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arxiv: 2604.14791 · v1 · submitted 2026-04-16 · 💻 cs.HC

Evaluating Encodings for Bivariate Edges in Adjacency Matrices

Jorge Acosta-Hern\'andez , Alexander Lex , Tingying He This is my paper

Pith reviewed 2026-05-10 10:58 UTC · model grok-4.3

classification 💻 cs.HC
keywords adjacency matrixbivariate encodingedge visualizationnetwork visualizationvisual channelsuser studymultivariate networks
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The pith

Empirical tests show area-based marks and bar charts best encode two quantitative values per edge in adjacency matrices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests four encodings for displaying both a central value and a spread value on each edge inside a grid-based network diagram. A study with 156 participants completed eight analysis tasks using each encoding. Area overlays and bar charts led to better accuracy and speed than angle marks or color scales alone. A sympathetic reader cares because real networks often carry multiple measurements per connection, and the grid format keeps overall structure visible but leaves very little room per cell for extra data.

Core claim

The evaluation compared four encodings for bivariate quantitative edge attributes in adjacency matrices: a bivariate color palette, embedded bar charts, color with overlaid area marks, and color with overlaid angle marks. Across eight analytical tasks, area-based overlaid marks and bar charts produced the highest performance, angle-based marks showed moderate but less stable results, and bivariate color consistently ranked lowest.

What carries the argument

Four candidate encodings for mapping central tendency and dispersion to visual channels inside the small cells of an adjacency matrix, evaluated through task accuracy, speed, and subjective ratings in a crowdsourced study.

If this is right

  • Designers should favor area overlays or bar charts when an adjacency matrix must show both center and spread for each edge.
  • Position and size channels remain more reliable than color or angle when cell space is tightly limited.
  • The performance gap between encodings holds across tasks that require value reading, comparison, and identification.
  • Bivariate color scales face consistent limits in conveying dispersion accurately within matrix cells.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar controlled comparisons could test encodings for three or more values per edge if more compact marks are found.
  • Real-world use might reveal whether the crowdsourced tasks match the priorities of practicing analysts.
  • Angle encodings might close the gap with refinements such as clearer reference lines or different angular ranges.

Load-bearing premise

The eight chosen tasks and the crowdsourced participants represent the questions and perceptual abilities of domain experts who analyze real multivariate networks.

What would settle it

A study with domain experts performing their own typical questions on actual multivariate network datasets in which bivariate color performs as well as area marks or bar charts.

Figures

Figures reproduced from arXiv: 2604.14791 by Alexander Lex, Jorge Acosta-Hern\'andez, Tingying He.

