How to Draw Clustered Weighted Graphs using a ... - CiteSeerX

How to Draw Clustered Weighted Graphs using a Multilevel Force-Directed Graph Drawing Algorithm Romain Bourqui, David Auber, Patrick Mary LaBRI, Universit´e Bordeaux I, France {romain.bourqui, david.auber, patrick.mary}@labri.fr Abstract Visualization of clustered graphs has been a research area since many years. In this paper, we describe a new approach that can be used in real application where graph does not contain only topological information but also extrinsic parameters (i.e. user attributes on edges and nodes). In the case of force-directed algorithm, management of attributes corresponds to take into account edge weights. We propose an extension of the GRIP algorithm in order to manage edge weights. Furthermore, by using Voronoi diagram we constrained that algorithm to draw each cluster in a non overlapping convex region. Using these two extensions we obtained an algorithm that draw clustered weighted graphs. Experimentation has been done on data coming from biology where the network is the genesproteins interaction graph and where the attributes are gene expression values from microarray experiments.

1. Introduction Visualization of clustered graphs has been a research area since many years [7, 27, 14, 23, 12, 19]. However, only few algorithms are able to manage edge weights. The main interest in managing edge weights is that we can simultaneously visualize a clustering induced by a topological algorithm or given by a taxonomy and still have a correlation between Euclidian distances in the drawing and weights of edges in the graph. In that work, we wanted to visualize genes-proteins interaction network and to be able to see both the relations, between genes and proteins, and the result of their expression according to the results of experimentation on the level of expression of each gene/protein on a micro array. The main difficulty of that problem is first the size of the network and, as we can see in figure 1, its connectivity that produces a mess effect when we draw it with a force directed algorithm or with a multi dimensional scaling algorithm. Such a problem is frequent in biological graph, where the

number of connections is often really large compare to the number of nodes and because a lot of these networks are scale free (i.e. the distribution of degree follows a power law). To give a useful representation of these data we need to be able to first decompose the data into groups and if necessary to decompose each group into sub-group and so on. That process of decomposition leads to a well-known structure called a cluster tree. It exists several ways of building such a decomposition [20, 22]. Moreover, when one works with real data, one often has domain specific decomposition (i.e. taxonomy) that one can use to produce a more meaningful cluster tree. The challenge of visualization of such networks is thus to be able to layout the graph according to a cluster tree and to take into account the strength of the interaction according to experimental results. All the requirements given above lead to an algorithm that should respect the following constraints : 1. forbid groups overlap in order to easily identify groups, 2. preserve groups inclusions in the drawing, to see the hierarchy produced by the clustering algorithm, 3. have a drawing in which each group can be surrounded by a convex polygon, and 4. respect weighted graph distances using energy minimization function. We experiment our technique on genes-proteins interaction network of the fly (see figure 1). In figure 1 (right), one can see the result obtained using our algorithm. We first simplify the data by removing vertices of degree 1 and repeating this process until there is no more vertex of degree 1 (that method is called peeling [25]). Removed nodes can be ”easily” added in the drawing at the end of the algorithm. Then, we decompose the data by constructing “balls” of similar vertices (i.e. aggregation of connected nodes with similar values) and repeat this decomposition process into

Figure 1. (left) Force directed layout of a protein-genes interaction graph, scale free nature of the data gives a messy effect, (middle) clusters obtained using an aggregative method are highlighted using convex hulls (right) drawing of the same graph using our algorithm.

these “balls”. Finally, we draw the clustered graph using our drawing algorithm. To improve visibility in the final drawing, we use edge bundling [13]. Inter-cluster edges are drawn so that if edges link the same clusters, they are ”merged” in the drawing. This operation enables to solve the problem of edge cluttering in the final drawing. This paper is organized as follow. In section 2, we introduce the terminology used in this paper. Then, we present related approaches to visualize similar graphs in section 3. In section 4, we give an overview of our method and finally in section 5, we give details of our algorithm.

In the following, we call height of the clustered graph the height of the hierarchy tree T .

