Initial Placement for Fruchterman--Reingold Force Model With Coordinate Newton Direction

The Fruchterman--Reingold (FR) force model is widely used in force-directed graph drawing, and multilevel approaches such as sfdp in Graphviz scale these methods effectively. A crucial step in multilevel schemes is refinement, which improves the graph layout typically performed with the FR algorithm. For this refinement, optimization-based methods, such as L-BFGS, often yield better layouts by exploiting past gradient information. However, they suffer from high per-iteration costs for large graphs and difficulty in handling highly twisted graphs. In this research, we propose a new initial placement based on the stochastic coordinate descent to accelerate the optimization process. We first reformulate the problem as a discrete optimization problem using a hexagonal lattice and then iteratively update a randomly selected vertex along the coordinate Newton direction with low per-iteration costs. We demonstrate the effectiveness of our method through numerical experiments, showing that our initial placement leads to faster convergence and higher-quality layouts compared to naive optimization approaches. We also discuss how the coordinate Newton direction can be applied to other graph-related problems from an optimization perspective.