# Designing a Glyph-Based Hierarchical Visualization for Orders of Magnitude Values

Katerina Batziakoudi, Ambre Assor, Christophe Bortolaso, Jean-Daniel Fekete

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Katerina Batziakoudi, Ambre Assor, Christophe Bortolaso, Jean-Daniel Fekete. Designing a Glyph-Based Hierarchical Visualization for Orders of Magnitude Values. Posters of IEEE VIS 2025 - Visualization & Visual Analytics, Nov 2025, Vienna, Austria. hal-05236011

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# Designing a Glyph-Based Hierarchical Visualization for Orders of Magnitude Values

Katerina Batziakoudi*
Université Paris-Saclay, CNRS,
Inria & Berger-Levrault

Ambre Assor†
Université Paris-Saclay,
CNRS, Inria

Christophe Bortolaso‡
Berger-Levrault

Jean-Daniel Fekete§
Université Paris-Saclay,
CNRS, Inria

<figure><img src="image_2.png" /></figure>

<table><caption>Figure 1: On the left is the glyph-based hierarchical visualization presented in this poster; on the right, a standard Icicle plot. Both represent the Education and Research budget of the French government. Yellow rectangles highlight two levels to illustrate how our approach facilitates reading and comparing values spanning several orders of magnitude within and across hierarchical levels.</caption><thead><tr><th>Education and Research</th><th>Education & Youth</th><th>School Education</th><th>Public Secondary Education</th><th>Middle School</th></tr></thead><tbody><tr><td>92,728,077,074</td><td>86,038,252,643</td><td>85,338,381,802</td><td>38,424,611,769</td><td>13,459,460,937</td></tr><tr><td></td><td></td><td></td><td></td><td>High School (Gen/Tech)<br>8,433,618,300</td></tr><tr><td></td><td></td><td></td><td></td><td>Vocational High School<br>8,432,078,420</td></tr><tr><td></td><td></td><td></td><td></td><td>Admin & Leadership<br>Post-Bacc in High School</td></tr><tr><td></td><td></td><td></td><td>Public Primary Education</td><td>Primary<br>13,184,623,426</td></tr><tr><td></td><td></td><td></td><td></td><td>Pre-primary<br>6,498,941,340</td></tr><tr><td></td><td></td><td></td><td></td><td>Special needs<br>Substitute Teaching</td></tr><tr><td></td><td></td><td></td><td>Private Primary & Secondary</td><td>Middle School (Private)</td></tr><tr><td></td><td></td><td></td><td></td><td>Student Life<br>7,941,021,922</td></tr><tr><td></td><td></td><td></td><td></td><td>Policy Support<br>School Life & Citizenship<br>Disability Inclusion</td></tr><tr><td></td><td>Higher Education & Research</td><td>Research & Higher Education</td><td>University Education & Research</td><td>Research<br>Undergrad (Bac-/License)</td></tr><tr><td></td><td>26,688,824,431</td><td>26,688,824,431</td><td>15,180,783,720</td><td>Master Level</td></tr><tr><td></td><td></td><td></td><td>Scientific Research</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>Student Life (Higher Ed)<br>Direct Aid</td></tr></tbody></table>

## ABSTRACT

We present a glyph-based hierarchical visualization, designed to support quantitative comparisons of values associated with each tree node spanning several orders of magnitude. Common hierarchical visualizations—such as Treemaps and Icicle plots struggle to represent such disparities, particularly across levels, and often rely on multi-view or interactive approaches that increase cognitive load by requiring users to recall values from previously seen nodes. To address these issues, we adopt a node-link layout and design glyphs for each node, inspired by symbolic number representation, to enable compact and comparable value representations in overview. We demonstrate the design on a sample of the French budget.