2. Terminology

In a clustered graph C = (G, T ), to each level i of T can correspond a quotient graph QG where VG is the partition represented by nodes of level i of T . For a weighted graph G = (V, E, w) and one partition (C1 , ..., Ck ), we define the weighted quotient graph QG = (VG , EG , w
) as follow :

In this section, we present the definition and notion used in this article. Let G = (V, E) be a simple graph (without multi-edge and self-loop), where V is the set of vertices and E the set of edge, and |V |=n and |E|=m. An edge-weighted graph Gw is defined as follow : Gw = (V, E, w) where V and E are respectively the sets of vertices and edges, and w: E → R, is the weight of an edge. In this paper, we only consider positive weights. Let S ⊆ V , the subgraph of G induced by S is G[S] = (S, ES ) where ES = {e = (u, v) ∈ E such that u ∈ S and v ∈ S}. Clustered graph: As defined in [7], a clustered graph C = (G, T ) consists of a graph G and a rooted tree T such that the leaves of T are exactly the vertices of G. Each node v of T represents a cluster of vertices of G that are the leaves of the subtree of T rooted at v. The root of T represents the whole graph G.

Quotient graph: In [24], R. Brockenauer and S. Cornelsen define the quotient graph QG = (VQ , EQ ) for a partition (C1 , ..., Ck ) of G = (V, E) as : • VQ = {C1 , ..., Ck } and • (Ci , Cj ) ∈ EQ iff i = j and ∃u ∈ Ci and v ∈ Cj such that (u, v) ∈ E.

• VQ = {C1 , ..., Ck } and • (Ci , Cj ) ∈ EQ iff i = j and ∃u ∈ Ci and v ∈ Cj such that (u, v) ∈ E. → R and ∀e = (Ci , Cj ) ∈ EG : • w
: E
G w
(e)=

u∈Ci ,v∈Cj ,(u,v)∈E

w((u,v))

|{u∈Ci ,v∈Cj ,(u,v)∈E}|

In this paper, we call metanode, a node of the quotient graph that represents a cluster Ci in the original graph. Weighted graph distances: In this article, we have to compute graph distances in a weighted graph G = (V, E, w). Let u and v be two vertices

of G and let p1 , p2 , ..., pl , where pi = (e1 , e2 , ..., er−1 , er ) is a sequence of distinct adjacent edges of G and e1 (resp. er ) is adjacent to u (resp. v), be all paths between u and v, then weighted graph distance
between u and v in G is : distG,w (u, v) = min1≤i≤l { e∈pi w(e)} (i.e. the shortest path between u and v).

collapsing. Each time a cluster is expanded, they draw the lower level using an optimized version of [2] (which is a force-directed placement algorithm), time computation of this algorithm allows an efficient interaction/navigation.

In this article, we use the following notations : G is the connected edge-weighted graph to draw, C is the corresponding clustered graph, Qi is the quotient graph corresponding to level i of C and T is a tree spanner of G.

In [12, 19], the authors present top-bottom force-directed algorithms that take into account edges weights to draw clustered graphs. After drawing a level i (using forcedirected algorithm), they compute a Voronoi diagram of this level. For a given node v of level i, children of v in the hierarchy tree are drawn in the region (Voronoi cell) of v. This process is repeated until the whole graph is drawn. In [12], to force the set S of children of a node v to stay in v’s cell, the authors first compute the placement of the subgraph G[S] induced by S and then re-scale positions of nodes of S in the Voronoi cell of v using a simple geometric transformation. This method leads to good results. The problem is that the authors do not consider that nodes of G[S] are forced to stay in the cell of their parent while computing force-directed placement. After several iterations of the force-directed algorithm, nodes of G[S] could have ”weak” interactions, whereas after re-scaling in the cell, they should have ”strong” repulsive forces. In [19], the process is almost the same as in [12]. The main modification is that, to improve time complexity, the authors only consider forces applied on a node v (of level i) by v’s siblings (i.e. nodes represented by the same metanode as cv in level i − 1). As for a given level of the hierarchy, intercluster edges are not considered to compute forces applied on a node, this process may conduct to have long edges.

