Graph Visualization

P. Eades1 and K. Klein2

1 University of Sydney peter.eades@sydney.edu.au2 Monash University karsten.klein@monash.edu

1 Introduction

Graph visualization is the process of making a drawing of a graph, so that a human canunderstand the graph. This is illustrated as the graph visualization pipeline in Fig. 1.A drawing function D takes a graph G from a graph data set and produces a graphdrawing D(G). A perception function P takes the drawing D(G) and produces someknowledge P (D(G)) in the human. The drawing function D can be executed with pen

Graph Graph drawing Human

𝑫 𝑷

(a) (b) (c)

Fig. 1. A graph visualization pipeline.

and paper by a human, but since the advent of computer graphics in the 1970s, therehas been increasing interest in executing this function on a computer; in this paper wediscuss computer algorithms that implement D. The perception function P is executedby the human’s perceptual and cognitive facilities.

As an illustration, consider the social network in Fig. 2; it describes a friendshiprelation between a group of people. It is represented in Fig. 2(a) as a table with the firstcolumn listing the people, and the second column listing the friends of each person. Forexample, the friends of Brian are Keith, John, Michael, and William. The drawing func-tion D produces the picture in Fig. 2(b); each person is represented by a box with text,and the friendship relation is represented by lines connecting the boxes. The perceptionfunction P takes the picture as input and produces some knowledge in the human. Thiscould be low-level knowledge such as “John and George are friends”, or higher-levelknowledge such as “Keith is important”.

Graphs (aka networks)are one of the most pervasive models used in technology;social networks are prime examples, but we also find graphs in areas as diverse as

William

Keith

George

Brian

Morgan

John

Riley Lee

Michael

Paul

Brian Keith, John, Michael, William

George John, Paul, William

John Brian, George, Paul

Keith Brian, Michael, Morgan, William

Lee Morgan, Riley

Michael Brian, Keith, Paul

Morgan Keith, Lee, Riley

Paul George, John, Michael

Riley Lee, Morgan

William Brian, George, Keith

Keith is important

𝑫 𝑷

(a) (b) (c)

Fig. 2. A graph, a drawing of that graph, and some knowledge perceived by the human.

biotechnology, in forensics, in software engineering, and epidemiology. For humans tomake sense of these graphs, a picture or graph drawing is helpful. In this paper, weintroduce the basic methods for creating pictures of graphs that are helpful for humans.

The graph data in Fig. 1(a) is a set of attributed graphs. Each such graph consistsof a set of vertices (sometimes called “nodes”) and a set of binary relationships (oftencalled “edges”) between the vertices. The vertices and edges usually have attributes.For example, the vertices in Fig. 2 have textual names. Edge attributes could include,for example, a number that quantifies the strength of a friendship.

The graph drawing in Fig. 1(b) is a “node-link” diagram: it consists of a glyphD(u) for each vertex u of the graph, and a curve segment D(e) connecting the glyphsD(u) and D(v) for each edge e = (u, v) of the graph. Each glyph D(u) has geometricattributes (such as position and size) and graphical attributes (such as colour). Similarly,each curve D(e) has geometric attributes (such as its route) and graphical attributes(such as colour and linestyle). Note that other kinds of graph drawing are possible;see Section 5.4 below. However, in this paper we will concentrate on the node-linkmetaphor, as it is the most commonly used.

In practice, it is relatively easy to find a good mapping from the vertex and edgeattributes to the graphical attributes of glyphs and lines, using well-established rules ofgraphic design (see, for example, [95]). A large variety of graphical notations exists incommon application areas, Figs. 3 and 4 show real world examples of attribute map-pings from biology. The representation in 3 uses the SBGN standard [70] and has beenproduced with SBGN-ED [21], an extension of the Vanted framework [83].

In contrast, it is difficult to find a good layout for a node-link diagram: if we chosethe location of each vertex and the route for each edge badly, then the resulting diagramis tangled and hard to read. In Section 2 below, we examine the geometric properties of“good” node-link diagrams. Then we describe methods for constructing good layouts ofnode-link diagrams. In particular we describe two important approaches: the topology-shape-metrics approach (Section 3), and the energy-based approach (Section 4).

2

Fig. 3. A part of a biological pathway drawn using the SBGN notation. Attributes are mapped tographical attributes. The network is a part of the visual representation describing the develop-ment of diabetic retinopathy, a condition which leads to visual impairment if left untreated.

2 Readability and faithfulness

We now consider the properties of “good” drawings of graphs. We concentrate on geo-metric properties, in particular the location of each vertex and the route for each edge.There are two aspects of the quality of a graph drawing: readability and faithfulness.

Readability concerns the quality of the perception function P in Fig. 1: how well doesthe human understand the picture? Two further drawings of the graph in Fig. 2(a) are inFig. 5. Intuitively, these two drawings are less readable than that in Fig. 2(b).

Geometric properties of readable drawings of a graph are commonly called “aes-thetic criteria”. Discussions of aesthetic criteria began in the 1970s. For example, Sugiyama(see [88]) and Tamassia et al. [91] produced structured lists of aesthetic criteria; a sam-ple is below.

C1 The number of edge crossings is minimized.C2 The total length of edges is minimized.C3 The ratio of length to breadth of the drawing is balanced.C4 The number of edge bends is minimized (using straight lines where possible).C5 Minimization of the area occupied by the drawing.

All these aesthetic criteria were based on intuition and introspection rather than any sci-entific evidence. Later, Purchase et al. [80] began the scientific investigation of aestheticcriteria, based on HCI-style human experiments. She measured the time to completetasks such as tracing a shortest path in a graph drawing, and errors made in such tasks.These variables were correlated with aesthetic criteria such as those above. Purchase

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Fig. 4. A combination of three metabolic pathways with vertex attributes represented by chartsand colors mapped onto the vertex representations. Bar charts represent the amount of a metabo-lite for four different plant lines.