Figure 1
Figure 1. Figure 1: Four representative techniques illustrated using example 3Œ3 AMs with the same dataset. Each encoding represents two measures: central tendency (CT) and dispersion (D). (a) Bivariate color: hue encodes CT, lightness encodes D. (b) Embedded bar charts: the green bar encodes CT, the purple bar encodes D. (c) Overlaid mark size: hue encodes CT, the size of the overlaid square encodes D. (d) Overlaid mark angl… view at source ↗
Figure 2
Figure 2. Figure 2: Examples of possible designs for jointly encoding two edge attributes in AM cells, beyond the four designs proposed by Alper et al. [ABHR∗ 13]. Designs (a)(c) are discussed in Sec. 3.1, (d) and (e) in Sec. 3.2, and (f) and (g) in Sec. 3.3. 3. Design Characterization of Techniques for Encoding Two Edge Attributes Jointly in AMs In this section, we characterize techniques for encoding two edge attributes per… view at source ↗
Figure 3
Figure 3. Figure 3: Study interface for an example Adjacency by Attribute Combination task. The task prompt appears in the upper-left corner, with the selected nodes shown below it as the users responses. In this example, the participant used three interaction mechanisms, indicated by the orange elements: (1) the Maryland column is highlighted as a result of a previous click on its column label; (2) the Missouri row is highli… view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparative evaluation of perceived data reading ability (PREVis DataRead subscale [CHII25]) and perceived aesthetics (Beau￾Vis [HIDI23]) across the four tested encoding techniques. Results for the remaining PREVis subscales are reported in the supplementary material. in T2, T3, T5, T6 and T8. However, the expected performance hier￾archy among the remaining techniques did not emerge (see [PITH_FULL_IMAGE:… view at source ↗
Figure 6
Figure 6. Figure 6: Stimuli used in the study. (a) Bivariate color palette encoding. (b) Embedded bar chart encoding. (c) Overlaid mark size encoding. (d) Overlaid mark angle encoding. This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.111… view at source ↗
Figure 8
Figure 8. Figure 8: Encoding techniques for bivariate AMs by Alper et al. Image reproduced from [ABHR∗ 13, [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Bar Chart encoding tested by Fuchs et al. against a similar encoding on the BioFabric Layout. Image reproduced from [FDH∗ 24, [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of eight bivariate representations by Retchless and Brewer. Image reproduced from [RB16, [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: A task testing one of Alper et al. encodings against NL by Nobre et al. Image reproduced from [NWHL20, [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Proposed encodings for multivariate AMs by Vogogias et al. Image reproduced from [VABK20, [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 16
Figure 16. Figure 16: Browser Usage Distribution 1 2 3 4 5 0 10 20 30 40 50 37 28 50 33 6 [PITH_FULL_IMAGE:figures/full_fig_p017_16.png] view at source ↗
Figure 13
Figure 13. Figure 13: Age Distribution Male Female CONSENT_REVOKED 0 10 20 30 40 50 60 70 80 78 71 4 [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Sex Distribution High school Bachelor Masters PhD Other Missing 0 10 20 30 40 50 60 70 80 28 81 38 5 2 2 [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Education Level Distribution Chrome Edge Firefox 0 20 40 60 80 100 120 131 17 8 [PITH_FULL_IMAGE:figures/full_fig_p017_15.png] view at source ↗
Figure 20
Figure 20. Figure 20: T1 This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy [PITH_FULL_IMAGE:figures/full_fig_p019_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: T2 - Task Prompt 1 This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 22
Figure 22. Figure 22: T2 - Task Prompt 2 This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 23
Figure 23. Figure 23: T3 - Task Prompt 1 This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 24
Figure 24. Figure 24: Task 3 - Task Prompt 2 This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy [PITH_FULL_IMAGE:figure… view at source ↗
Figure 25
Figure 25. Figure 25: T4 - Task Prompt 1 This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 26
Figure 26. Figure 26: T4 - Task Prompt 2 This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 27
Figure 27. Figure 27: T5 - Task Prompt 1 This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 28
Figure 28. Figure 28: T5 - Task Prompt 2 This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 29
Figure 29. Figure 29: T6 - Task Prompt 1 This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 30
Figure 30. Figure 30: T6 - Task Prompt 2 This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 31
Figure 31. Figure 31: T7 - Task Prompt 1 This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 32
Figure 32. Figure 32: T7 - Task Prompt 2 This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 33
Figure 33. Figure 33: Task 8 - Task Prompt 1 This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy [PITH_FULL_IMAGE:figure… view at source ↗
Figure 34
Figure 34. Figure 34: T8 - Task Prompt 2 This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 35
Figure 35. Figure 35: Comparative evaluation of the Reading Data Item of the PREVis Scale across the four tested encoding techniques. This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commer… view at source ↗
Figure 36
Figure 36. Figure 36: Comparative evaluation of the Understand Item of the PREVis Scale across the four tested encoding techniques. This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commerci… view at source ↗
Figure 37
Figure 37. Figure 37: Comparative evaluation of the Layout Item of the PREVis Scale across the four tested encoding techniques. This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commercial p… view at source ↗
Figure 38
Figure 38. Figure 38: Comparative evaluation of the Reading Features Item of the PREVis Scale across the four tested encoding techniques. This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-co… view at source ↗
Figure 39
Figure 39. Figure 39: Comparative evaluation of perceived aesthetics score across the four tested encoding techniques. This is the accepted version of the following article: J. Acosta-Hernández, A. Lex, T. He. Evaluating Encodings for Bivariate Edges in Adjacency Matrices. Computer Graphics Forum / EuroVis, 2026, which has been published in final form at 10.1111/cgf.70475. This article may be used for non-commercial purposes i… view at source ↗
read the original abstract

We present the first empirical evaluation of techniques for encoding distributions of quantitative edge values within adjacency matrices. In many real-world networks, edges represent not a single value but a set of measurements. While adjacency matrices preserve structural clarity, their compact cells limit the simultaneous display of multiple values. To address this, we explore edge encodings that represent distributions by two values: a measure of central tendency (mean, median, mode) and a measure of dispersion (standard deviation, variance, IQR). We select four possible encodings for evaluation that prior work has suggested are suitable for the limited space available in matrices: a bivariate color palette, embedded bar charts, and two overlaid-mark designs mapping the primary attribute to color and the secondary attribute to area or angle. In a preregistered crowdsourced study with 156 participants, we assessed performance of these encodings across eight analytical tasks and collected readability and aesthetic ratings. Results reveal clear performance regimes: area-based overlaid marks and bar charts achieved the highest overall performance; angle-based marks show moderate but less stable performance,and bivariate color consistently underperforms these alternatives. These findings clarify how visual channels behave under strict constraints and delineate the strengths and limitations of key design choices for multivariate edge visualization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents the first empirical evaluation of four encodings for bivariate quantitative edge attributes in adjacency matrices: bivariate color palettes, embedded bar charts, and two overlaid-mark designs (color with area or color with angle). Using a preregistered crowdsourced study with 156 participants who completed eight analytical tasks, the authors report clear performance regimes in which area-based overlaid marks and bar charts achieve the highest overall performance, angle-based marks show moderate but less stable performance, and bivariate color consistently underperforms; subjective readability and aesthetic ratings were also collected.

