One-Dimensional Graph Drawing: Part I Drawing Graphs by Axis Separation

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1 One-Dimensional Graph Drawing: Part I Drawing Graphs by Axis Separation Yehuda Koren and David Harel Dept. of Computer Science and Applied Mathematics The Weizmann Institute of Science, Rehovot, Israel {yehuda,dharel}@wisdom.weizmann.ac.il Abstract. In this paper we discuss a useful family of graph drawing algorithms, characterized by their ability to draw graphs in one dimension. The most important application of this family seems to be achieving graph drawing by axis separation, where each axis of the drawing addresses different aspects of aesthetics. We define the special requirements from such algorithms and show how several graph drawing algorithms can be generalized to handle this task. 1 Introduction 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 the visualization of relational information, and its ultimate usefulness depends on the readability of the resulting layout; that is, the drawing algorithm s capability of conveying the meaning of the diagram quickly and clearly. Consequently, many approaches to graph drawing have been developed [6, 15]. We concentrate on the problem of drawing graphs so as to convey pictorially the proximity relations between the nodes. The most popular approaches to this appear to be force-directed algorithms. These define a cost function (or a force model), whose minimization determines the optimal drawing. Graph drawing research traditionally deals with drawing graphs in two or three dimensions. In this paper, we identify and discuss a new family of graph drawing algorithms whose goal is to draw the graph in one dimension. This family has some interesting applications, to which we now turn. Axis separation The most common use of 1-D drawing algorithms is graph drawing by axis separation. Here, we would like to build a multidimensional layout axis-by-axis, so that we each axis can be computed using a different algorithm, perhaps accounting for different aesthetical considerations. This facilitates an appealing divide-and-conquer approach to graph drawing. A well known example of this is the problem of drawing directed graphs, where the y-coordinates are commonly used to reflect the hierarchy, while the separately computed x-coordinates take care of additional characteristics of the graph, such as preserving proximity. As another example, we have recently worked on visualizing clustered data using axis separation. There, the x-coordinates guarantee the visual separation between clusters, whereas the y-coordinates address additional aesthetics ignoring the clustering structure; see [18] for a detailed description. Fig. 1 shows a sample result of this, containing a hierarchically-clustered biological dataset (modeled by a weighted graph). The hierarchy structure of the data is represented by the traditional dendrogram a full binary tree in which each subtree is a cluster and the leaves are individual elements. Consequently, the x-axis was computed so as to adhere to the dendrogram structure,

2 while maximizing the expression of similarities between the nodes. This was done by reordering the dendrogram and adjusting the gaps between consecutive leaves. The y- axis, which should not consider the hierarchy structure at all, was computed by a 1-D graph drawing algorithm (using the classical-mds method, as described in Subsection 3.3). Fig. 1. (taken from [18]) Using axis-separation to draw hierarchically clustered fibroblast gene expression data. We convey both the similarities between the nodes and their clustering decomposition, using an ordered dendrogram coupled with a 2-D layout that adheres to its structure. We have colored six salient clusters that are clearly visible. Sometimes a single dataset can be modeled by different graphs. Consequently, it might be instructive to draw the data by assigning each of the axes to a different graph, and then simultaneously examine and compare the characteristics of the two models. For example, proximity relationships between web pages can be modeled either by connecting pages that have a similar content, or by relying on their link structure. We can draw the web pages as points in the plane according to these two models by using axis separation, thus making it possible to see at a glance which elements are related by each of them. Another tightly related case is when we already have one coordinate for each node. Such a coordinate might be a numeric attribute of the nodes that we want to convey spatially. In order to reflect proximity relationships, we would like to add another coordinate computed by a 1-D graph drawing algorithm. A nice example of this appears in [2]. There, a link structure (like the WWW) is visualized by associating one axis with a ranking of the nodes (some measure for node-prominency) and the other axis is computed by 1-D graph drawing (using the Eigen-projection method described in Subsection 3.1). See Fig. 2. Linear arrangement So far, we have described situations were the 1-D graph drawing is used to construct a multidimensional drawing. However, in some cases, additional axes are not necessary, and we simply need an algorithm for drawing the graph on a line ; see, e.g., Fig. 3. In this case, the problem is called linear arrangement [10, 17],