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Keith

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Morgan

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Riley Lee Michael

Paul

(a) (b)

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Paul

Fig. 5. Poor quality drawings of the graph in Fig. 2(a).

found significant evidence that both time and errors increase with the number of edgecrossings and with the number of edge bends, and less significant evidence for other

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aesthetic criteria. Further experiments [79, 101, 57] confirmed, refined and extendedPurchase’s original work.

Faithfulness [75] concerns the quality of the drawing function D in Fig. 1. The draw-ing D(G) of a graph G is faithful if it uniquely represents the graph G. In other words,D is faithful it has an inverse; that is, if the graph G can be recovered uniquely fromthe drawing D(G). This concept may seem strange at first, because it may seem that allgraph drawings are faithful. However, the concept is significant for very large graphs.As a simple example, consider the graph in Fig. 6. This drawing uses a technique re-cently called edge bundling [54] (originally called edge concentration [74]) to copewith the large number of edges. While this drawing may be readable, it is not faithful:it does not uniquely represent a graph (because there are many graphs that could havethis drawing).

While readability has a long history of investigation, faithfulness has only arisensince the advent of very large data sets, and it is currently not well understood. Onefaithfulness criterion that has been proposed [75] is based on the intuition that in afaithful graph drawing, the distance between u and v in the graph G should be reflectedby the geometric distance between the positions D(u) and D(v) of u and v in the draw-ing. To make this notion more precise, suppose that �G(u, v) is the distance between uand v in G (for example, �G(u, v) could be the length of a graph-theoretic shortest pathbetween u and v). For a drawing function D that maps vertices of a graph G = (V,E)

to points in R2, we define

�(D(G)) = ⌃u,v2V (�G(u, v)��R2(D(u), D(v))

2 (1)

where �R2 is a distance function in R2 (for example, Euclidean distance). In otherwords, � is the sum of squared errors between distances in the graph G and distancesin the drawing D(G). In this way, � measures the faithfulness of the drawings insofaras distances are concerned. In the 1950s, Torgerson [94] employed a similar criterionwhen he proposed the Multidimensional Scaling method for psychometrics, a projectiontechnique that allows to represent distance information as a 2D or 3D picture. Over thefollowing decades, such methods were refined and extended into distance-based graphdrawing methods. With the success of the stress minimization approach, these methodshave recently gained more and more interest; see Section 4.

3 The topology-shape-metrics approach

Motivated by the need to database diagrams, Batini et al. [7] introduced a method fordrawing graphs. The method has been refined and extended many times, and it is nowknown as the “topology-shape-metrics” approach. An example of a graph drawing com-puted with this approach is in Fig. 7. The method has three phases:

1. Topology: first we compute an appropriate topological arrangement of vertices,edges, and faces. In this phase, we aim for a small number of edge crossings.

2. Shape: next we compute the general shape of each edge of the drawing. In thisphase, we aim for a small number of edge bends.

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Fig. 6. Money movements; an unfaithful graph drawing.

3. Metrics: Finally we compute the precise location of each vertex, each edge bend,and each edge crossing. In this phase, we aim for a drawing with high resolution.

Before we describe each of these phases, we outline the concepts of orthogonal griddrawings and planar graphs; these concepts are needed to understand the approach.

3.1 Orthogonal grid drawings

In a grid drawing of a graph, each vertex is located at an integer grid point, that is, ithas integer coordinates. Examples of grid drawings are in Fig 8. Grid drawings are usedto ensure that the drawing has adequate vertex resolution, that is, vertices do not lie tooclose to each other. Suppose that we have a grid drawing in which x

max

, xmin

, ymax

and ymin

are the maximum and minimum x and y coordinates of a vertex respectively.The area of the grid drawing is

A = (xmax

� xmin

)(ymax

� ymin

)

and the aspect ratio is

R =

ymax

� ymin

xmax

� xmin

.

The drawing in Fig 8(a) has area A = 35 and aspect ratio R =

5

7

. If the drawing isrendered on an X ⇥ Y screen, then the distance between two vertices is at least

A�0.5min

�XR0.5, Y R�0.5

�;

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Fig. 7. An orthogonal drawing of a UML class diagram, computed using the topology-shape-metrics method in OGDF (as published in [47], ACM Symposium on Software Visualization2003).

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(b)

Fig. 8. (a) A grid drawing. (b) An orthogonal grid drawing.

thus to obtain good vertex resolution, we want a grid drawing in which the area A is assmall and the aspect ratio R is close to Y

X .

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A graph drawing is orthogonal if each edge is a polyline consisting of vertical andhorizontal line segments. Fig 8(b) shows an orthogonal grid drawing. Note that an or-thogonal drawing of a graph with a vertex of degree greater than 4 is necessarily un-faithful; thus for the moment we assume that every vertex has degree at most 4. Inpractice this restriction can be overcome by a variety of methods (such as the Kandin-sky approach [32]).

Orthogonal drawings are widely used in software design diagrams, such as Fig. 7.From the aesthetic criterion C4 in Section 2 want an orthogonal graph drawing with fewbends.

3.2 Planarity and topology

A drawing of graph is planar if it has no edge crossings; a graph G is planar if there is aplanar drawing of G. Examples of planar and non-planar graphs are in Fig. 9. The theory

(a) (b) (c) (d) (e)

Fig. 9. The graph drawings (a) and (b) are planar, and the graph drawings (c), (d), and (e) arenon-planar. However, the graph in (c) is planar, because there is a planar drawing of this graph.The graphs in (d) and (e) are non-planar.

of planar graphs has been developed by mathematicians for hundreds of years. Forexample, Kuratowski [69] gave an elegant characterisation of the class of planar graphs:a graph is planar if and only if it does not contain a subgraph that is a subdivision of thecomplete graph K

5

on five vertices or the complete bipartite graph K3,3 on six vertices

(a subdivision of a graph is formed by adding vertices along edges). The classical butinelegant linear-time algorithm of Hopcroft and Tarjan [55] can be used to test whethera graph is planar; simpler linear time algorithm have been developed more recently [12,86].

