[Paper Review] INKA: An Ink-based Model of Graph Visualization

Quan Nguyen|arXiv (Cornell University)|Jan 22, 2018

Data Visualization and Analytics23 references3 citations

TL;DR

This paper introduces INKA, an ink-based model that mathematically formalizes the relationship between layout factors (edge crossings, total edge length, drawing area) and rendering factors (node size, edge width) in graph visualization. By modeling ink usage as a function of these factors, INKA enables quantitative assessment of ink-effectiveness and clarity trade-offs, offering a foundation for balanced, readable, and render-optimized graph layouts.

ABSTRACT

Common quality metrics of graph drawing have been about the readability criteria, such as small number of edge crossings, small drawing area and small total edge length. Bold graph drawing considers more realistic drawings consisting of vertices as disks of some radius and edges as rectangles of some width. However, the relationship that links these readability criteria with the rendering criteria in node-link diagrams has still not been well-established. This paper introduces a model, so-called INKA (Ink-Active), that encapsulates mathematically the relationship between all common drawing factors. Consequently, we investigate our INKA model on several common drawing algorithms and real-world graphs.

Motivation & Objective

• To establish a formal mathematical relationship between common layout quality metrics and rendering parameters in graph visualization.
• To address the gap in understanding how abstract readability criteria (e.g., low crossings, short edges) interact with physical rendering factors (e.g., node size, edge width).
• To propose a model that quantifies ink usage in graph drawings to guide the design of more effective and perceptually clear visualizations.
• To provide a framework for evaluating and optimizing graph layouts based on both aesthetic and rendering considerations.

Proposed method

• Proposes the INKA model (Ink-Active) as a mathematical framework to express total ink usage in graph drawings based on node radius (r), edge width (w), and layout characteristics.
• Defines total ink as the sum of ink used for nodes (clarity(V)) and edges (clarity(E)), minus overlap to account for ambiguity.
• Derives the ink difference equation for varying edge width: $ ink(D') - ink(D) = (w' - w)[L - 2mr - (w + w') imes cr(D)] $, showing dependency on edge length and crossings.
• Derives the ink difference for varying node radius: $ ink(D') - ink(D) = n heta(r'^2 - r^2) $, revealing quadratic impact on ink usage.
• Introduces the concept of 'ink-effectiveness' as the inverse of data-ink ratio, emphasizing minimal ink use for maximum information clarity.
• Uses the INKA-total equation and INKA-area inequality to formalize constraints linking layout and rendering factors.

Experimental results

Research questions

• RQ1 How do node size and edge width affect the total ink usage in a graph drawing, and what is the trade-off between ink minimization and visual clarity?
• RQ2 What is the mathematical relationship between layout factors (edge crossings, total edge length, drawing area) and rendering factors (node radius, edge width) in graph visualization?
• RQ3 Can a unified model quantify the combined impact of layout and rendering choices on the perceptual quality and ink efficiency of a graph drawing?
• RQ4 How do changes in edge width or node size influence the ambiguity (overlap) and readability of a graph layout?
• RQ5 To what extent can the INKA model guide the optimization of graph layouts for both aesthetic quality and rendering efficiency?

Key findings

• Increasing node radius quadratically increases total ink usage, as shown by $ ink(D') - ink(D) = n heta(r'^2 - r^2) $, indicating a strong trade-off between node visibility and ink efficiency.
• Thicker edges may reduce ink-effectiveness if they increase overlap, especially in drawings with many edge crossings, as per the edge width difference equation.
• The model reveals that minimizing total ink can reduce clarity and increase ambiguity, particularly when overlap is high, highlighting a fundamental trade-off between efficiency and readability.
• For a fixed layout, the total ink remains unchanged between two drawings if $ L - 2mr = (w + w') imes cr(D) $, indicating a balance point between edge length, node size, and crossings.
• The INKA model successfully approximates total ink usage before rendering, providing a fast and practical tool for evaluating layout feasibility.
• Empirical evaluation on real-world graphs shows that the INKA model supports meaningful comparisons across layout algorithms and reveals insights into rendering trade-offs.