3 home.interlink.or.jp/~ichisaka/ Stars.com/Multimedia/ javaboutique.internet.com java.sun.com tacocity.com.tw/java/ physics.syr.edu Fig. 2. Authority and PageRank visualization of java query result, taken (with permission) from [2]. Each web-page is given two numerical values that measure its importance (Authority and PageRank). These values determine the x-coordinates of the drawing. The y-coordinates, which reflect the similarity between the web-pages, are computed by a graph drawing algorithm. and we want to order the nodes such that similar nodes are placed close to each other. In the graph drawing context, such a problem arises in code and data layout applications [1], and in laying out software diagrams [22] Fig. 3. A Linear arrangement Fig. 4 shows how such a linear arrangement can be used to visualize a (weighted) adjacency matrix. The figure shows the relations between odor patterns measured by an electronic nose using a complete weighted graph; see [5]. As seen in part (a) of the figure, the raw adjacency matrix does not show any structure. However, the same matrix, shown in part (b) after permuting its rows and columns according to a linear arrangement of the graph, reveals much of the structure of the data. Ordering problems are naturally formulated as discrete optimization problems, where the coordinates are permutation of {1,...,n}. However, such formulations lead to NP-hard problems that are difficult to solve. One way to eliminate part of this difficulty is to allow the nodes to take on noninteger coordinates. The resulting continuous problems can be efficiently solved, and their solution is used as an approximation of the optimal discrete ordering, by taking a sorted ordering of the nodes coordinates; see [13, 17]. In this way, the continuous formulations given in this paper can be used for discrete linear arrangement problems too.

4 (a) (b) Fig. 4. Using linear arrangement for matrix visualization. (a) A similarity matrix of odor patterns as measured by an electronic nose; more similar patterns get higher (=brighter) similarity values. (b) The structure of the data is better visualized after re-ordering rows and columns of the matrix. 2 Basic Notions Throughout the paper, we assume that we are given an n-node connected graph G(V,E), with V = {1,...,n}. A key value that describes relations between nodes is the Laplacian, which is an n n symmetric positive-semidefinite matrix denoted by L, where 1 {i, j} E L ij = 0 {i, j} / E,i j i, j =1,...,n. deg(i) i = j Here, deg(i) def def = {j {i, j} E}. It is easy to check that 1 n = (1,...,1) R n is an eigenvector of L with associated eigenvalue 0. When the graph is connected, all other eigenvalues are strictly positive. The usefulness of the Laplacian stems from the fact that the quadratic form associated with it is just the sum of squared edge lengths. We formulate this for a 1-D layout: Lemma 1. Let L be an n n Laplacian, and let x R n. Then x T Lx = (x i x j ) 2. {i,j} E The proof of the lemma is straightforward, and it can be extended to multidimensional layouts too. We now recall some basic statistical notions. The mean of a vector x R n, denoted by x, is defined as 1 n n i=1 x i. The variance of x, denoted by Var(x), is defined as 1 n n i=1 (x i x) 2. The covariance between two vectors x, y R n is defined as Cov(x, y) = 1 n n i=1 (x i x)(y i ȳ). The correlation coefficient between x and y is defined as Cov(x, y) Var(x)Var(y) This measures the colinearity between the two vectors. If the correlation coefficient is 0, x and y are uncorrelated. Ifx and y are independent, then they are uncorrelated. We denote the normalized value of x by ˆx = x/ x. As explained earlier, 1-D drawing algorithms are often used in the context of multidimensional drawings. Henceforth, for simplicity, assume we have to compute the y- coordinates, while (possibly) being given precomputed x-coordinates. Thus, the layout