A planar graph drawing divides the plane into regions, called faces. The drawing Ain Fig. 10 has seven faces f

0

, f1

, . . . , f6

(note that f0

is the outside face of the drawing).Two drawings of a graph G are topologically equivalent if there is a continuous defor-mation of the plane that maps one to the other; an equivalence class under topologicalequivalence is a topological embedding of G. To illustrate this, consider the three planardrawings A, B, and C of a graph in Fig. 10. Here A and B have the same topologicalembedding: it is possible to deform the plane so that A transforms to B. It can be seenthat C has a different topological embedding, because while A and B both have a facewith eight vertices, C has no such face.

It is well-known that two planar drawings of the same graph are topologically equiv-alent if and only if the clockwise circular order of edges around each vertex is the same.

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𝒇𝟒 𝒇𝟐 𝒇𝟏

𝒇𝟓 𝒇𝟑

𝒇𝟎

𝒇𝟔

𝒇𝟒

𝒇𝟐

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𝑨 𝑩 𝑪

Fig. 10. The graph drawings A and B are have the same topology. The graph drawing C displaysthe same graph, but the topology is different.

One can check this property for the examples in Fig. 10. This property is a combina-torial characterization of a topological embedding, and can be used to construct datastructures that implement operations on topological embeddings efficiently (see, for ex-ample, [19]). Variations of the Hopcroft-Tarjan algorithm (see, for example, [72]) canbe used to construct a topological embedding (as a clockwise circular ordering of edgesaround each vertex) of a planar graph in linear time.

3.3 Computing the topology, using planarization

A non-planar graph drawing can be converted into a planar graph drawing simply byadding new “dummy” vertices at each crossing point, as illustrated in Fig. 11. This

(a) (b)

Fig. 11. Adding dummy vertices to the non-planar graph drawing (a) gives the planar drawing(b).

simple process of adding dummy vertices gives the intuition for the “topology” phase ofthe topology-shape-metrics method. However, graph visualization begins with a graph,not with a graph drawing, and converting to a planar graph is not so straightforward.Further, from the aesthetic criterion C1 in Section 2, we want to ensure that the numberof crossings is small.

The “topology” phase, sometimes called a planarization process, takes a non-planargraph is G = (V,E) as input, and produces a planar topological embedding G000

=

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(V 000, E000) as output. The first step is to find planar subgraph G0

= (V,E0) (E0 ⇢ E)

where |E0| is as large as possible. This is a non-trivial problem; in fact, finding a maxi-mum planar subgraph of a given graph is NP-hard [42]. However, a number of heuristicmethods are available [60–62]. The next step is to find a topological embedding G00

of the planar graph G0. This step is relatively easy, and can be accomplished in lineartime using a variation of the Hopcroft-Tarjan algorithm, or perhaps one of the simpleralgorithms developed more recently (for example, see [11, 86]). The third step is to in-sert the edges of E � E0. The aim in this step is to minimize the number of crossings;although this is NP-hard, it is common to use the simple strategy of inserting one edgeat a time, locally minimizing crossings at each insertion. The local minimization canbe done by using a shortest path algorithm on the graph of faces of G00. This gives ourplanar topological embedding G000

= (V 000, E000).

We can illustrate the planarization process with an example. We begin with a graphG, represented as a table in Fig. 12(a); note that this is a combinatorial graph, withno topology or geometry. A (bad) drawing of this graph is in Fig. 12(b). Further, Gis a non-planar graph by Kuratowski’s Theorem, because there is a subgraph (shownin Fig. 12(c)) that is a subdivision of the complete graph K

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on 5 vertices. Next we

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Nodes Adjacent nodes

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A graph 𝑮 A drawing of 𝑮

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2 6

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A subdivision of 𝑲𝟓 in 𝑮

Fig. 12. A non-planar graph.

identify a large planar subgraph G0 of G, using one of the heuristic methods available.In this case, we can delete the edges (2, 6) and (6, 7) to give a planar subgraph. Usinga variation of the Hopcroft-Tarjan algorithm, we can find a topological embedding G00

of G0. Such an embedding is illustrated in Fig. 13(a). Finally, we re-insert the edges(2, 6) and (6, 7), and place dummy vertices a, b and c at the crossing points, to give atopological embedding G000, as in Fig. 13(b).

3.4 Computing the shape

The output of the topology phase is a topological embedding, which we shall nowdenote as G. Some of the vertices of G are dummy vertices, representing crossingsbetween edges in the original graph. The final drawing output from topology-shape-

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Fig. 13. (a) A topological embedding G00 of G0. (b) The edges (2, 6) and (6, 7) are re-inserted,with dummy vertices a, b, and c, to form a topological embedding G000.

metrics method is an orthogonal drawing, in that each edge is a sequence of horizon-tal and vertical line segments. The shape phase chooses “shape” of each edge, in thefollowing sense. Suppose that the edge (u, v) is directed from u to v, and it consistson a sequence (u

0

, u1

), (u1

, u2

), . . . , (uk�1

, uk) of k segments, where u0

= u anduk = v. Each line segment (ui, ui+1

) has a compass direction: either north, south,east, or west. The sequence (d

0

, d1

, . . . , dk�1

), where di is the compass direction of(ui, ui+1

), is the shape of the edge (u, v). As examples, the edge (0, 1) in Fig. 14(a)has shape (north, east, south), and the edge (1, 2) has shape (west, south, west).

Two orthogonal drawings A and B have the same shape if each edge in A has thesame shape as the corresponding edge in B. The drawing in Fig. 14(a) has the sameshape as that in Fig. 14(b). However, Fig. 14(c) has a different shape.