3. Related Works

4. Our approach

In [7], P. Eades and Q.-W. Feng introduce the definition of clustered graphs and two main ways to represent these kind of graphs. First, they present plane drawings (i.e. G is drawn as points and curves in the plane) for which they give two visualizations : straight-line convex and orthogonal rectangular drawings. Second, they explain multilevel drawings : a sequence of plane drawings, one for each level i in the hierarchy and a 3-dimensional drawing of the tree hierarchy (i.e. a superposition of planes of each level and edges of the tree between these planes). In [27, 14, 23], the authors give force-directed methods to visualize clustered graphs. In [14, 23], the technique allows navigation in the resulting drawing, while in [27], X. Wang and I. Miyamoto integrate a force-directed method and a constraint solver to improve the layout quality. In [15], the authors give a way to visualize very large graph (i.e. graph that cannot be totally loaded into RAMMemory). They propose a visualization similar to the approach of [23]. They also present several techniques of interaction and navigation, like cluster expanding or

In this section, we present an overview of our technique. Our algorithm is based on the method presented in [12]. This is a top-bottom multilevel drawing algorithm. We first compute the layout of the upper level graph of the hierarchy and then go down to draw lower levels, until the whole graph has been drawn. The input of our algorithm is a clustered graph with height h. Figure 2.a gives an example of graph and the figure 2.b gives a possible hierarchy tree of that graph with h = 3. First of all, we draw the top level quotient graph Q1 using a force-directed algorithm that takes edges weight into account. We obtain the layout of the highest level of abstraction of G. Figure 2.c gives the result of this step applied on the clustered graph presented in figures 2.a and b. In order to guarantee the constraints 1, 2 and 3 (presented in section 1), we must draw all nodes of G represented by a unique metanode in Q1 in a convex region. Moreover, we must ensure that this region do not overlap other regions. To determine these regions, we compute the Voronoi diagram [26] of the quotient graph Q1 . Figure 3.a shows the

k-spanner: For a graph G = (V, E) and k ≥ 1, a subgraph H = (VH , EH ) is a k-spanner of G if : • VH = V and EH ⊆ E and • ∀u, v ∈ V : distH (u, v) ≤ k × distG (u, v) And a tree k-spanner T of G, is a k-spanner of G that is a tree. For a weighted graph, k-spanner (and tree k-spanner) is defined using weighted graph distances. Voronoi diagram: Let P = {u1 , u2 , ..., un } be a set of n distinct points in a 2-dimensional Euclidean plane, called the sites. The Voronoi diagram of P is defined as the subdivision of the plane into n cells, one for each site in P . We note cell of ui by V or(ui ). V or(ui ) contains all the points at least as close to ui as to any other site.

Figure 2. (a) Left, an example of graph; (b) Center, an example of tree hierarchy of the graph presented in (a), in this example the height of the clustered graph is 3; (c) Right, drawing of the quotient graph Q1 corresponding to level 1 of the clustered graph presented in (a) and (b).

computed regions of the graph given in figure 2.c. It allows us to know the exact convex regions in which we can draw nodes represented by a metanode of Q1 . Next, our algorithm draws quotient graphs of each level i, 1 < i ≤ h using the result of level i−1. To draw Qi , we draw each subgraph G
of Qi represented by a metanode uG in Qi−1 . The drawing algorithm is the same forcedirected algorithm as for Q1 . Positions of nodes of each subgraph G
are initially set to the position of uG . During the algorithm, nodes’ positions are forced to stay in the cell of uG . In figure 3.b, one can see the result of this step on the metanode 1 of figure 3.a. Thus, our algorithm is a trade-off between [12] and [23] : we use a similar approach to [12], but they consider that inter-cluster and intra-cluster edges have same influence on the layout, whereas we consider that intra-cluster edges are at least as important as inter-cluster edges. This is similar to what is done in [23] when the authors add a virtual node in each cluster. Finally, new cells are computed in order to be able to apply our technique recursively. Figure 3.c shows the drawing of the graph Q2 computed by our method, this layout will then be used to draw the graph Q3 .

5. Algorithm In this section, we givre details of the main steps described above. First, we present how to draw first level of the hierarchy, and then, how to draw all other levels.

5.1. Drawing quotient graph Q1 The algorithm presented in [11] and later improved in [10] has two main steps: first the algorithm computes a hierarchy of the graph, then for each level an intelligent

placement and a force-directed algorithm are performed. In these two steps, graph distances must be computed. Thus, computation time increases when considering edgeweighted graphs. In order to take edges weight into account, we use a tree spanner T of the graph to ”approximate” graph distances. Below we give the method used for computing a tree spanner, then we describe steps of [11, 10] and how to adapt them to edge-weighted graphs, and finally, we present the technique used to prevent cluster overlaps.