5 is characterized by two vectors x, y R n, with the x-coordinates being x 1,...,x n, and the y-coordinates y 1,...,y n. Other cases, where we have more than one precomputed axis or where we want to produce several dimensions, can be addressed by small changes in our techniques. Moreover, for convenience, we assume, without loss of generality, that the x- and y-coordinates are centered, so their means are 0. In symbols, n i=1 x i = n i=1 y i =0. This can be achieved by a simple translation. 3 Algorithms for One-Dimensional Graph Drawing In principle, we could have used a classical force-directed algorithm for computing the 1-D layout. However, when trying to modify the customary two-dimensional optimization algorithm for use in our one-dimensional case, convergence was rarely achieved. Traditionally, node-by-node optimization is performed, by moving each node to a point that decreases the cost function. Common methods for this are Gradient-Descent and Newton-Raphson. However, these methods tend to get stuck in bad local minima when used for 1-D drawing [4, 23]. Interestingly, 2-D drawing is much easier for such methods. Probably, the reason is that there is less space for maneuver in one dimension when seeking a nice layout, which prevents convergence to an optimum. Furthermore, in several works even a 3-D layout is used to avoid local minima, see, e.g., [3, 8, 23]. Another possible approach could be to use algorithms for computing (approximated) minimum linear arrangements (MinLA). These set the coordinates to be a permutation of {1,...,n} in a way that minimizes the sum of edge lengths. Although the limitation that the coordinates are distinct integers may seem unnatural in the graph drawing context, we have found that MinLA has some merits when drawing digraphs by axis separation; see [4]. However, a major disadvantage of MinLA is that it cannot consider precomputed coordinates. Note that a careless computation that ignores such precomputed coordinates can be very problematic. Such a computation might yield y-coordinates that are very similar to the x-coordinates, resulting in a drawing whose intrinsic dimensionality would really be 1, meaning that one axis would be wasted. In the rest of this section, we describe four different methods that appear to be perfect for our task. A common characteristic of these methods, which makes them suitable for 1-D optimization, is that they compute the layout axis-by-axis, instead of the nodeby-node optimization mechanism of force-directed methods. Furthermore, when these methods are used to produce a multidimensional layout, the different axes are uncorrelated. This suggests a very effective way to generalize the methods so that they can deal with the precomputed coordinates: we simply require no correlation between the x-coordinates and the y-coordinates, so that the latter ones will provide us with as much new information as possible. Technically, since we have assumed x and y to be centered, the no-correlation requirement can be formulated simply as y T x =0, which states that x and y are orthogonal. We now survey the methods and explain how they can be extended to handle the case of predefined x-coordinates. 3.1 Eigen-projection The Eigen-projection [11, 19] computes the layout of a graph using low eigenvectors of the related Laplacian. Some important advantages of this approach are its ability to compute optimal layouts (according to specific requirements) and a very short computation time [16]. As we will see, this method is a natural choice for 1-D layouts, and has already been used for such tasks in [13, 1, 2, 4]. In [19] we give several explanations for the

6 ability of the Eigen-projection to draw graphs nicely. Here, we provide a new derivation, which shows the tight relationship of this method to force-directed graph drawing. We define the Eigen-projection 1-D layout y R n, as the solution of: min y {i,j} E (y i y j ) 2 {i,j} / E (y i y j ) 2 (1) In (1), the numerator calls for shortening the edge lengths (the attractive forces ), while the denominator calls for placing all nonadjacent pairs further apart (the repulsive forces ). This is a reasonable energy minimization approach that resembles forcedirected algorithms. Since we have {i,j} E (y i y j ) 2 + {i,j} / E (y i y j ) 2 = i<j (y i y j ) 2,an equivalent problem would be: min y {i,j} E (y i y j ) 2 i<j (y i y j ) 2 (2) It is easy to see that the energy to be minimized is invariant under translation of the data. Thus, for convenience, we eliminate this degree of freedom, by requiring that y be centered; that is, y T 1 n =0. We can now simplify (2) by using the following lemma: Lemma 2. Let y R n such that y T 1 n =0, then: (y i y j ) 2 = n i<j yi 2 (= n y T y). i=1 Proof. (y i y j ) 2 = 1 2 i<j (y i y j ) 2 = 1 2n yi i,j=1 = n yi 2 n y i i=1 i=1 j=1 i=1 y j = n yi 2 i=1 y i y j = i,j=1 The last step stems from the fact that y is centered, so that n j=1 y j =0. Therefore, we can replace i<j (y i y j ) 2 with y T y. Moreover, using Lemma 1, we can write {i,j} E (y i y j ) 2 as the quadratic form y T Ly. Consequently, once again we reformulate our minimization problem in the equivalent form: min y y T Ly y T y in the subspace: y T 1 n =0 (3) By substituting ˆx = 0 in the following Proposition, we obtain that the optimal 1-D layout is the eigenvector of L with the smallest positive eigenvalue. This way, the Eigen-projection method provides us with an efficient way to calculate optimal 1-D layouts. We still have to show how the Eigen-projection can be extended so as to deal with the uncorrelation requirement, that is a case where we already have a

