[Paper Review] MLSEB: Edge Bundling using Moving Least Squares Approximation

Jieting Wu, Jianping Zeng+2 more|arXiv (Cornell University)|Sep 5, 2017

Data Visualization and Analytics39 references3 citations

TL;DR

This paper proposes MLSEB, a novel edge bundling method that uses moving least squares (MLS) approximation to generate high-quality, scalable graph visualizations with reduced distortion and clutter. By projecting edge sample points onto local regression curves based on neighborhood density, MLSEB achieves superior quality scores—outperforming FDEB and FFTEB—while maintaining computational efficiency suitable for large graphs.

ABSTRACT

Edge bundling methods can effectively alleviate visual clutter and reveal high-level graph structures in large graph visualization. Researchers have devoted significant efforts to improve edge bundling according to different metrics. As the edge bundling family evolve rapidly, the quality of edge bundles receives increasing attention in the literature accordingly. In this paper, we present MLSEB, a novel method to generate edge bundles based on moving least squares (MLS) approximation. In comparison with previous edge bundling methods, we argue that our MLSEB approach can generate better results based on a quantitative metric of quality, and also ensure scalability and the efficiency for visualizing large graphs.

Motivation & Objective

• To address the trade-off between clutter reduction and distortion in edge bundling for large graph visualization.
• To develop a scalable and efficient edge bundling method that maintains high visual quality.
• To introduce a new metric-based evaluation framework that balances readability and faithfulness in edge bundle quality.
• To leverage moving least squares approximation for local curve fitting to minimize distortion in bundled edges.
• To demonstrate that MLSEB outperforms existing methods like FDEB and FFTEB in quantitative quality assessment.

Proposed method

• MLSEB samples graph edges into a point cloud and applies moving least squares (MLS) projection to iteratively align points toward local regression curves based on neighborhood density.
• The method uses a local regression model to minimize distortion by projecting each point onto a curve that best fits its neighboring points.
• Edge bundling is achieved through iterative refinement: each point is projected toward the local MLS curve until convergence.
• The approach uses a distance-minimizing approximation function inspired by thinning unorganized point clouds into curve-like structures.
• A quality metric Q = ΔP / T̄ is used, where ΔP is pixel reduction and T̄ is average distortion per sample point, favoring high clutter reduction with low distortion.
• The method is implemented efficiently and runs at the same order of magnitude as the fastest existing method, FFTEB.

Experimental results

Research questions

• RQ1 Can moving least squares approximation produce edge bundles with lower distortion while maintaining high clutter reduction?
• RQ2 How does MLSEB compare quantitatively to FDEB and FFTEB in terms of visual quality, measured by the ratio of clutter reduction to distortion?
• RQ3 Does MLSEB scale effectively to large graphs with tens of thousands to millions of edges?
• RQ4 Can the proposed quality metric effectively capture the trade-off between readability and faithfulness in edge bundling results?
• RQ5 How does MLSEB preserve the mental map and structural fidelity of original graphs compared to force-directed and density-based methods?

Key findings

• MLSEB achieves the highest quality score Q across all tested datasets, with Q = 26.0 on the France airlines dataset, outperforming FDEB (3.7) and FFTEB (21.3).
• On the large US migrations dataset (545,881 edges), MLSEB maintains a high quality score of 13.3, comparable to FFTEB and significantly better than FDEB.
• For the US migrations graph (9,780 edges), MLSEB achieves Q = 9.20, surpassing FDEB (8.9) and FFTEB (7.60).
• MLSEB produces lower average distortion (T̄ = 0.88K for US migrations) than FDEB (1.10K) and FFTEB (1.2K), indicating better preservation of original edge structure.
• The method scales efficiently, with computational time on the same order of magnitude as FFTEB, the current fastest method.
• MLSEB demonstrates robustness on large graphs, maintaining high quality even when edge counts increase by orders of magnitude, unlike FDEB which suffers from high distortion at scale.