A multi-dimensional approach to force-directed layouts of large graphs

We present a novel hierarchical force-directed method for drawing large graphs. Given a graph G = (V, E), the algorithm produces an embedding for G in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higher-dimensional embedding into a two or three dimensional subspace of E. Such projections typically result in drawings that are "smoother" and more symmetric than direct drawings in 2D and 3D. In order to obtain fast placement of the vertices of the graph our algorithm employs a multi-scale technique based on a maximal independent set filtration of vertices of the graph. While most existing force-directed algorithms begin with an initial random placement of all the vertices, our algorithm attempts to place vertices "intelligently", close to their final positions. Other notable features of our approach include a fast energy function minimization strategy and efficient memory management. Our implementation of the algorithm can draw graphs with tens of thousands of vertices using a negligible amount of memory in less than one minute on a 550 MHz Pentium PC. (C) 2004 Elsevier B.V. All rights reserved.