Drawing Graphs by Eigenvectors: Theory and Practice*

K e y w o r d s G r a p h drawing, Laplaclan, Eigenvectors, Fledler vector, Force-directed layout, Spectral graph theory 1. I N T R O D U C T I O N A graph G(V, E) is an abstract structure that is used to model a relation E over a set V of entities. Graph drawing is a standard means for visualizing relational information, and its ultimate usefulness depends on the readability of the resulting layout, that is, the drawing algorithm's ability to convey the meaning of the diagram quickly and clearly. To date, many approaches to graph drawing have been developed [2,3]. There are many kinds of graph-drawing problems, such as drawing di-graphs, drawing planar graphs, and others. Here, we investigate the problem of drawing undirected graphs with straight-line edges. In fact, the methods that we utilize are not limited to traditional graph drawing and are intended, also, for general low dimensional visualization of a set of objects according to their pairwise similarities (see, e.g., Figure 1). We have focused on spectral graph drawing methods, which construct the layout using eigenvectors of certain matrices associated with the graph To get some feeling, we provide results for three graphs in Figure 2. This spectral approach is quite old, originating with the work of Hall [4] in 1970. However, since then, it has not been used much. In fact, spectral graph drawing algorithms are almost absent in graph-drawing literature (e.g., they are not mentioned in the two books [2,3] that deal with graph drawing). It seems that, in most visuahzation research, the spectral approach is difficult to grasp, in terms of aesthetics. Moreover, the numerical algorithms for computing the eigenvectors do not possess an intuitive aesthetic interpretation. *An early and short version of this work appeared m [1] 0898-1221/05/$ see front matter (~) 2005 Elsevmr Ltd All rights reserved Typeset by .AMe-TEX dol.1