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Toroidal Graph Drawing by Multidimensional Scaling (Poster Abstract)

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

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Subject Classification

ACM Subject Classification
  • Human-centered computing → Graph drawings
  • Mathematics of computing → Graphs and surfaces
Keywords
  • Graph layout
  • non-Euclidean geometry
  • multidimensional scaling

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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 Get BibTex

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) https://doi.org/10.4230/LIPIcs.GD.2026.44

Author Details

Timo Brand
  • Technical University of Munich, Heilbronn, Germany
Sebastian Damrich
  • Hertie Institute for AI in Brain Health, University of Tübingen, Germany
Stephen Kobourov
  • Technical University of Munich, Heilbronn, Germany
Jacob Miller
  • Technical University of Munich, Heilbronn, Germany

References

  1. Kun-Ting Chen, Tim Dwyer, Benjamin Bach, and Kim Marriott. It’s a wrap: Toroidal wrapping of network visualisations supports cluster understanding tasks. In CHI '21: CHI Conference on Human Factors in Computing Systems. ACM, 2021. URL: https://doi.org/10.1145/3411764.3445439.
  2. Kun-Ting Chen, Tim Dwyer, Kim Marriott, and Benjamin Bach. Doughnets: Visualising networks using torus wrapping. In CHI '20: CHI Conference on Human Factors in Computing Systems. ACM, 2020. URL: https://doi.org/10.1145/3313831.3376180.
  3. Kun-Ting Chen, Tim Dwyer, Yalong Yang, Benjamin Bach, and Kim Marriott. Gan’sda wrap: Geographic and network structured data on surfaces that wrap around. In Simone D. J. Barbosa, Cliff Lampe, Caroline Appert, David A. Shamma, Steven Mark Drucker, Julie R. Williamson, and Koji Yatani, editors, CHI '22: CHI Conference on Human Factors in Computing Systems, pages 135:1-135:16. ACM, 2022. URL: https://doi.org/10.1145/3491102.3501928.
  4. Kun-Ting Chen, Quynh Quang Ngo, Kuno Kurzhals, Kim Marriott, Tim Dwyer, Michael Sedlmair, and Daniel Weiskopf. Reading strategies for graph visualizations that wrap around in torus topology. In Enkelejda Kasneci, Frédérick Shic, and Mohamed Khamis, editors, Proceedings of the 2023 Symposium on Eye Tracking Research and Applications, ETRA 2023, pages 67:1-67:7. ACM, 2023. URL: https://doi.org/10.1145/3588015.3589841.
  5. Sabin Devkota, Abu Reyan Ahmed, Felice De Luca, Katherine E. Isaacs, and Stephen G. Kobourov. Stress-plus-x (SPX) graph layout. In Daniel Archambault and Csaba D. Tóth, editors, Graph Drawing and Network Visualization - 27th International Symposium, GD 2019, volume 11904 of Lecture Notes in Computer Science, pages 291-304. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-35802-0_23.
  6. Emden R. Gansner, Yifan Hu, and Stephen C. North. A maxent-stress model for graph layout. IEEE Trans. Vis. Comput. Graph., 19(6):927-940, 2013. URL: https://doi.org/10.1109/TVCG.2012.299.
  7. Emden R. Gansner, Yehuda Koren, and Stephen C. North. Graph drawing by stress majorization. In János Pach, editor, Graph Drawing, 12th International Symposium, GD 2004. Revised Selected Papers, volume 3383 of Lecture Notes in Computer Science, pages 239-250. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-31843-9_25.
  8. Richard J Gardner, Erik Hermansen, Marius Pachitariu, Yoram Burak, Nils A Baas, Benjamin A Dunn, May-Britt Moser, and Edvard I Moser. Toroidal topology of population activity in grid cells. Nature, 602(7895):123-128, 2022. URL: https://doi.org/10.1038/s41586-021-04268-7.
  9. Tomihisa Kamada and Satoru Kawai. An algorithm for drawing general undirected graphs. Inf. Process. Lett., 31(1):7-15, 1989. URL: https://doi.org/10.1016/0020-0190(89)90102-6.
  10. Jacob Miller, Dhruv Bhatia, Helen C. Purchase, and Stephen G. Kobourov. Euclidean, hyperbolic, and spherical networks: An empirical study of matching network structure to best visualizations. Comput. Graph. Forum, 44(3), 2025. URL: https://doi.org/10.1111/CGF.70126.
  11. Jacob Miller, Vahan Huroyan, and Stephen Kobourov. Spherical graph drawing by multi-dimensional scaling. In Graph Drawing and Network Visualization, pages 77-92. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-22203-0_7.
  12. Jacob Miller, Stephen G. Kobourov, and Vahan Huroyan. Browser-based hyperbolic visualization of graphs. In 15th IEEE Pacific Visualization Symposium, PacificVis 2022, pages 71-80. IEEE, 2022. URL: https://doi.org/10.1109/PACIFICVIS53943.2022.00016.
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