We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE finds an optimal drawing by minimizing a quadratic energy function due to Hall, using a novel algebraic multigrid technique. The algorithm exhibits an improvement of something like two orders of magnitude over the fastest algorithms we are aware of; it draws graphs of a million nodes in less than a minute. Moreover, the algorithm can deal with more general entities, such as graphs with masses and negative weights (to be defined in the text), and it appears to be applicable outside of graph drawing too.

. We report on our experiments with five graph drawing algorithms for general undirected graphs. These are the algorithms FR introduced by Fruchterman and Reingold [5], KK by Kamada and Kawai [11], DH by Davidson and Harel [1], Tu by Tunkelang [13] and GEM by Frick, Ludwig and Mehldau [6]. Implementations of these algorithms have been integrated into our GraphEd system [9]. We have tested these algorithms on a wide collection of examples and with different settings of parameters. Our examples are from original papers and by our own. The obtained drawings are evaluated both empirically and by GraphEd's evaluation toolkit. As a conclusion we can confirm the reported good behaviour of the algorithms. Combining time and quality we recommend to use GEM or KK first, then FR and Tu and finally DH. 1 Introduction Graph drawing has become an important area of research in Computer Science. There is a wide range of applications including data structures, data bases, software engineering, VLSI te...

Dynamic graph layout refers to the layout of graphs that change over time. These changes are due to user interaction, algorithms, or other underlying processes determining the graph. Typically, users spend a noteworthy amount of time to get familiar with a layout, i.e.

Networks have remained a challenge for information visualization designers because of the complex issues of node and link layout coupled with the rich set of tasks that users present. This paper offers a strategy based on two principles: (1) layouts are based on user-defined semantic substrates, which are non-overlapping regions in which node placement is based on node attributes, (2) users interactively adjust sliders to control link visibility to limit clutter and thus ensure comprehensibility of source and destination. Scalability is further facilitated by user control of which nodes are visible. We illustrate our semantic substrates approach as implemented in NVSS 1.0 with legal precedent data for up to 1122 court cases in three regions with 7645 legal citations.

The vertices of most graphs that appear in real applications are nonuniform. They can be circles, ellipses, rectangles, or other geometric elements of varying shapes and sizes. Unfortunately, current force directed methods for laying out graphs are suitable mostly for graphs whose vertices are zero-sized and dimensionless points. It turns out that naively extending these methods to handle nonuniform vertices results in serious deficiencies in terms of output quality and performance. In this paper we try to remedy this situation by identifying the special characteristics and problematics of such graphs and offering several algorithms for tackling them. The algorithms can be viewed as carefully constructed extensions of force-directed methods, and their output quality and performance are similar.

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.

This paper describes a multi-perspective software visualization environment, SHriMP, which combines single view and multi-view techniques to support software exploration at both the architectural and source code levels. SHriMP provides three different views: a primary nested view and two subsidiary views. The primary nested view employs fisheye views of nested graphs, provides contextual cues, and supports general exploration activities. In SHriMP, subsidiary views exist as a searching tool and a relation tracer. These views complement each other and allow programmers to examine a software system from multiple perspectives.

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, e#cient 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. 1 Introduction Graphs are common in many applications, from data structures to networks, from software engineering...

Sensor network applications frequently require that the sensors know their physical locations in some global coordinate system. This is usually achieved by equipping each sensor with a location measurement device, such as GPS. However, low-end systems or indoor systems, which cannot use GPS, must locate themselves based only on crude information available locally, such as inter-sensor distances. We show how a collection of sensors, capable only of measuring distances to close neighbors, can compute their locations in a purely distributed manner, i.e. where each sensor communicates only with its neighbors. This can be viewed as a distributed graph drawing algorithm. We experimentally show that our algorithm consistently produces good results under a variety of simulated real-world conditions, and is relatively robust to the presence of noise in the distance measurements.

We propose a novel visualization approach that facilitates graphical exploration and communication of relative actor status in social networks. The main idea is to map, in a drawing of the entire network, actor status scores to vertical coordinates. The resulting problem of determining horizontal positions of actors and routing of connecting lines such that the overall layout is readable is algorithmically difficult, yet well-studied in the literature on graph drawing. We outline a customized approach. The advantages