Significance. If the results hold, the work provides useful empirical guidance on visual channel effectiveness under the strict spatial constraints of matrix cells, helping designers choose encodings for multivariate network data. The preregistered protocol, substantial participant sample, multi-task design, and collection of both objective performance data and subjective ratings are notable strengths that support the reliability of the identified performance ordering.

major comments (2)
  1. [§4 (Study Design)] §4 (Study Design): The eight analytical tasks are presented as generic probes, yet the abstract motivates the work with 'real-world networks'; without explicit mapping or validation showing that these tasks elicit the same perceptual and cognitive demands as those faced by domain experts inspecting bivariate edge distributions, the generalizability of the performance regimes to the motivating use cases is not demonstrated.
  2. [§5 (Results)] §5 (Results): The abstract states that 'clear performance regimes' were observed but provides no details on the statistical tests, effect sizes, confidence intervals, or exclusion criteria used to establish differences between encodings; these elements are load-bearing for interpreting the stability claims (e.g., for angle-based marks) and should be foregrounded even if present in the full methods section.
minor comments (2)
  1. [Abstract] Abstract: 'performance,and' contains a missing space after the comma.
  2. [§3 (Encodings)] The manuscript should clarify how the overlaid marks (area and angle) are scaled and rendered within the limited cell space of the adjacency matrix to ensure reproducibility of the encodings.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and positive recommendation. We address the major comments point by point below, indicating where revisions will be made to the manuscript.

read point-by-point responses

  1. Referee: [§4 (Study Design)] §4 (Study Design): The eight analytical tasks are presented as generic probes, yet the abstract motivates the work with 'real-world networks'; without explicit mapping or validation showing that these tasks elicit the same perceptual and cognitive demands as those faced by domain experts inspecting bivariate edge distributions, the generalizability of the performance regimes to the motivating use cases is not demonstrated.

    Authors: The tasks were carefully chosen to represent key analytical operations for bivariate edge data in matrices, informed by the network visualization literature. To better connect them to real-world networks, we will revise §4 to include an explicit mapping of each task to example use cases from domains such as social networks and transportation systems. This addition will clarify how the tasks align with the perceptual demands of domain experts. A comprehensive validation study with experts is beyond the current scope but could be addressed in future work. revision: yes

  2. Referee: [§5 (Results)] §5 (Results): The abstract states that 'clear performance regimes' were observed but provides no details on the statistical tests, effect sizes, confidence intervals, or exclusion criteria used to establish differences between encodings; these elements are load-bearing for interpreting the stability claims (e.g., for angle-based marks) and should be foregrounded even if present in the full methods section.

    Authors: We agree that the abstract would benefit from more transparency on the statistical foundation. The results section details the preregistered analyses, including ANOVA, effect sizes, and confidence intervals, along with exclusion criteria. We will update the abstract to briefly reference these, for example by noting the significant effects and effect sizes that support the identified performance regimes. This change will strengthen the abstract without altering its overall length substantially. revision: yes

Circularity Check

0 steps flagged

No circularity: central claims rest on new experimental data

full rationale

The paper reports results from a preregistered crowdsourced study with 156 participants across eight analytical tasks. Performance rankings (area-based overlaid marks and bar charts highest, angle-based moderate, bivariate color lowest) are measured directly from participant accuracy, time, and subjective ratings rather than derived from equations, fitted parameters, or prior results. No mathematical derivations, self-referential uniqueness theorems, or ansatzes appear. Citations to prior work serve only to motivate encoding choices and are not load-bearing for the empirical ordering. The evaluation is self-contained against the collected data.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

As an empirical user study, the claims rest on experimental design choices rather than mathematical derivations. No free parameters or invented entities are introduced. The key assumptions concern task representativeness and participant generalizability.

axioms (2)
  • domain assumption The selected encodings are suitable for the limited space available in matrix cells
    The paper selects four encodings that prior work has suggested are suitable for compact cells.
  • domain assumption Crowdsourced participants can interpret the encodings in ways that generalize to target analyst populations
    The study relies on a general online participant pool rather than domain experts.

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Reference graph

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