**Index Terms:** hierarchies, glyphs, orders of magnitude

## 1 INTRODUCTION

In this work, we present a glyph-based hierarchical visualization designed to support quantitative comparisons of values that span several orders of magnitude, both within and across hierarchical levels. A common example of such data is public budgets, which often contain Orders of Magnitude Values (OMVs)—values that differ by four or more orders of magnitude [1]. For instance, in the French national budget, each node in the hierarchy (ministry, program, action, sub-action) is associated with a monetary allocation, ranging from hundreds of billions to tens of thousands of euros, spanning seven orders of magnitude. Understanding these magni-

tude differences is essential for interpreting government priorities and contextualizing the budgeting proposals of different parties.

To support this kind of quantitative understanding, users must identify nodes and their hierarchical relationships, compare values both across and within branches, and detect outliers and similarities in overview. These tasks require multiple branches to be displayed simultaneously, with all relevant nodes visible. Common hierarchical visualization techniques can represent two or more levels of depth, either by inclusion, as in Treemaps and circle packing charts, or by adjacency, as in Icicle plots and Sunburst charts. However, the visibility of the displayed information is heavily influenced by the differences in orders of magnitude between nodes. Interaction techniques such as semantic or geometric zooming reallocate space to the selected hierarchical level, increasing the range of visible magnitudes. Yet these methods result in multi-view visualizations, requiring users to mentally recall previously seen nodes to compare across levels, thereby increasing cognitive load.

To address these issues, we focus on node-link representations, which support the selective display of multiple branches simultaneously, and we encode each node's value using custom-designed glyphs. Our approach draws on a visual metaphor inspired by how people symbolically express numbers, building on the concept of mantissa-exponent visualizations [1]. In this poster, we present the rationale behind our design, demonstrate its application within a node-link tree visualization of a sample from the French national budget (see Figure 1), and outline directions for future research.

## 2 VISUALIZATION DESIGN

### A node-link Tree for Comparisons Across Hierarchies

Node-link trees are a widely used and familiar representation of hierarchical structures, reflecting conventions established in domains such as genealogy, file systems, and biological taxonomy. Although researchers have criticized their suboptimal use of space compared to space-filling techniques such as Treemaps and Icicle plots, interaction techniques can mitigate these limitations [6]. For example,

<footnote>*e-mail: kbatziakoudi@gmail.com</footnote>

<footnote>†e-mail: ambre.assor@inria.fr</footnote>

<footnote>‡e-mail: christophe.bortolaso@berger-levrault.com</footnote>

<footnote>§e-mail: Jean-Daniel.Fekete@inria.fr</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 2: Glyph design process</figcaption></figure>

enabling users to expand or collapse branches dynamically allows for selective exploration, allocating space on demand to areas of interest. This flexibility is particularly beneficial for supporting comparisons across different hierarchical levels within a single view. However, while node-link trees are effective for navigating and understanding hierarchical structures, they are not inherently designed to support quantitative comparisons between nodes. Prior work has attempted to address this limitation by mapping quantitative values to nodes using visual variables such as size and color. Yet, these approaches are inadequate for supporting quantitative comparisons of OMVs: color intensity is impossible to discriminate over large ranges, and size encodings render smaller magnitudes invisible. To the best of our knowledge, no existing design effectively supports the visual encoding of values in a way that enables quantitative comparisons across nodes differing by several orders of magnitude. To address this gap, we design a glyph for each node, leveraging glyphs' ability to convey complex information in a compact and comparable format [2]. In the following paragraph, we explore the question: **How can we design a glyph that enables quantitative comparisons of OMVs within a node-link tree?**

**A Glyph for Orders of Magnitude Values** To design glyphs that effectively encode OMVs—making both large and small magnitudes visible and comparable—we draw on insights from numerical cognition [4] and visualization research [1]. We round values to reduce cognitive effort [5] and decompose OMVs into three components: (1) the **mantissa** and (2) the **exponent**, following scientific notation, and (3) **scale words** (e.g., billions, millions), which reflect the role of categorical information in aiding the perception of large numbers [4]. Figure 2 illustrates our design process using numbers spanning five orders of magnitude.