Note that drawings in Figs. 14(a) and (b) have 19 edge bends each, but Fig. 14(c)has only 8. The aim of the shape phase is to choose a shape with few bends.

A surprising result of Tamassia [90] gives a polynomial-time algorithm for choosinga shape with a minimum total number of edge bends. Tamassia’s algorithm is based ona reduction to the maximum flow problem; the best implementation [43] runs in timeO(|V |1.75). A simpler algorithm of Tamassia and Tollis [92], based so-called “visibilitygraphs”, runs in linear time and results in a drawing with at most 4 bends per edge (butnot necessarily giving a minimum total number of bends).

A naıve routing of orthogonal edges for the topological embedding illustrated inFig. 13 is given in Fig. 15(a). A better shape for this embedding is in Fig. 15(b).

3.5 Computing the metrics

The shape phase described above determines the sequence of horizontal and verticalline segments that make up each edge. The metrics phase chooses integer coordinatesfor each vertex, each bend, and each crossing point. Each of these points is located

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Fig. 14. Three orthogonal drawings with the same topological embedding. The drawings in (a)and (b) have the same shape, and each has 19 edge bends; the drawing in (c) has a differentshape and has 8 edge bends.

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Fig. 15. Two orthogonal drawings with the topological embedding in Fig. 13. The shape (a) hasmore bends than (b).

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Fig. 16. Two orthogonal grid drawings with the same shape. The drawing (a) has a larger areathan (b).

at an integer grid point, and thus we have an orthogonal grid drawing. The main aimof the metrics phase is to give a drawing of small area; the phase is sometimes called

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compaction. An example is in Fig. 16. The problem of constructing a layout with smallarea has a long history in the literature of VLSI layout, and methods can be borrowed.

As the final step of the metrics phase, the graph is rendered without rendering thedummy vertices. The final drawing for the graph in Fig. 12 is in Fig. 17.

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Fig. 17. final drawing for the graph in Fig. 12.

3.6 Remarks and open problems for the topology-shape-metrics approach

The topology-shape-metrics approach has been improved, refined, and extended manytimes since its first inception. Figs. 3 and 7 are examples of the output of such algo-rithms. The methods work well for small-to-medium sized orthogonal graph drawing,but are less successful on large graphs.

Nevertheless, a number of open problems remain:

1. Clustered planarity. A common method for dealing with very large data sets is tocluster the data, then treat each cluster as a data item. This method is also used ingraph drawing: the vertices of a very large graph can be clustered to form “super-vertices”; these super-vertices can be clustered to form ”super-super-vertices”, andso on, in a hierarchical fashion. More formally, a clustered graph C = (G, T )consists of a graph G and a rooted tree T such that the leaves of T are the verticesof G. The tree T forms a cluster hierarchy on the graph. A drawing of a clusteredgraph C = (G, T ) consists of a drawing of the graph G and a region r(t) of theplane for each vertex T of the tree T , such that(a) if t

1

is a child of t0

in T , then r(t1

) ⇢ r(t0

).(b) if t

1

is not a descendent of t0

and t0

is not a descendent of t1

in T , then r(t1

)\r(t

0

) = ;.

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(c) if u is a vertex of G (and thus a leaf of T ) then the location of u in the drawingof G is inside r(u).

(d) if (u, v) is an edge of G and the curve representing (u, v) intersects r(t) forsome vertex t of T , then either u or v is a descendent of t.

(e) if (u, v) is an edge of G and both u and v are descendants of t, then the curverepresenting (u, v) is inside r(t).

Further, the drawing of C is clustered-planar if the drawing of G is planar. Anexample of a drawing of a clustered graph C = (G, T ) is in Fig. 18(a); note thatthis drawing is not clustered-planar. The tree T is illustrated in Fig. 18(b). Note that,

r1 r2 r3 1

3 2

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6 5

7

9 8 r1 r2 r3

1 3 2 4 6 5 7 9 8

root

(a) (b)

Fig. 18. (a) A drawing of a clustered graph C = (G, T ); (b) the tree T .

although the underlying graph G is planar (see Fig. 19), there is no clustered-planardrawing of this clustered graph. (Observe that in any planar embedding of G, the3-cycle (4, 5, 6) must have either at least one of the 3-cycles (1, 2, 3) or (7, 8, 9)inside; thus the cluster region r

2

would have to contain either cluster region r1

orr2

, contradicting the above rules for clustered drawings.)

1

3 2

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6 5

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9 8

Fig. 19. The underlying graph G for the clustered graph in Fig. 18.

A clustered graph is clustered-planar if it has a clustered-planar drawing. Clus-tered planarity is a significant problem: shape/metrics steps for clustered graphs

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are well established, but despite much investigation [27, 31, 58, 22, 20, 46, 20], theplanarization step is still unsolved.

2. Different ways to count edge crossings. In the mid-1990s, Mutzel performed an in-formal experiment during a lecture at a conference. She showed the audience twodrawings of the same graph, shown in Fig. 20(a) and (b). The audience overwhelm-

Fig. 20. The Mutzel experiment.

ingly preferred (a), despite the fact that it has significantly more edge crossings than(b). In fact, most of the audience mistakenly assumed that (b) had fewer edges than(a). Mutzel’s experiment challenged the conventional wisdom that simply count-ing the number of edge crossings gives a good metric for the quality of a graphvisualization, and led to a number of new directions for research:

– A number of researchers [4, 13, 45, 89, 30, 28] have begun to investigate k-planar drawings where the number of crossings on each edge is at most k. Forexample, the drawing in Fig. 21 is 1-planar. Most of this research concentrateson mathematical properties of k-planar drawings with k = 1 or k = 2; a goodpractical algorithm for finding a k-planar drawing with minimum k remainsunknown.