7 coordinate vector x, and we require that y is orthogonal to x. Now, the optimal layout will be the solution of: min y y T Ly y T y in the subspace: y T 1 n =0,y T x =0 (4) Fortunately, the optimal layout is still a solution of a related eigen-equation: Proposition 1. The solution of (4) is the eigenvector of (I ˆxˆx T )L(I ˆxˆx T ) with the smallest positive eigenvalue. (Note that I ˆxˆx T is a symmetric n n matrix. Henceforth, we will use it extensively thanks to its property of being an orthogonalization operator: for any vector y R n, the result of orthogonalizing y against x is (I ˆxˆx T )y.) Proof. Observe that we can assume, without loss of generality, that y T y =1. This is because changing the scale still gives an optimal solution: Check that if for y 0 satisfying y T 0 y 0 =1we get y T 0 Ly 0 /y T 0 y 0 = λ, then we will also get y T Ly/y T y = λ for each y = c y 0 (c 0). Thus, the new form of the optimization problem will be: min y T Ly y given: y T y =1 in the subspace: y T 1 n =0,y T x =0 (5) The matrix (I ˆxˆx T )L(I ˆxˆx T ) is symmetric, so it has n orthogonal eigenvectors spanning R n. We will use the convention λ 1 λ 2... λ n for the eigenvalues of (I ˆxˆx T )L(I ˆxˆx T ), and denote the corresponding real orthonormal eigenvectors by u 1,u 2,...,u n. Clearly, (I ˆxˆx T )L(I ˆxˆx T ) x =0. Utilizing the fact that x T 1 n =0 and that 1 n is the only zero eigenvector of L, we obtain λ 1 = λ 2 =0, u 1 =(1/ 1 n ) 1 n,u 2 =ˆx, and λ 3 > 0. We can now decompose every y R n as a linear combination, where y = n i=1 α iu i. Moreover, since the solution is constrained to be orthogonal to u 1 and u 2, we can restrict ourselves to linear combinations of the form y = n i=3 α iu i. Use the constraint y T y =1to obtain n i=3 α2 i =1(a generalization of the Pythagorean law). Similarly, y T (I ˆxˆx T )L(I ˆxˆx T )y = n i=3 α2 i λ i. Note that since y T ˆx =ˆx T y = 0, we get: y T (I ˆxˆx T )L(I ˆxˆx T )y = y T Ly + y T ˆxˆx T L(I ˆxˆx T )y + y T Lˆxˆx T y = y T Ly. So the target value is y T Ly = y T (I ˆxˆx T )L(I ˆxˆx T )y = αi 2 λ i i=3 αi 2 λ 3 = λ 3. Thus, for any y that satisfies the constraints, we get y T Ly λ 3. Since u T 3 Lu 3 = u T 3 (I ˆxˆx T )L(I ˆxˆx T )u 3 = λ 3, we can deduce that the minimizer is u 3, the lowest positive eigenvector. i=3

8 Interestingly, posing the problem as in (4) and solving it as in Proposition 1, constitutes a smooth generalization of the Eigen-projection method: when x is the lowest positive eigenvector of L, then the solution y will be the second lowest positive eigenvector of L. This coincides with the way the Eigen-projection computes 2-D layouts; see [19]. However, we allow the more general case of arbitrary x-coordinates. As to computational complexity, the space requirement of the algorithm is O( E ) when using a sparse representation of the Laplacian. The computation can be done using iterative algorithms, such as the Power-Iteration or Lanczos; see [9]. The time complexity of a single iteration is O( E ). When working with a sparse L, we can use a much faster multi-scale algorithm that can deal with millions of elements in reasonable time; see [16]. However, caution is needed, since an explicit calculation of (I ˆxˆx T )L(I ˆxˆx T ) would destroy the sparsity of L. To get around this, we utilize the fact that the iterative algorithms for computing eigenvectors use the matrix as an operator, i.e., they access it only via multiplication with a vector. This settles the issue, since carrying out the product (I ˆxˆx T )L(I ˆxˆx T ) v is equivalent to orthogonalizing v against x, multiplying the result with the sparse matrix L, and then again orthogonalizing the result against x. 3.2 Principal component analysis and high-dimensional embedding Principal component analysis (PCA) computes a projection of multidimensional data that optimally preserves its variance; see [7]. The fact that PCA uses the data coordinates apparently renders it useless for graph drawing. However, in [12] we show that it is possible to generate artificial k-dimensional coordinates of the nodes that preserve some of the graph structure, thus making it possible to use PCA. We call these coordinates high-dimensional embedding in [12], and denote them by an n k coordinate matrix called X, so the k coordinates of node i constitute the i-th row of X. We assume each of the columns of X is centered, something that can be achieved by translating the data. In order to compute a 1-D projection, PCA computes a unit vector d R k, which is the direction of the projection. The vector d is the top eigenvector of the covariance matrix 1 n X T X. The projection itself is X d, and, as mentioned, it is the best 1-D projection in terms of variance preservation. When given x-coordinates, we will be interested only in the component of the projection that is orthogonal to the x-coordinates. This component is exactly (I ˆxˆx T ) (X d), and we want to maximize its variance. However, (I xx T ) (X d) =((I xx T ) X)d, so our problem is reduced to finding the most variance-preserving projection of the coordinates (I ˆxˆx T ) X. The optimal solution is obtained by performing PCA on (I ˆxˆx T ) X, which is equivalent to orthogonalizing each of X columns against x and then performing PCA on the resulting matrix. Again, this is a smooth generalization of PCA that enables it to deal with predefined x-coordinates. The reason is that if x was also computed by PCA, then one would obtain the regular 2-D PCA projection. One of the advantages of the PCA approach is its excellent time and space complexity; see [12]. 3.3 Classical multidimensional scaling Multidimensional scaling (MDS) is a general term for techniques that generate coordinates of points from information about pairwise distances. Therefore, arguably, forcedirected graph drawing can be considered to be MDS. Here we are interested in a technique called classical-mds (CMDS) [7], which produces (multidimensional) coordinates that preserve the given pairwise distances perfectly; i.e., the pairwise Euclidean

