Cell Size Optimization for Graph Drawing on Torus

Recent research revealed that embedding a node-link diagram in a flat torus avoids edge crossings, improves aesthetic metrics, and has advantages in tasks such as path tracking and understanding network structures. However, the effectiveness of torus layouts using graphs and the influence of the cell size, the size of the torus space used during torus layouts on the drawing results have not been investigated. In this study, we clarified that the optimal cell size that minimizes the stress of drawing results differs depending on the graph through computational experiments using benchmark graph data. Moreover, we uncovered that in graphs with enabled torus layouts, the stress function depending on the cell size, is close to an unimodal function. We focused on this unimodal property and proposed an algorithm to determine whether a torus is valid from the drawing results and the optimal cell size using the golden-section search. Reportedly, the aesthetic metrics of graphs with optimal cell sizes outperformed empirically used cell sizes through computational experiments.