We present a multi-scale layout algorithm for the aesthetic drawing of undirected graphs with straight-line edges. The algorithm is extremely fast, and is capable of drawing graphs that are substantially larger than those we have encountered in prior work. For example, the paper contains a drawing of a graph with over 15,000 vertices. Also we achieve "nice" drawings of 1000 vertex graphs in about 1 second. The proposed algorithm embodies a new multi-scale scheme for drawing graphs, which was motivated by the earlier multi-scale algorithm of Hadany and Harel [HH99]. In principle, it could significantly improve the speed of essentially any force-directed method (regardless of that method's ability of drawing weighted graphs or the continuity of its cost-function).

Recent years have seen an immense increase in the complexity of geometric data sets. Today's gigabyte-sized polygon models can no longer be completely loaded into the main memory of common desktop PCs. Unfortunately, current mesh formats do not account for this. They were designed years ago when meshes were orders of magnitudes smaller. Using such formats to store large meshes is inefficient and unduly complicates all subsequent processing.

We present a novel approach to the aesthetic drawing of undirected graphs. The method has two phases: first embed the graph in a very high dimension and then project it into the 2-D plane using PCA. Experiments we have carried out show the ability of the method to draw graphs of 10 nodes in few seconds. The new method appears to have several advantages over classical methods, including a significantly better running time, a useful inherent capability to exhibit the graph in various dimensions, and an effective means for interactive exploration of large graphs.

The spectral approach for graph visualization computes the layout of a graph using certain eigenvectors of related matrices. Some important advantages of this approach are an ability to compute optimal layouts (according to specific requirements) and a very rapid computation time. In this paper we explore spectral visualization techniques and study their properties. We present a novel view of the spectral approach, which provides a direct link between eigenvectors and the aesthetic properties of the layout. In addition, we present a new formulation of the spectral drawing method with some aesthetic advantages. This formulation is accompanied by an aesthetically-motivated algorithm, which is much easier to understand and to implement than the standard numerical algorithms for computing eigenvectors.

We introduce an energy model whose minimum energy drawings reveal the clusters of the drawn graph. Here a cluster is a set of nodes with many internal edges and few edges to nodes outside the set. The drawings of the bestknown force and energy models do not clearly show clusters for graphs whose diameter is small relative to the number of nodes. We formally characterize the minimum energy drawings of our energy model. This characterization shows in what sense the drawings separate clusters, and how the distance of separated clusters to the other nodes can be interpreted.

Finding a linear ordering of the vertices of a graph is a common problem arising in diverse applications. In this paper we present a linear-time algorithm for this problem, based on the multi-scale paradigm. Experimental results are similar to those of the best known approaches, while the running time is significantly better, enabling it to deal with much larger graphs. The paper contains a general multi-scale construction, which may be used for a broader range of ordering problems.

We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE exhibits a vast improvement over the fastest algorithms we are currently aware of; using a serial PC, it draws graphs of millions of nodes in less than a minute. ACE finds an optimal drawing by minimizing a quadratic energy function. The minimization problem is expressed as a generalized eigenvalue problem, which is solved rapidly using a novel algebraic multigrid technique. The same generalized eigenvalue problem seems to come up also in other fields, hence ACE appears to be applicable outside graph drawing too.

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We present a novel sampling-based approximation technique for classical multidimensional scaling that yields an extremely fast layout algorithm suitable even for very large graphs. It produces layouts that compare favorably with other methods for drawing large graphs, and it is among the fastest methods available. In addition, our approach allows for progressive computation, i.e. a rough approximation of the layout can be produced even faster, and then be refined until satisfaction.

Methods for ranking World Wide Web resources according to their position in the link structure of the Web are receiving considerable attention, because they provide the first e#ective means for search engines to cope with the explosive growth and diversification of the Web. Closely related methods have been used in other disciplines for quite some time.