Graph-of-Text View

• Graph-of-text view is a paradigm where textual data is modeled as nodes and edges, supporting integrated visual analytics and NLP applications.
• It employs unified overlap removal using Delaunay triangulation and iterative optimization to balance label clarity with structural preservation.
• The approach is effective in visual analytics, geographical mapping, and complex network analysis, ensuring compact layouts and straight edge representation.

A graph-of-text view refers to the explicit or implicit representation of textual data—words, sentences, or labels—as topological elements of a graph, enabling integrated visual analytic, structural, or algorithmic processing. In this paradigm, textual information is associated with nodes and/or edges, which may contain literal text, semantic annotations, or linguistic features. The graph-of-text approach is foundational in visual analytics, NLP, and graph learning, where the dual challenges are maintaining legibility of textual components and preserving the structural fidelity of the original graph. This article provides a comprehensive overview of the principles, algorithms, optimization strategies, and application domains underpinning the graph-of-text view.

1. Unified Overlap Removal Algorithm for Node and Edge Labels

The unified overlap removal framework solves the challenge of preventing text or graphic overlaps in graphs by leveraging a proximity stress model constructed from the Delaunay triangulation of node positions. Each proximity edge $(i, j)$ receives an overlap factor

$$t_{ij} = \max\left(\min\left(\frac{W_i + W_j}{|x_i^0(1) - x_j^0(1)|}, \frac{H_i + H_j}{|x_i^0(2) - x_j^0(2)|}\right), 1\right)$$

where $W_i$ and $H_i$ denote half-widths and half-heights, and $x_i^0$ are node coordinates. The layout optimization seeks to minimize

$$\text{Stress} = \sum_{(i,j)\in E_p} w_{ij}(\|x_i - x_j\| - d_{ij})^2$$

with $d_{ij}$ incorporating the overlap factor and a capped scaling $s_{ij} = \min(t_{ij}, s_\text{max})$ (often $s_\text{max}=1.5$), and $w_{ij}=1/d_{ij}^2$.

When edge labels are present, the vPRISM algorithm augments the graph with additional "label nodes" but can induce edge bends. The ePRISM variant addresses this by enforcing label nodes to remain close to edge midpoints via the penalty term

$$\rho\sum_{k \in V_e; \{i,k\},\{j,k\}\in E_a} w_k\,\|x_k - (x_i + x_j)/2\|^2$$

with $\rho=4$, $w_k=1/\|x_i-x_j\|^2$. The final quadratic system

$(L_w + \rho L_e)x = L_{w,d} x^0$

is solved iteratively, ensuring straight edges and minimal layout deformation.

2. Initial Layout Strategies vs. Post-Processing

Two scenarios are distinguished for label placement:

• Initial Layout Incorporation: Some systems prevent overlaps by incorporating label sizes during initial node placement, but this can lead to substantial area expansion and non-practical visuals.
• Post-Processing Removal: The PRISM/ePRISM strategy applies overlap removal after computing an initially aesthetic, structurally informative layout, thereby preserving the local geometry and wider arrangement of the graph. The proximity graph scaffolding ensures structural retention.

This methodological distinction underscores the importance of preserving both global shape and local arrangement when introducing textual information into graphs.

3. Structural Information Retention

Preserving the original structural semantics of the graph is paramount when merging textual labels with topological features:

• Proximity Graph Constraint: Using the Delaunay triangulation provides local neighborhood anchors, reducing unstructured drift during node movement.
• Stress Function Calibration: Ideal edge lengths are set close to original values, unless forced to expand by overlap, keeping inter-node relationships intact.
• Penalty for Edge Labels: In ePRISM, centering edge labels preserves the visual straightness of edges and supports accurate interpretation of graph relationships.

This strategy ensures that information-rich, annotated graphs retain interpretability across both macro (graphwide) and micro (subgraph, local) scales.

4. Area Efficiency in Text-Enriched Graph Layouts

Minimizing additional area required for label accommodation is achieved through:

• Controlled Expansion: Dampening the overlap factor limits excessive stretching, preserving compactness.
• Iterative Solvers: Successive adjustments minimize total distortion at each iteration, balancing overlap elimination with area preservation.

Table: Comparison of area growth and edge bend for methods

Method	Area Expansion	Edge Bend Severity
Initial	High	Minimal
PRISM	Moderate	Possible
ePRISM	Minimal	Minimal

Maintaining a compact and clear graph is crucial for downstream analysis, especially in visualization-rich domains.

5. Edge Straightness and Label Placement

Edge straightness is a key measure of visual clarity:

• Treating edge labels as ordinary nodes (vPRISM) can lead to pronounced bends, reducing interpretability.
• The penalty term in ePRISM aligns edge labels to midpoints, mathematically pulling positions to near colinearity, which is substantiated by fewer, milder bends in experimental layouts (cf. Olympic torch relay and country relationship graphs in the paper).

This design choice is validated via practical visualizations, demonstrating its utility for complex, data-rich graphs.

6. Applications, Use Cases, and Implications

The unified algorithm is applicable across domains requiring the merging of textual data with graph structures:

• Visual Analytics: Overlap removal and edge straightness directly improve legibility in networks with rich annotation (e.g., social graphs, transport maps, bioinformatics).
• Geographical and Relationship Mapping: As shown in the Olympic torch relay and country relationship use cases, dense topological structures containing textual information remain interpretable post-layout adaptation.
• Complex Information Spaces: In social network visualization, urban infrastructure, or intercity connection analysis, preservation of structure and text clarity supports both exploratory analysis and decision-making tasks.

The approach is adaptable wherever both structural fidelity and textual clarity are necessary requisites for effective graph-based reasoning.

7. Significance in Graph Visualization Research

The proximity-based stress model, with its capacity for synchronous node/edge label overlap removal, represents a foundational advance for graph-of-text applications. Its iterative optimization, combined with explicit retention of geometric and relational properties, enables robust, interpretable visualizations of information-rich, annotated graphs. The algorithm’s innovations in penalty modeling for label placement and straight-line preservation are substantiated with both formal evaluation and visual demonstration, confirming its centrality in the landscape of technical graph visualization methodologies (0911.0626).