– Huang et al. [57] showed that edge crossings are tolerable if the crossing angleis large. For example, all crossings in the drawing in Fig. 21 at right angles.For orthogonal drawings, all crossing angles are right angles, and perhaps thenumber of crossings is not significant at all! (See [10] for an orthogonal draw-ing algorithm that ignores edge crossings). Current research has been mostlymathematical (see [2, 3, 23]), and the investigation of good practical methodsfor drawing with large crossing angles is just beginning.

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Fig. 21. A 1-planar drawing. Note that all edge crossings are at right angles.

4 Energy-based approaches and stress minimization

The most popular approach to create a layout for undirected graphs is based on so-calledenergy-based layout methods. This popularity is due to the intuitive underlying model ofthe basic versions, and the fact that these methods can be reasonably easy to implement.In addition, the resulting layouts are often aesthetically pleasing, drawings are describedto have a more “organic” or natural appearance than drawings from other methods, andthat they show symmetries well. Edges are normally represented as straight lines, whichmakes bend minimisation unnecessary. Fig. 22 shows an example drawing created withan energy-based method in comparison to an orthogonal drawing.

(a) (b)

Fig. 22. Drawing of a Sierpinski triangle, a fractal defined as a recursively subdivided triangle.(a) Drawing created by an energy-based method (b) Drawing created with the topology-shape-metrics approach described in Section 3.

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The underlying concept of energy-based methods is to model the graph as a sys-tem of objects that contribute to the overall “energy” of the system, and energy-basedmethods then try to minimize the energy in the system. A basic assumption for thesuccess of such an approach is that a low energy state of the system corresponds to agood drawing of the graph. In order to achieve such an optimum, an energy functionis minimized. Energy-based methods thus consist of two main components: a model ofobjects and their interactions (a virtual physical model), and an algorithm to computea configuration with low energy (an energy minimization method). There are variousmodels and algorithms under this approach, and the flexibility in the definition of theenergy model and energy function implies a wide range of both optimization methods.

Energy-based drawing methods have a long history. Tutte [97, 98] used such anapproach in one of the earliest graph drawing methods, based on barycentric represen-tations that are obtained by solving a system of linear equations. Tutte proposed thebarycenter algorithm to draw a triconnected planar graph G = (V,E), and showed thatthe result is a planar drawing where every face is convex. The algorithm proceeds byfirst selecting a subset A of the vertices of the graph G to constitute the outer face ofthe topological embedding of G. The vertices of A are placed on the apices of a convexpolygon, and are fixed. Each remaining vertex is placed so as to minimise an energyfunction that simulates a system of elastic bands. In fact, minimum energy is obtainedwhen each vertex in V �A is at the barycenter of its graph-theoretic neighbors. This set-ting can be modelled by a non-degenerate system of linear equations, where the positionof each vertex is determined as a convex combination of its neighbors’ positions. Sucha system has a unique solution that can be computed in polynomial time. Barycentredrawings can be very beautiful. However, many barycentre drawings have poor vertexresolution, in the sense that vertices can be placed very close to each other. See Fig. 23for an example.

Implementations of energy-based layout methods can be found in a large numberof software tools and web-services, and most drawings published in both a scientificand non-scientific context are computed using such methods. Some of the more so-phisticated energy-based methods allow us to compute layouts for graphs with severalhundreds of thousands vertices in seconds on a standard desktop machine.

A classical, and still frequently used, example for energy-based methods are the so-called force-directed models, where the graph objects are modeled as physical objectsthat mutually exert forces on each other. In the most simple model, unconnected verticesrepel each other, and vertices linked by edges attract each other. Force-directed methodshave also been applied early for printed circuit board design, where a system of elasticleads and repulsive forces was described for the construction of circuit board drawings.For example, the spring embedder model [25] models vertices as electrically chargedsteel rings and edges as springs, such that the electrical repulsion between vertices andthe mechanical forces exerted by the springs in a given layout define the energy of thesystem, see Fig. 24. A minimization of the overall system energy is associated witha layout that optimizes the Euclidean distances between the vertices with respect to agiven ideal distance.

The minimization is done in an iterative fashion, moving towards a local energyminimum. First, the vertices are placed in an initial layout, and then in each iteration

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Fig. 23. A barycentre drawing.

a displacement is computed for each vertex based on the forces exerted on it by vertexrepulsion and edge attraction. At the end of the iteration, the positions of all verticesare updated, and a new iteration is started unless the overall displacement falls under acertain threshold. The spring between vertices u and v has an ideal length `, and in agiven layout this spring has a current length �R2

(u, v) (the Euclidean distance betweenu and v). A variation of Hooke’s law is applied to compute the force exerted by a spring,based on the relation between ` and �R2

(u, v): if �R2(u, v) is larger than `, then the

vertices are attracted to each other, and if it is smaller then they are repelled.

The iterations can be continued until the total force on each vertex converges tozero. In practice the number of iterations may be limited to a bound K that depends onthe size of the graph; then the runtime is O(K|V |2)).

While the first energy-based methods and models were intuitive and rather simple,and the corresponding methods were widely successful in practice, they also exhibitcertain disadvantages. First of all they are rather slow and do not scale well to graphswith more than a few hundred vertices. They are thus not well suited to cope with themuch larger graphs that are visualised today, such as protein-protein interactions inbiology or social interactions in social network analysis, with thousands to millions ofvertices and edges. Secondly, they rely on an initial drawing and tend to get stuck in alocal energy minimum during optimisation; see Fig. 25 for examples.

Recent approaches, discussed in Section 4.1 below, are more complex and makeuse of more advanced mathematical methods for the optimization. This development

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(a) (b)

Fig. 24. Applying the spring embedder model (b) to the graph in (a). Vertices are modeled bysteel rings, edges by springs. Springs exert a force when their length deviates from the naturalspring length, which is a parameter for the model, and vertices repel each other. In (b), repulsionforces from the darker shaded steel ring are represented by arrows. The force decreases with thesquare of the distance between vertex pairs.