5.1.1. Tree t-spanner computation A tree t-spanner is a spanning tree that approximates graph distances with a factor t (cf. section 2). Many results have been found on tree t-spanner (for an overview see [4, 3]). For any fixed t > 1, the problem of determining the existence of a tree t-spanner in a weighted graph is NPcomplete. Due to the NP-completeness of this problem, we must use an heuristic. Our heuristic works as follows : we search a spanning tree such that graph distances between any two nodes are as short as possible. To compute such a spanning tree, we use the well known Kruskal’s algorithm [18]. However, minimum spanning tree computed with weight w does not necessary give a good spanner, we have to make a trade-off between weights and graph topology. To achieve this, we use strength metric (describe in [1, 5]) on edges. This metric counts the number of cycles of size three or four an edge belongs to. The intuition we have is that the less cycles (of size three or four) an edge belongs to, the more it belongs to a shortest path. For instance, an isthmus e (i.e. an edge such that G
= (V, E \ {e}) is not connected) has a strength equal to zero, and e is in all shortest paths between any two nodes of different connected components of G
= (V, E \ {e}). Our spanner T is obtained by comput-

Figure 3. (a) Left, in green, Voronoi diagram of vertices of the graph Q1 of figure 2.c; (b) Center, result after drawing subgraph corresponding to the metanode 1 of figure (a); (c) Right, result after drawing the quotient graph Q2 corresponding to level 2 and computation of its regions.

ing a minimum spanning tree with the metric ms defined as follow : ∀e ∈ E : ms (e) = w(e) × strength(e).

5.1.2. Maximal Independant Set Filtering The first step of the algorithm of [11, 10] computes a of the set Maximal Independent Set Filtration (MISF) : V0 ⊃ V1 ⊃ V2 ⊃ ... ⊃ Vk ⊃ ∅ of vertices V of G, where V0 = V and ∀0 < i ≤ k, Vi is a maximal subset of Vi−1 such that for all pairs u, v of vertices of Vi the graph distance between u and v is greater or equal to 2i . The algorithm then places each node of each filtration set beginning with the smallest one.

ν

ν

Computing directly this filtering on weighted graph has no significance. For instance, if we consider a large graph having very small weights of edges, we could have only one vertex in V1 (if all graph distances to this node are less than 2). The optimal solution of this problem is to find graph distances (of each level) that optimize the distribution of the vertices in levels. Finding these distances is not straightforward. Our method consists of computing the MISF on our tree spanner T without considering edge weights. In T , “long” edges of the graph have been removed if a shorter path between its two extremities exists. Thus, two nodes u and v linked by one of these edges are no more considered as adjacent, whereas they are adjacent in G. In G, u (resp. v) would represent v (resp. u) in the first filtration set V1 . At the opposite, in T , u (resp. v) cannot represent v (resp. u) in V1 since unweighted graph distance between u and v is greater than 1.

5.1.3. Placement In the second step of [11, 10], the process first places nodes of Vk and then each filtration set Vk−1 , Vk−2 , ..., V0 . For each level, the placement is splited in two phases : an

intelligent initial placement (i.e., nodes are placed closed to theirs final positions), and then a refinement placement that uses a force directed algorithm (either Kamada-Kawai [17] or Fruchterman-Reingold [9] algorithm). Intelligent Placement : For the smallest set Vk , the authors consider that |Vk | = 3 (if it is not the case, nodes of Vk−1 are included in Vk ). Let u1 , u2 and u3 be the three nodes of Vk , they are placed on a triangle such that :   distR (u1 , u2 )=distG (u1 , u2 ) distR (u1 , u3 )=distG (u1 , u3 )  distR (u2 , u3 )=distG (u2 , u3 ) where distR (u, v) is the Euclidean distance between u and v. For each other level i, 0 ≤ i < k, let u be a vertex of Vi , and v1 , v2 and v3 be the three closest and already placed neighbors of u, the authors lay out u at the barycenter of v1 , v2 and v3 using graph distances between u and v1 , v2 and v3 as weights. That method can be directly extended to edge-weighted graphs by using distG,w instead of distG . Refinement : In order to refine placement, the authors use force directed algorithms. For each level i, 1 ≤ i < k, they use the well known Kamada-Kawai algorithm [17]: this algorithm takes into account graph distances in the computation of the forces. For the last filtering set V0 , computing all graph distances would take too much time and/or memory space, so they use the Fruchtermann and Reingold algorithm [9] that uses only Euclidean distances. For a vertex v of Vi , the authors closest compute forces between v and its avgDeg(G).n |Vi | neighbors in Vi (where avgDeg(G) is the average degree

