}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{21:1--21:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.21},
  URN =		{urn:nbn:de:0030-drops-213051},
  doi =		{10.4230/LIPIcs.GD.2024.21},
  annote =	{Keywords: Stress, Graph Drawing, Visual Perception}
}
Document
DOI: 10.4230/DagRep.13.1.216
Perception in Network Visualization (Dagstuhl Seminar 23051)

Authors: Karsten Klein, Stephen Kobourov, Bernice E. Rogowitz, Danielle Szafir, and Jacob Miller

Published in: Dagstuhl Reports, Volume 13, Issue 1 (2023)

Abstract
Networks are used to model and represent data in many application areas from life sciences to social sciences. Visual network analysis is a crucial tool to improve the understanding of data sets and processes over many levels of complexity, such as different semantic, spatial and temporal granularities. While there is a great deal of work on the algorithmic aspects of network visualization and the computational complexity of the underlying problems, the role and limits of human perception are rarely explicitly investigated and taken into account when designing network visualizations. To address this issue, this Dagstuhl Seminar raised awareness in the network visualization community of the need for more extensive theoretical and empirical understanding of how people perceive and make sense of network visualizations and the significant potential for improving current solutions when perception-based strategies are employed. Likewise, the seminar increased awareness in the perception community that challenges in network research can drive new questions for perception research, for example, in identifying features and patterns in large, often time-varying networks. We brought together researchers from several different communities to initiate a dialogue, foster exchange, discuss the state of the art at this intersection and within the respective fields, identify promising research questions and directions, and start working on selected problems.
Cite as

Karsten Klein, Stephen Kobourov, Bernice E. Rogowitz, Danielle Szafir, and Jacob Miller. Perception in Network Visualization (Dagstuhl Seminar 23051). In Dagstuhl Reports, Volume 13, Issue 1, pp. 216-244, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Article{klein_et_al:DagRep.13.1.216,
  author =	{Klein, Karsten and Kobourov, Stephen and Rogowitz, Bernice E. and Szafir, Danielle and Miller, Jacob},
  title =	{{Perception in Network Visualization (Dagstuhl Seminar 23051)}},
  pages =	{216--244},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2023},
  volume =	{13},
  number =	{1},
  editor =	{Klein, Karsten and Kobourov, Stephen and Rogowitz, Bernice E. and Szafir, Danielle and Miller, Jacob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.13.1.216},
  URN =		{urn:nbn:de:0030-drops-191220},
  doi =		{10.4230/DagRep.13.1.216},
  annote =	{Keywords: Network Visualization, Graph Drawing, Perception, Cognition}
}

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