(a) (b)

Fig. 25. Typical unfolding and convergence problems for iterative force-directed algorithms: ASierpinski triangle with 1095 vertices (a) and a tree with 1555 vertices (b). Even though bothgraphs are planar and sparse, no planar drawing with good structure representation was computed.

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allowed large improvements both in the drawing quality and in the computational effi-ciency.

4.1 Scaling to large graphs

Three major directions can be identified which in recent years lead to large improve-ments both regarding the layout quality and the runtime performance. The first one isthe emergence of so-called multilevel methods. The second one is the improvement inoptimization process, in particular fast approximations for the energy computation. Thethird one is the identification of better energy functions.

Multilevel methods. The multilevel paradigm is a generic approach to handle largedatasets by reducing the complexity over a number of hierarchically ordered levels. Itis well suited for graph algorithms and can be used to improve energy-based layoutmethods, regarding both the layout quality and computation time. The first use of themultilevel approach is commonly credited to Barnard and Simon [5], where it was usedto speedup the recursive spectral bisection algorithm. In the context of graph partition-ing, Karypis and Kumar [64] showed that the quality of the multilevel approach canalso be theoretically analyzed and verified.

Multilevel layout methods consist of three components, coarsening, single-levellayout, and placement. The main idea is to construct a sequence of increasingly smallergraph representations (“coarsening levels”) that approximately conserve the global struc-ture of the input graph G, and to then compute a sequence of approximate solutions,starting with the smallest representation. Intermediate results can be used on the subse-quent level to speed up the computation and to achieve a certain quality.

The graph representations are typically created by series of graph contractions,where a set of vertices on one level is collapsed to a single representative on the next,smaller level. A contraction operation can for example simply be an edge contraction,that is, a set of two adjacent vertices is collapsed, and the coarsening for one levelthen includes contractions of all edges from a maximum independent edge set. Thesecontractions are repeated until the graph size reduces to a given threshold. For graphdrawing purposes, usually a threshold of 10-25 vertices is chosen, where force-directedmethods can achieve a high quality drawing quickly. For the resulting levels of thecoarsening phase now single-level layouts are calculated. After computing a layout forthe coarsest level from scratch, for each of the intermediate levels a force-directed lay-out method is applied. As each vertex v from the coarser level li represents a set ofvertices s on the current, finer level li�1

, the layout for li can be used to create an ini-tial drawing for li�1

that is then iteratively improved. Initial positions for vertices canbe derived from li during this placement phase by simple strategies, such as placingvertices at the barycenter of their neighbors. Experiments indicate that the influence ofdifferent placement strategies is marginal [6].

Although several layouts have to be computed, including one for the original graph,the reuse of intermediate drawings leads to less required work and much better conver-gence on each level than for single level methods.

Walshaw [100], Harel and Koren [52], and Gajer et al. [35], introduced the multi-level paradigm to graph drawing, after a closely related concept, the multi-scale method,

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Fig. 26. Several levels during multilevel layout computation for a graph with 11143 vertices and32818 edges. The leftmost drawing shows the coarsest level, the rightmost the final drawing. Thegraph is part of the 10th DIMACS implementation challenge, and available from the UFL SparseMatrix Collection [76].

was proposed by Hadany and Harel [50]. Another related method, the FADE paradigm [81],used a geometric clustering of the vertex locations for coarsening.

Multilevel approaches can help to overcome local minima and slow convergenceproblems, by improving the unfolding process due to a good coarsening and subsequentplacement. However, while multilevel methods can cope even with very large graphs, itmay still happen that the resulting layout represents a local minimum far from the opti-mum. Hachul and Junger [49] presented an experimental study of layout algorithms forlarge graphs, including energy-based multilevel approaches. Bartel et al. [6] presentedan experimental comparison of multilevel layout methods within a modular multilevelframework.

Fast Approximations. Another important concept for the practical improvement ofenergy-based methods is the approximation of the forces to speed up the force cal-culation. Typically, the repulsive forces are computed approximately whereas the at-traction forces are computed exactly. This means that all edges are taken into account,but individual forces are not calculated for every pair of vertices, as this would meana runtime of ⌦(|V |2). One of the first such approximations was the grid-based vari-ant of the Fruchterman-Reingold algorithm [33], which divides the display space intoa grid of squares and for each vertex restricts repulsive forces to vertices within nearbysquares. More sophisticated versions [48, 81] involve the application of space decom-position structures such as quadtrees (see Fig. 27) for geometric clustering, as well asefficient approximation schemes such as the multipole method (for example, see [103].

Hachul and Junger combine a multilevel scheme with the multipole approximation,leading to a very fast layout algorithm. With an asymptotic runtime of O(|V | log |V |+|E|), in practice the algorithm is capable to create high quality drawings of graphs upto 100000 vertices in around a minute.

Spectral graph drawing methods are fast algebraic methods that compute a layoutbased on so-called eigenvectors, sets of vectors associated with matrices defined by theadjacency relations in a graph. Spectral drawing methods were introduced by Hall [51]in the 1970s. Later developments include the algebraic multigrid method ACE [68] andthe high-dimensional embedding approach HDE [53]. These methods show that alge-braic methods are very fast and can create reasonable layouts for a variety of graphclasses. However, these methods tend to hide details of the graph and are prone to de-

21

Fig. 27. Use of a quadtree for space partitioning, as shown in [48]. First the drawing space isrecursively partitioned into four squares, until each square only contains a few vertices (left). Theresulting hierarchy can be efficiently represented by a quadtree structure (right), which in turn canbe used to allow an efficient force approximation. Forces that act on a vertex are only calculateddirectly for close-by vertices, whereas the force contribution of a group of vertices that is in a faraway partition is only approximated, replaced by a group force. In the left drawing, the impact ofvertices 9,10, and 11 on vertex 1 is combined in an approximated group force.

generative effects for some graph classes, such as where large subgraphs are projectedon a small strip of the drawing area. Hachul and Junger’s experimental study of largegraph layout methods, which also includes algebraic approaches, consequently showsthat, while being very fast, algebraic methods often fail to compute reasonable draw-ings.