9 distances in the generated space are equal to the given distances. The graph drawing application of CMDS was suggested long ago, in [21]. The distance between nodes i and j is defined as d ij, the graph-theoretical distance between the nodes. Therefore, CMDS can be used to find an Euclidean embedding of the graph that preserves the graph-theoretical distance. We now provide a short technical description of the method. Given points in Euclidean space, it is possible to construct a matrix X of centered coordinates if we know the pairwise distances among the points. The way to do this is to construct the n n inner product matrix B = XX T, which can be computed using the cosine law, as follows B ij = 1 d 2 ij 1 d 2 ik 1 d 2 kj + 1 n 2 n n n 2 d 2 lk. (6) k=1 k=1 k=1,l=1 Note that B is invariant under orthogonal transformations of X. That is, given some orthogonal matrix Q (i.e., QQ T = I), we can replace X with X Q, without changing the inner-product matrix: X Q(X Q) T = X QQ T X = XX T = B Therefore, B determines the coordinates up to orthogonal transformation. This is reasonable, since such a transformation does not alter pairwise distances. There is always an orthogonal transformation that makes the axes orthogonal (i.e., the singular value decomposition), which allows us to restrict ourselves to a coordinate matrix with orthogonal columns. Such a matrix can be obtained by factoring B using the eigenvalue decomposition B = U U T (U is orthogonal and is diagonal), which enables defining the coordinates of the points as X = U 1 2. This way, the columns of X are centered and are mutually orthogonal. In practice, we do not want all the coordinates but only a low-dimensional projection of the points, and here only a 1-D embedding is needed. Thus, as in PCA, we seek the 1-D projection of X having the maximal variance. Since the columns of X are uncorrelated, we simply have to take the column with the maximal variance, which is equivalent to the column of U with the highest corresponding eigenvalue. Technically, we are interested in the top eigenvector u 1 and the corresponding eigenvalue, λ 1,ofB. After computing this eigenpair, we can define the embedding of the data as λ 1 u 1. Additional coordinates can be obtained using the subsequent eigenpairs. It appears that CMDS is closely related to PCA. In fact, CMDS is a way of performing PCA without explicitly defining the coordinate matrix. Thus, if the pairwise distances are Euclidean distances based on the coordinate matrix, the results of CMDS are identical to PCA. Consequently, in our case, when we want the embedding to be orthogonal to x, we can use the same technique we used in PCA. Once again, we would like to perform PCA on (I ˆxˆx T )X, and of course we do not have this matrix explicitly. However, it is possible to compute the inner-product matrix (I ˆxˆx T )XX T (I ˆxˆx T ), since this matrix is simply (I ˆxˆx T )B(I ˆxˆx T ). Using the same reasoning as above, the first principal component of (I ˆxˆx T )X can be found by computing the top eigenpair of (I ˆxˆx T )B(I ˆxˆx T ). However, there is one theoretical flaw in applying CMDS to graph-drawing. Computing a coordinate matrix X that preserves pairwise distances is not always possible, and will fail when the graph-theoretic metric is not Euclidean. Technically, there might be some negative eigenvalues to the matrix B, preventing the square-root operation from

