A Multilevel Force-Directed Graph Drawing Algorithm Using Multilevel Global Force Approximation Carl Crawford, Chris Walshaw, Alan Soper School of Computing and Mathematical Sciences University of Greenwich London, UK C.J.Crawford@gre.ac.uk, C.Walshaw@gre.ac.uk, A.J.Soper@gre.ac.uk Abstract—In this paper we discuss an efficiency saving for multilevel force directed placement algorithms. Typically such algorithms use a Barnes Hut octree (or sometimes a grid) in order to approximate global repulsive forces. Here we instead exploit the graph coarsening structure, already in place to facilitate the multilevel scheme, in order to provide a hierarchical approximation to the global forces. Not only is this more efficient, but also it takes better account of the graph structure than an octree or a grid. Keywords-graph; drawing; force directed placement; multilevel refinement I. INTRODUCTION Graph drawing is a technique used to represent collections of data in a graphical way so a reader can better understand the structure and relationships held within the information. Typical types of graphs drawn include website structures, communication networks, biological maps and various physical meshes. Large datasets are impossible to draw by hand [1] [2] [6] and therefore, graph drawing algorithms, which embed vertices in R² or R³, are used. Although research in this area has been fruitful, due to the complexity of calculations involved with generating a good drawing, many existing algorithms still struggle to reach a layout within a reasonable time frame. It is therefore our goal to decrease the runtime required without losing quality. II. A BRIEF HISTORY A graph G, consists of a number of vertices, V, representing data and a number of edges, E, of which each edge connects two vertices and describes a relationship between data elements. A graph drawing algorithm assigns a position in 2D or 3D space to each vertex, v, within V. A. Aesthetics The aesthetics of a graph drawing are the qualities of the layout which aid in the reading of the information. Research by Purchase [5] has shown that the aesthetics which help readers absorb information are; • fewer edge crossings; • high symmetry; • fewer bends and tangles Most algorithms developed in recent years aim to achieve one or more of these aesthetics criteria, in order to maximize the usefulness and readability of the drawings they compute. B. Force Directed Placement Force Directed Placement (FDP) is a method of drawing graphs which models the data as a mechanical system of connected points which have an energy based on their placement. High energy configurations result from vertices being too close together, or too far apart, and high energy is therefore assumed to represent poor layout (bad aesthetics). Consequently, force directed placement algorithms aim to find the lowest possible energy by iteratively changing the position of the vertices. 1) The Spring Embedder is the particular method of FDP used here, which models the placement dynamics using a mechanical system of rings and springs (vertices and edges respectively), which exhibits the spring-like forces to keep connected vertices from drifting too far apart, and repulsive forces to keep vertices drifting too close to one another – therefore reducing the energy of the graph. After an initial placement the graph is “released”, allowing the rings to move about freely until comfortable, low energy positions are found. The computationally intensive part of a spring embedder is the calculation of global repulsive forces, potentially an O(n²) calculation, which must take place every iteration. C. Multilevel Paradigm In larger graphs of more than 100 vertices, the FDP algorithms alone are not enough to find a good layout due to so many contradicting movements. Walshaw implemented a means of simplifying the structure of a graph by generating a series of approximations through coarsening [2] The result is a hierarchy of increasingly coarse graphs, each with approximately the same structural information as the previous, but with fewer vertices, each of which may represent several vertices in the original graph. These approximations to the original graph coarsely represent the original data provided for visualization. 2012 16th International Conference on Information Visualisation 1550-6037/12 $26.00 © 2012 IEEE DOI 10.1109/IV.2012.78 454