Better Energy Functions. Distance-based drawing methods constitute an alternativeperspective to the graph layout problem. They aim to create a faithful projection froma high dimensional space to 2D or 3D, with the dimensions simply arising from somenotion of dissimilarity, or distance, of each vertex to all other vertices. To this end,distance-based methods usually minimize the stress in the drawing, which measures thedeviation of the vertex pair distances in the drawing to their dissimilarity. The objectivefunction for stress minimization is

⌃u,v2V wuv (�⇤(u, v)��R2(D(u), D(v))

2 (2)

which sums up the stress for all pairs of vertices u,v located at positions D(u) andD(v) respectively, where duv�⇤(u, v) is the desired distance between u and v. Notethe similarity between equations (1) and (2); we can regard stress minimisation methodsas faithfulness maximisation methods. The value wuv is a normalization constant whichis often set to 1

�G(u,v)2 , where �G(u, v) is the length of the shortest path between u

and v. This emphasizes the influence of deviations from the desired distance for pairs ofvertices that have a short graph theoretic distance, and the influence is dampened withincreasing distances between pairs of vertices.

The energy-based approach by Kamada and Kawai [63] uses the shortest graph-theoretic paths as ideal pairwise distance values and subsequently tries to obtain a draw-ing that minimizes the overall difference between ideal and current distances in an iter-ative process. Kamada and Kawai propose to use a two-dimensional Newton-Raphsonmethod to solve the resulting system of non-linear equations in a process that moves onevertex at a time to achieve a local energy minimum. As the cost function involves theall-pairs shortest-path values, the complexity is at least O(|V |2log|V |+ |V ||E|) or |V |3

22

for weighted graphs, depending on the algorithm used, and O(|V |2) for the unweightedvariant. Following the Kamada-Kawai model, stress majorization was introduced as analternative and improved solution method [40]. Several improvements were proposedto make such methods more scalable, for example by approximation of the distances[65], adding an entropy model [38], or to respect non-uniform edge lengths [37].

Each of these improvements led to methods that clearly outperform their prede-cessors in runtime and layout quality. The combination of the multilevel approach andthe multipole method for force approximation in the Fast Multiple Multilevel MethodFMMM [48], or the use of the maxent-stress model [39], efficiently computing drawingsthat clearly depict the structure of many graphs up to a size of several thousand verticesand edges.

4.2 Constraint-based layout using stress

Even though not suited for drawings of large graphs, constraint-based drawing meth-ods constitute the most flexible approach to draw graphs in practice. Constraint-basedmethods take a declarative approach to graph drawing [71], that is, instead of giving analgorithm that describes how to compute a drawing, requirements for the drawing aredefined, for example, by geometric constraints. They often resort to generic solutiontechniques, such as integer linear programming methods, to solve the resulting systemof constraints, and are thus often significantly slower than other approaches.

The big advantage of constraint-based methods is that they are able to create highquality drawings for small to medium graphs while taking into account user-definedconstraints, such as requirements from a drawing convention or personal preferences.

As a result, such methods are getting more and more popular in application areaswhere highly constrained drawings are required; for example, where representation ofstructural and semantical information beyond the basic structure is needed. Applica-tions include the drawing of technical and flow diagrams to depict hardware systemsor biological processes [85, 84] In the last years great progress has been made in suchmethods to allow interactive graph layout, in commonly used environments such as webbrowsers [99].

While constraint-based methods are well suited for relatively easy extension of adrawing approach by additional drawing constraints, this comes at the cost of decreasedcomputational efficiency of the resulting approach. While there have been approaches tospeed up the optimization for specific constraint types [24], the runtime performance isstill an impediment for a more wide-spread use of contraint-based methods. In addition,to guarantee a certain quality of the computed drawings a good compromise has tobe found to prioritize more important soft constraints over less important ones, and aconflict-solving strategy has to be employed.

For further reading, see the surveys in [66, 56].

4.3 Remarks and open problems for energy-based methods

Energy-based methods, in one form or another, are well established tools for graphvisualization. Nevertheless, many open problems remain:

23

1. Animation. An important advantage of energy-based methods, based on the itera-tive nature of the numerical methods to compute the layout, is that they allow asmooth animation of the change from one drawing to another (since the energyfunction is smooth). They provide one kind of solution to the so-called “mentalmap problem” [73]. Although existing graph drawing tools often use this kind ofanimation, it has received little attention form researchers. A thorough investigationof energy-based animation methods would be useful.

2. Why are some graphs hard to draw? Energy-based methods are successful on manygraphs, but unsuccessful on many others. Intuitively, some graph-theoretic proper-ties are behind success or failure; for example:

– Dense graphs (that is, graphs with many edges) often lead to a “hairball” draw-ing that makes it hard to perceive the graph structure.

– Low diameter graphs seem to become cluttered with distance-based methods.It would be useful to justify this intuition with mathematical theorems.

3. Edge crossings and energy-based methods. It is commonly claimed (without justi-fication) that energy-based methods reduce edge crossings. However, in some casesit is seems impossible to avoid crossings with energy-based methods, even on edgesof planar subgraphs [1]. The relationship between low energy drawings and edgecrossings needs investigation, both mathematically and empirically. (Note that re-cent results seem to indicate however that crossings might not be the dominatingfactor regarding readability of large graphs; see [29, 67]).