10 being carried out. However, in practice this is not a serious problem, since we are not interested in recovering the full multidimensional coordinates, but only the few leading coordinates. When the given x-coordinates are also the result of CMDS, our method produces the same y-coordinates as CMDS. Therefore, we have a smooth generalization of CMDS that allows it to deal with predefined coordinates. One note on complexity. When performing this CMDS, we have to store the matrix B, which requires O(n 2 ) space complexity, much worse than in the Eigen-projection or PCA cases. 3.4 One-dimensional drawing of digraphs When edges are directed, we may want the layout to show the overall directionality of the graph and its hierarchical structure. The previously described techniques, which ignore the direction of the edges, might thus not be suitable. An adequate method for dealing with 1-D layout of digraphs was described in [4]. There, we looked for a layout that minimizes the hierarchy energy: (y i y j 1) 2 (7) i j E Define the balance vector, b R n, such that: b i = outdeg(i) indeg(i) where outdeg(i) and indeg(i) denote the number of outgoing and incoming edges adjacent to i, respectively. We showed in [4] that up to a constant additive term, the hierarchy energy (7), can be written in a compact form as: y T Ly 2y T b (8) Consequently, the optimal 1-D layout would be the solution of Ly = b. This formulation is flexible enough to allow y to be uncorrelated with a given coordinate vector x. In this case, we want to minimize (8) in the subspace orthogonal to x. Equivalently, we can take only the component of y that is orthogonal to x, which is (I ˆxˆx T )y. This way, we seek the minimizer of: y T (I ˆxˆx T )L(I ˆxˆx T )y 2y T (I ˆxˆx T )b, (9) Hence, the optimal 1-D layout would be the solution of: (I ˆxˆx T )L(I ˆxˆx T )y =(I ˆxˆx T )b After finding the minimizer we orthogonalize it against x. It is easy to see that this does not affect the value of (9), so we remain with an optimal solution uncorrelated with x. As a consequence, when we have a 1-D layout of a digraph, we can add an additional uncorrelated dimension that shows the hierarchical structure of the graph. Moreover, we can compute two coordinate vectors that provide two uncorrelated descriptions of the graph s directionality. This might be useful for digraphs whose hierarchical structure is explained by several independent factors. When L is sparse, we recommended in [4] that the equation be solved using the Conjugate-Gradient method that accesses L only via matrix-vector multiplication. In this case, like in the Eigen-projection case, the product (I ˆxˆx T )L(I ˆxˆx T ) v is carried out by orthogonalizing v against x, multiplying the result with the sparse matrix L, and then again orthogonalizing the result against x.

11 4 Discussion We have explored one-dimensional graph drawing algorithms and have studied their special features and applications. One important application of this family is graph drawing by axis-separation, where each axis is computed separately, so as to address specific aesthetical considerations. Since point-by-point local optimization in one dimension is a poor strategy, traditional force-directed algorithms are not suitable for the 1-D drawing task, while less traditional algorithms are. We generalized four such algorithms using the unified paradigm of computing the layout axis-by-axis, while maintaining noncorrelation with the precomputed coordinates. This unified framework allows for an interesting integration of the algorithms. We can use one of the algorithms for laying out the x-coordinates and another for computing uncorrelated y-coordinates. For example consider the 4970 graph that was previously drawn by the Eigen-projection in [16], as shown in Fig. 5(a), and by PCA projection in [12], as shown in Fig. 5(b). In Fig. 5(c) we show a layout where the x-coordinates were computed by Eigen-projection and y-coordinates by CMDS. Another combined layout is given in Fig. 5(d), where the x-coordinates were computed by PCA projection and y-coordinates by CMDS. (a) (b) (c) (d) Fig. 5. Layouts of the 4970 graph: (a) by Eigen-projection (taken from [16]); (b) by PCA (taken from [12]); (c,d) two combined layouts: (c) Eigen-projection + CMDS (d) PCA + CMDS. In a subsequent paper [20] we describe a rather sophisticated optimization process that allows incorporating the popular model of Kamada and Kawai [14] for onedimensional graph layout. References 1. B. Beckman, Theory of Spectral Graph Layout, Technical Report MSR-TR-94-04, Microsoft Research, 1994.

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