1. We begin with the piecewise linear EplusM scale, where exponents are evenly spaced along the axis—similar to a logarithmic scale—while the mantissa is linearly interpolated between adjacent exponents [1]. Bars are right-aligned to reflect how large numbers are typically read and written, with higher magnitude categories appearing to the left of lower ones.

2. We incorporate scale words using a redundant encoding strategy: faceting by column—aligned with how people group digits in threes for readability (e.g., using commas, as in 200,000,000)—and color hue and intensity, to enhance the pre-attentive perception of different magnitudes in overview while maintaining the ordinal perception of scale words.

3. We remove the axis and treat each bar as an independent glyph. The combination of scale word faceting and EplusM grid lines forms groups of rectangles, with each rectangle representing a distinct exponent. This design serves as a visual metaphor for digit grouping, preserving the familiar structure of multi-digit numbers. To ensure a consistent reference frame across glyphs, we allocate a fixed length based on the highest scale word and use gray to indicate missing exponents.

4. Finally, we divide the mantissa into discrete units and vertically stack them at the position corresponding to the exponent

of the value. Colored units encode the value of the mantissa, while gray units represent the number of missing steps to the next exponent, supporting a form of subitizing [3]. These design choices enable a more accurate and discriminable encoding of the mantissa, facilitating comparisons between values in the same and neighboring magnitudes.

## 3 CONCLUSION AND FUTURE WORK

In this poster, we leverage insights from numerical perception research [3, 4] and design guidelines for mantissa-exponent visualizations [1] to create glyphs that support quantitative comparisons of OMVs. We integrate these glyphs into a node-link tree layout to overcome the limitations of common hierarchical visualizations in enabling overview and quantitative comparisons for OMVs. We illustrate the technique using a sample of the French national budget. We are currently working on refining the design, collecting user feedback, and planning user studies to evaluate the effectiveness of the proposed visualization in supporting quantitative understanding of budget data and identifying potential limitations. To demonstrate our approach at a real-world data scale and facilitate community adoption, further refinement, and evaluation, we provide an Observable notebook containing the full hierarchy of the French budget, our glyph-based hierarchical visualization, and additional baseline designs—including two treemaps (one nested, one zoomable) and an icicle plot. These alternatives are included to highlight differences in how each technique supports the visibility and comparison of OMVs, both within and across hierarchical levels.

## REFERENCES

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[2] J. Fuchs, P. Isenberg, A. Bezerianos, and D. A. Keim. A systematic review of experimental studies on data glyphs. *IEEE Trans. Vis. Comput. Graph.*, 23(7):1863–1879, 2017. doi: 10.1109/TVCG.2016.2549018 2

[3] E. L. Kaufman and M. W. Lord. The discrimination of visual number. *Am J Psychol.*, 62(4):498–525, Oct. 1949. 2

[4] D. Landy, A. Charlesworth, and E. Ottmar. Categories of Large Numbers in Line Estimation. *Cognitive Science*, 41(2):326–353, 2017. doi: 10.1111/cogs.12342 2

[5] H. A. Nguyen, J. M. Hofman, and D. G. Goldstein. Round numbers can sharpen cognition. In *Proceedings of the 2022 CHI Conference on Human Factors in Computing Systems*, CHI ’22. Association for Computing Machinery, New York, NY, USA, 2022. doi: 10.1145/3491102.3501852 2

[6] C. Plaisant, J. Grosjean, and B. B. Bederson. Spacetree: Supporting exploration in large node link tree, design evolution and empirical evaluation. In P. C. Wong and K. Andrews, eds., *2002 IEEE Symposium on Information Visualization (InfoVis 2002)*, 27 October - 1 November 2002, *Boston, MA, USA*, pp. 57–64. IEEE Computer Society, 2002. doi: 10.1109/INFVIS.2002.1173148 1