A Multi-dimensional Approach to Force-Directed Layouts of Large Graphs

Abstract

Abstract. We present a novel hierarchical force-directed method for drawing large graphs. The algorithm produces a graph embedding 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. Projecting high-dimensional drawings onto two or three dimensions often results in drawings that are "smoother" and more symmetric. Among the other notable features of our approach are the utilization of a maximal independent set filtration of the set of vertices of a graph, a fast energy function minimization strategy, efficient memory management, and an intelligent initial placement of vertices. 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 mid-range PC.

Large-Graph Layout Algorithms at Work: An Experimental Study

“…Gajer and Kobourov [9] and Gajer et al [8] developed the force-directed multilevel algorithm GRIP. In general, multilevel algorithms are based on two phases.…”

“…However, in order to generate drawings of graphs that contain thousands or hundreds of thousands of vertices more efficient force-directed techniques have been developed [21,20,9,8,13,23,12,11]. Besides fast force-directed algorithms other very fast methods for drawing large graphs (see e.g., [14,17]) have been invented.…”

“…In particular, we investigate the force-directed algorithm GRIP of Gajer and Kobourov [9] and Gajer et al [8], the Fast Multi-scale Method (FMS) of Harel and Koren [13], and the Fast Multipole Multilevel Method (FM 3 ) of Hachul and Jünger [12,11]. We did not test other efficient force-directed methods like FADE of Quigley and Eades [20] and JIGGLE of Tunkelang [21] since the original implementations were not available to us or the implementations did not allow to import test graphs, respectively.…”

Force-Directed Lombardi-Style Graph Drawing

“…So, rather than review all of this previous related work, let us refer the reader to some of the excellent books and articles that survey the subject (e.g., see [1,2,7,[17][18][19]), and focus here on the most related papers on this topic. Holten and van Wijk [24] give a force-directed method for producing an edge-bundled drawing that is similar to a confluent drawing.…”

“…Force-directed layout algorithms, also known as "spring embedders," are known for the balanced types of drawings they produce, in terms of vertex and edge placement, using straight-line edges (e.g., see [1,2,7,[17][18][19]). Still, straight-line segments rarely occur in nature; hence, it is not clear that humans prefer straight-line segments for the sake of graph readability, and, indeed, the work of Mark Lombardi suggests that they don't.…”