Graph Drawing Stress Model with Resistance Distances

Abstract

This paper challenges the convention of using graph-theoretic shortest distance in stress-based graph drawing. We propose a new paradigm based on resistance distance, derived from the graph Laplacian's spectrum, which better captures global graph structure. This approach overcomes theoretical and computational limitations of traditional methods, as resistance distance admits a natural isometric embedding in Euclidean space. Our experiments demonstrate improved neighborhood preservation and cluster faithfulness. We introduce Omega, a linear-time graph drawing algorithm that integrates a fast resistance distance embedding with random node-pair sampling for Stochastic Gradient Descent (SGD). This comprehensive random sampling strategy, enabled by efficient pre-computation of resistance distance embeddings, is more effective and robust than pivot-based sampling used in prior algorithms, consistently achieving lower and more stable stress values. The algorithm maintains $O(|E|)$ complexity for both weighted and unweighted graphs. Our work establishes a connection between spectral graph theory and stress-based layouts, providing a practical and scalable solution for network visualization.