Figure 4. (a) Left, in blue Voronoi diagram of the subgraph is computed; (b) Right, new cells are computed.

of G). It enables them to improve time computation. The method used to find this closest neighborhood is to apply a BFS (Breath-First Search) algorithm. For an edge-weighted graph, it is not possible to use this method because closest neighbors are not necessary those found using a BFS. In the worst case, to find the closest neighborhood of a given vertex, we have to compute weighted graph distances between it and all other nodes. It conducts to have a O(mn+n2 log(n)) time complexity and O(n2 ) space complexity by solving the well-known all-pair shortest paths problem [6, 21], or a O(nm+n2 log(n)) time complexity and O(n) space complexity by solving n singlesource shortest paths problem [8, 16]. For an overview on graph distances computation, reader can refer to [28]). Our algorithm computes the nearest neighbors using Dijkstra’s algorithm on each node of the tree spanner, so that the time complexity is O(n2 log(n)) and the space complexity O(n).

5.1.4. Regions computations Our algorithm has to force nodes of a cluster to be close to each other (cf. constraint 1). Last step of the quotient graph drawing is the computation of its Voronoi diagram. The cell of each metanode uc is the region in which nodes of G represented by uc in the quotient graph have to be placed (for instance, see figure 3.a). Moreover, it ensures that each cluster can be surrounded by a convex hull (cf constraint 3). Indeed, even if a Voronoi cell is infinite, one can consider that this is a convex polygon where a segment has been removed. Thus, the use of a Voronoi diagram ensures that we are able to compute independent convex hulls for all clusters of a given level.

5.2. Drawing quotient graph Qi The process presented in this section is repeated for each level of the clustering hierarchy (i.e. until the whole graph G is drawn). To draw the quotient graph corresponding to level i, we first place its nodes at the same position than their father in the hierarchy. And then, we consider subgraphs Guc represented by metanode uc in level i − 1 individually.

5.2.1. Drawing one subgraph Guc To draw a subgraph Guc , we use the algorithm presented in section 5.1. In order to take into account distances between nodes of Guc and nodes of Qi , we modify forcedirected algorithms. When computing forces applied on a node, we consider a constant number of nodes of Qi that do not belong to Guc . It enables us to improve the layout by decreasing the number of abnormally ”long” edges. During force directed algorithm, we constraint nodes in their parent’s cell at each iteration of the algorithm. Each time a node is moved outside its parent’s cell, we force the node to stay in it. Using this method, we give more importance to intra-cluster than to inter-cluster interactions. This technique is ”equivalent” to those used by Eades and Huang in [23] when they add a virtual node and virtual edges in each cluster to prevent overlap.

5.2.2. Voronoi diagrams computation As in section 5.1, we then compute the Voronoi diagram of each subgraph, to ensure that nodes represented by a metanode in the subgraph are drawn close to the position of the metanode (see figure 4.a). Finally, to ensure that these nodes are drawn in the region of all its ancestors, we constraint Voronoi diagram of the subgraph in the cell of the metanode that represents this subgraph (see figure 4.b).

6. Conclusion and future works In this paper, we presented an algorithm to lay out clustered weighted graphs. Our method addresses a challenging problem which consists in simultaneously visualizing groups inclusions avoiding overlap and respecting weighted graph distances. This leads to pleasant drawings that ease the identification of groups and of adjacency relations. Moreover taking into account lengths of edges in the drawing algorithm allows to visually recognize weights of the edges. We presented an experimentation of our method on real data coming from biology. Usually, due to their scale free property, drawing of such data is messy. We have shown that using our method can improve significantly the readability of interaction graphs. In the near future, we plan to improve graph distance approximation to obtain a drawing in which Euclidean distances are closer to graph distances. This could be done by computing several tree spanner and considering the shortest distance as our approximation. Moreover, we are doing experiments on different kind of scale free network in order to generalize that approach to that class of graphs.

Acknowledgements The authors would like to thank Christian N´eri who gave us the data coming from his own researches in biology.

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