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 simulation-based algorithm. For this refinement, we can utilize optimization-based methods such as L-BFGS, or combine them with initial placement methods such as Simulated Annealing to achieve better layouts. However, they have several limitations, such as suffering from high per-iteration costs for large graphs or having difficulty with weighted and structurally complex graphs, leaving room for improvement. In this research, we propose a new initial placement based on 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 applications of our method, such as drawing for Hooke--Coulomb and Eades force models.