Abstract — MatrixExplorer is a network visualization system that uses two representations: node-link diagrams and matrices. Its design comes from a list of requirements formalized after several interviews and a participatory design session conducted with social science researchers. Although matrices are commonly used in social networks analysis, very few systems support the matrix-based representations to visualize and analyze networks. MatrixExplorer provides several novel features to support the exploration of social networks with a matrix-based representation, in addition to the standard interactive filtering and clustering functions. It provides tools to reorder (layout) matrices, to annotate and compare findings across different layouts and find consensus among several clusterings. MatrixExplorer also supports Node-link diagram views which are familiar to most users and remain a convenient way to publish or communicate exploration results. Matrix and node-link representations are kept synchronized at all stages of the exploration process. Index Terms — social networks visualization, node-link diagrams, matrix-based representations, exploratory process, matrix ordering, interactive clustering, consensus. Fig. 1. MatrixExplorer showing two synchronized representations of the same network: matrix on the left and node-link on the right. 1

The minimum linear arrangement problem is widely used and studied in many practical and theoretical applications. In this paper we present a linear-time algorithm for the problem inspired by the algebraic multigrid approach which is based on weighted edge contraction rather than simple contraction. Our results turned out to be better than every known result in almost all cases, while the short running time of the algorithm enabled experiments with very large graphs.

Abstract—Current computer architectures employ caching to improve the performance of a wide variety of applications. One of the main characteristics of such cache schemes is the use of block fetching whenever an uncached data element is accessed. To maximize the benefit of the block fetching mechanism, we present novel cache-aware and cache-oblivious layouts of surface and volume meshes that improve the performance of interactive visualization and geometric processing algorithms. Based on a general I/O model, we derive new cache-aware and cache-oblivious metrics that have high correlations with the number of cache misses when accessing a mesh. In addition to guiding the layout process, our metrics can be used to quantify the quality of a layout, e.g. for comparing different layouts of the same mesh and for determining whether a given layout is amenable to significant improvement. We show that layouts of unstructured meshes optimized for our metrics result in improvements over conventional layouts in the performance of visualization applications such as isosurface extraction and view-dependent rendering. Moreover, we improve upon recent cache-oblivious mesh layouts in terms of performance, applicability, and accuracy. Index Terms—Mesh and graph layouts, cache-aware and cache-oblivious layouts, metrics for cache coherence, data locality. 1

High-dimensional collections of 0-1 data occur in many applications. The attributes in such data sets are typically considered to be unordered. However, in many cases there is a natural total or partial order # underlying the variables of the data set. Examples of variables for which such orders exist include terms in documents, courses in enrollment data, and paleontological sites in fossil data collections. The observations in such applications are flat, unordered sets; however, the data sets respect the underlying ordering of the variables. By this we mean that if A # B # C are three variables respecting the underlying ordering #, and both of variables A and C appear in an observation, then, up to noise levels, variable B also appears in this observation. Similarly, if A1 # A2 # # A l-1 # A l is a longer sequence of variables, we do not expect to see many observations for which there are indices i < j < k such that A i and Ak occur in the observation but A j does not.

In this paper we introduce a simple probabilistic model, hierarchical tiles, for 0-1 data. A basic tile (X,Y,p) specifies a subset X of the rows and a subset Y of the columns of the data, i.e., a rectangle, and gives a probability p for the occurrence of 1s in the cells of X x Y. A hierarchical tile has additionally a set of exception tiles that specify the probabilities for subrectangles of the original rectangle. If the rows and columns are ordered and X and Y consist of consecutive elements in those orderings, then the tile is geometric; otherwise it is combinatorial. We give a simple randomized algorithm for finding good geometric tiles. Our main result shows that using spectral ordering techniques one can find good orderings that turn combinatorial tiles into geometric tiles. We give empirical results on the performance of the methods.

Linear ordering problems are combinatorial optimization problems which deal with the minimization of different functionals in which the graph vertices are mapped onto (1, 2,..., n). These problems are widely used and studied in many practical and theoretical applications. In this paper we present a variety of linear-time algorithms for these problems inspired by the Algebraic Multigrid approach which is based on weighted edge contraction. The experimental result for four such problems turned out to be better than every known result in almost all cases, while the short running time of the algorithms enables testing very large graphs.

We present an algorithm for drawing directed graphs, which is based on rapidly solving a unique one-dimensional optimization problem for each of the axes. The algorithm results in a clear description of the hierarchy structure of the graph. Nodes are not restricted to lie on fixed horizontal layers, resulting in layouts that convey the symmetries of the graph very naturally. The algorithm can be applied without change to cyclic or acyclic digraphs, and even to graphs containing both directed and undirected edges. We also derive a hierarchy index from the input digraph, which quantitatively measures its amount of hierarchy.

We describe a novel approach to the visualization of hierarchical clustering that superimposes the classical dendrogram over a fully synchronized low-dimensional embedding, thereby gaining the benefits of both approaches. In a single image one can view all the clusters, examine the relations between them and study many of their properties. The method is based on an algorithm for lowdimensional embedding of clustered data, with the property that separation between all clusters is guaranteed, regardless of their nature. In particular, the algorithm was designed to produce embeddings that strictly adhere to a given hierarchical clustering of the data, so that every two disjoint clusters in the hierarchy are drawn separately.

In this paper we introduce a direct motivation for solving the minimum 2-sum problem, for which we present a linear-time algorithm inspired by the Algebraic Multigrid approach which is based on weighted edge contraction. Our results turned out to be better than previous results, while the short running time of the algorithm enabled experiments with very large graphs. We thus introduce a new benchmark for the minimum 2-sum problem which contains 66 graphs of various characteristics. In addition, we propose the straightforward use of a part of our algorithm as a powerful local reordering method for any other (than multilevel) framework.

Abstract. The construction of linear mesh layouts has found various applications, such as implicit mesh filtering and mesh streaming, where a variety of layout quality criteria, e.g., width and span, can be considered. Similar linear sequencing problems have also been studied in the context of sparse matrix reordering and graph labeling, where width and span correspond to vertex separation and bandwidth, respectively. One of the best-known heuristics for generating width-minimizing orderings is spectral sequencing, which is derived from the Fiedler vector. In terms of span however, other heuristics, such as the Cuthill-Mckee (CM) scheme, generally outperform spectral sequencing. In this paper, we study the general linear sequence generation as a problem of preserving graph distances and propose to use for sequencing the subdominant eigenvector of a kernel (affinity) matrix, defined by graph distances and appropriately chosen transfer functions. The use of Laplacians can then be seen as a special case, where a step transfer function of unit width is applied. Despite the non-sparsity of the kernel matrix we use, the sequences can be computed efficiently for problems of large size through subsampling and eigenvector extrapolation. When applied to mesh layouts generation, we show experimentally that the sequences obtained using our algorithm outperform those derived from the Fiedler vector, in terms of spans, and those obtained from CM, in terms of widths and other important quality criteria. Therefore, in applications where several such quality criteria can influence algorithm performance simultaneously, e.g., mesh streaming and implicit mesh filtering, the new mesh layouts could potentially provide a better trade-off. 1