COAST: A Convex Optimization Approach to Stress-Based Embedding

TL;DR

This paper introduces COAST, a convex optimization method using semi-definite programming and eigenvector re-parameterization to generate graph layouts that respect non-uniform edge lengths efficiently.

Contribution

It reformulates stress-based graph embedding as a convex problem and proposes a scalable SDP-based approach with a novel eigenvector re-parameterization.

Findings

Method scales well to large graphs

Produces reasonable layouts with edge length constraints

Outperforms traditional force-based approaches in constrained scenarios

Abstract

Visualizing graphs using virtual physical models is probably the most heavily used technique for drawing graphs in practice. There are many algorithms that are efficient and produce high-quality layouts. If one requires that the layout also respect a given set of non-uniform edge lengths, however, force-based approaches become problematic while energy-based layouts become intractable. In this paper, we propose a reformulation of the stress function into a two-part convex objective function to which we can apply semi-definite programming (SDP). We avoid the high computational cost associated with SDP by a novel, compact re-parameterization of the objective function using the eigenvectors of the graph Laplacian. This sparse representation makes our approach scalable. We provide experimental results to show that this method scales well and produces reasonable layouts while dealing with the edge length constraints.