5 Further topics

5.1 Directed graphs

The methods described in Sections 3 and 4 can be used to draw directed graphs (as inFig. 7), but these methods ignore the directions on the edges. It can be useful to havethe arrows representing directed edges laid out so that the general “flow” is from oneside of the screen to another. For example, in Fig. 28, the “flow” is mostly from the topto the bottom.

Sugiyama et al. [88] described a method for drawing directed graphs. The vertexset is divided into in “layers”, and each layer is drawn on a horizontal line; in Fig. 28there are 4 layers. The layers are chosen so that there are not too many layers, thenumber of vertices in each layer is not too large, and the edges are mostly directedfrom a higher layer to a lower layer. Then the vertices are ordered inside each layer inan attempt to minimise the number of edge crossings. Finally each vertex is given alocation (within its layer and respecting the ordering of that layer) so that edges are asstraight as possible. The Sugiyama method involves a number of NP-hard combinatorialoptimisation problems, but each of these problems has heuristic solutions that workreasonably well in practice. The method has been refined and improved significantlysince its original conception, most notably by Gansner et al.[41].

5.2 Trees

Rooted trees are normally drawn with the root at the top of the screen, and parents abovetheir children, as in Fig. 29(a). Rooted trees are directed graphs, and the Sugiyama

24

Fig. 28. A directed graph.

method described above can be used; however, simpler methods are available. A naıvealgorithm to draw trees in this way is a simple exercise; however, a naive approachoften leads to drawings which are too wide. Reingold and Tilford [82] defined a linear-time algorithm which moves subtrees together to avoid excessive width. The Reingold-Tilford algorithm has been improved and extended many times (see, for example, [16]).Unrooted trees can be drawn using energy-based methods. However, simple algorithmsusing drawing vertices on layers of concentric circles, as in Fig 29(b), are described in[9].

5.3 Interaction

For large and complex graphs, interactive exploration is necessary. Interactive opera-tions include:

– Computer-supported filtering. For example, the system may detection salient struc-tural or semantic features, and filter out all vertices and edges that are not related tothese features.

– Zoom and pan. If the user wishes to concentrate on a specific part of the graph,focus+context methods can be used in combination with a number of pan methods.These include screen-space methods such as fish-eye views (see, for example, [34])and slider bars, as well as graph-space methods (see, for example, [26]).

5.4 More metaphors

The node-link metaphor described in this paper is most common, but other visual rep-resentations of graphs are used as well. Examples include:

25

(a) (b)

Fig. 29. (a) A rooted tree drawing; (b) an unrooted radial tree drawing.

– The map metaphor is used in Fig. 30(a) to show another picture of the graph inFig. 1(a). In this case, each vertex is represented by a region of the plane, andfriendship between two people is represented as adjacency between regions. Thismetaphor has been developed extensively, from “treemaps” [59] to “Gmaps” [36].

– The adjacency matrix metaphor, illustrated in Fig. 30(b), has been shown to beuseful in some cases [44].

William

Keith

George

Brian

Morgan

John

Riley Lee

Michael

Paul

Brian

Geo

rge

Joh

n

Keith

Lee

Mich

ael

Mo

rgan

Paul

Riley

Willaim

Brian

George

John

Keith

Lee

Michael

Morgan

Paul

Riley

William

(a) (b)

Fig. 30. Drawings of the graph in Fig. 2: (a) using the “map metaphor”, (b) using the adjacencymatrix metaphor.

– Edge bundling. For a dense and complex graph, edges can be bundled together asin Fig. 6. This method reduces edge clutter at the cost of reduced faithfulness; itseems to improve human understanding of global structural aspects of the graph.

– Edge stubs. One interesting way to reduce edge clutter is by removing the parts ofthe edges where crossing occur, as in Fig. 31. For the case of directed graphs, Burch

26

et al. [17, 15] show that for complex tasks the error rate increases with decreasingstub length, while for simple tasks (such as detection of the vertex with highestdegree) the stub drawing can be beneficial in completion time and error rate.

Fig. 31. Representation of edges as stubs to overcome readability problems due to crossings andclutter. Example as given in [15]. (a) Original drawing with crossings (b) Maximum stub sizewhere both stubs have the same length. (c) Same as (b) with the additional constraint that the stublength to edge length ratio is the same for all edges – nearly nothing is left of the original edgelines.

6 Concluding remarks

The two graph drawing approaches described in Sections 3 and 4 cover the main graphdrawing algorithms in research and in practice. A large number of companies and or-ganisations distribute graph drawing software. Some examples:

1. Tom Sawyer Software – This commercial enterprise [87] currently dominates themarket for graphing software. Energy-based methods, orthogonal grid drawings,directed graph methods, and tree drawing methods are included. The methods arepackaged in many different ways to handle a variety of data sources.

2. OGDF – The Open Graph Drawing Framework [18] is a self-contained open sourcelibrary of graph algorithms and data structures, freely available under the GPL at[77]. It is implemented in C++ and offers a wide variety of efficient graph drawingalgorithm implementations, in particular covering planarization- and energy-basedapproaches. OGDF is maintained and used by several university research groupsaround the world.

3. Tulip – Tulip [96] is an information visualisation framework which is freely avail-able under the LPGL. It provides a Graphical User Interface and a C++ API. TheGUI allows import of a variety of graph file formats and provides a range of layoutalgorithms.

4. WebCoLa – WebCoLa is an open-source JavaScript library for arranging HTML5documents and diagrams using constraint-based optimization techniques. It sup-ports interactive layout generation in a browser and works well with the well-knownD3 library.

All of the above systems have one or more energy-based methods. In contrast, thetopology-shape-metrics approach is seldom implemented in practical systems, despite

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significant attention from researchers and scientific evidence of readability. There are anumber of possible reasons for the lack of commercial interest in the topology-shape-metrics approach: it is much more complex than the energy-based approach, and theapproach does not seem to scale visually to larger graphs. Further research is needed tounderstand why energy-based methods are dominant in practice.

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