hypergraph drawing by force-directed placementnaheed/files/hypergraphdrawing.pdfthe task of...

Hypergraph Drawing by Force-directedPlacement

Naheed Anjum Arafat and Stephane Bressan

School of Computing, National University of Singapore, 13 Computing Drive,Singapore, 117417

{[email protected],[email protected]}

Abstract. We address the problem of drawing hypergraphs. We proposeand evaluate a family of algorithms that transform a hypergraph drawingproblem into a graph drawing problem. Each algorithm in the familyrelies on a different encoding of a hyperedge into a set of edges. Thealgorithms can then use any graph drawing algorithms. However, wehere focus on the force-directed algorithms.We propose and discuss a number of criteria to evaluate the quality ofthe drawings from the points of view of aesthetics and of visualizationand analytics. We also devise several metrics to quantify selected aspects.We empirically and comparatively evaluate the quality of the drawingswith both synthetic and real data sets. The experiments reveal that theapproach is practical, efficient and generally effective, and, more impor-tantly, the drawings generated are not only aesthetically pleasing butalso readable for the purpose of visualization and analytics.

1 Introduction

Fruchterman and Reingold, in their 1991 seminal paper, presented the popularforce-based graph layout method that bears their names. Since then, many im-provements on Fruchterman-Reingold’s algorithm [17][14] as well as other force-based layout algorithms [24][19] have been proposed.

The task of Visualizing a graph can be decoupled into the following tasks -Designing a Graph Layout algorithm and Rendering the Graph according to theLayout Algorithm in the drawing canvas. A graph layout algorithm generates a2D or 3D realization of the Graph and a graph renderer is usually an API whichhandles the rendering of the realization on its drawing canvas using the drawingprimitives available in it. Vertices are typically represented by filled circles andedges as line segments (straight line drawing) or curves. Though the 2D or 3Ddrawing of the graph is not unique, certain aesthetic criterion exists character-izing a good drawing, for instance, minimum edge crossing, symmetry, uniformedge length etc.We address the problem of drawing hypergraphs. A Hypergraph is a generaliza-tion of a Graph where n-ary relations exists among the vertices comprising theedges including binary relations as in graph. Edges of a hypergraph are calledhyperedges. The ability to capture n-ary relationships among the data points

makes hypergraphs particularly useful as a structure to represent multi-ary re-lations. Consequently, hypergraphs have gained a lot of interest in relationaldatabase [12], computational biology [21][25], social networks [5] and VLSI cir-cuit visualization [11]. A key challenge in visualizing higher order relations is theexponential increase in the number of potential relations with the number of itsconstituent objects [2]. The need to perform visual analytics on data with multi-ary relations and the challenges imposed by their visualization entails practicallyefficient, effective and visually pleasing drawings of hypergraphs.

We propose and evaluate a family of algorithms for drawing hypergraphs.Each algorithm in the family transforms the hypergraph into a graph and canleverage especially but not limited to force-directed graph drawing algorithmsto draw the graph. The final positions of the vertices of the graph are then usedto draw closed convex curves enveloping the hyperedges.

Inspired by the desirable properties of hypergraph drawing proposed byMakinen [22], we propose and discuss some criteria for good hypergraph draw-ing. We also devise several metrics to empirically and comparatively evaluateour algorithms based on both synthetic and real data sets.

2 Related Work

2.1 Force-directed Graph Drawing

Force directed methods are one class of commonly used graph drawing meth-ods. In general, these methods model the graph as a physical system wherevertices are attracted and/or repelled according to some force function, eventu-ally resulting in an equilibrium configuration. Tutte’s barycentric method [29] isconsidered [3] as one of the seminal force-based algorithms. The algorithm fixesat least three vertices in the drawing canvas whereas the other free vertices arepositioned according to the averaged position of its neighboring vertices. Theseresults in a system of linear equation which can be solved using Newton-Raphsoniteration. One of the main advantages of this drawing is if the input graph isa 3-connected planar graph then it yields a crossing free drawing of the graphas guaranteed by Tutte’s theorem [29]. One drawback of this algorithm is theresulting drawing has a poor vertex resolution (exponential in the number ofvertices)[10] making it unsuitable for large graphs.Another class of force-based graph layout algorithms models the graph as SpringsSystem and Electrical Forces. For instance, Peter Eades’s Spring Embedder algo-rithm [9] and Fruchterman-Reingold’s algorithm [15]. The Spring Embedder al-gorithm considers the vertices as steel rings connected with each other by springsforming a mechanical system. Adjacent vertices apply attractive forces to eachother and all pair of vertices apply repulsive forces to each other. Fruchterman-Reingold’s algorithm, however, is inspired by atomic particles. They introducetemperature parameter in their algorithm that inverse-linearly decays to 0 froman initial value and this parameter controls the displacement of the vertices.

2.2 Hypergraph Drawing

A Hypergraph H is a pair (X,E) where X is a set of elements called verticesand E is a non-empty collection of subsets E1, E2, . . . , En of X called Hyper-edges.Two classes of hypergraph drawing have been frequently used in the literatureas mentioned by Makinen [22] namely, Subset based and Edge based. In bothof them, vertices of the hypergraphs are drawn as points in the plane. In Edgebased drawings, hyperedges are drawn as smooth curves connecting their ver-tices. In Subset based drawings, hyperedges are drawn as closed curves envelop-ing their vertices. However, determining whether a planar (crossing-free) drawingfor general hypergraphs exists in both classes of methods has been proven to beNP-complete [18][22].

Makinen [22] is the first to formulate the desirable properties of a subset basedhypergraph drawing. Bertault F. and Eades P. [4] are the first to demonstratea method for drawing hypergraph in subset standard. The drawing algorithmconstructs a Euclidean Steiner tree from the position of the vertices of eachhyperedge at each iteration of the drawing and force-directed graph drawing al-gorithm is applied to get the location of the vertices. Eventually, a curve denotinga hyperedge is constructed by taking contour around the edges of the Steinertree for that hyperedge. Their method fails miserably when a vertex belongsto many hyperedges [4]. Moreover, the paper also lacks discussion about howwell the drawings are in light of desirable properties of a subset based drawingmentioned by Makinen [22].Kritz and Perlin [20] proposed the QUAD scheme for drawing hypergraphs. Eachedge is represented by a column denoted by a rectangle called quad and eachvertex is visualized as a unique point along a row. A bead is in the quad along arow if the vertex in that row is a vertex of the hyperedge denoted by that quad.Though the drawing is fast and simple, hyperedge connectivity is not readilycomprehensible from it.

Since hyperedges are sets, visualizing hypergraphs is closely connected to ap-proaches for visualizing sets, in general. Euler and Venn diagrams are amongstthe earliest set visualization methods. Closed curves such as ellipses, circles orpolygons represent sets and curve overlaps represent set relations. All possiblepair of curve overlaps must be drawn in a Venn diagram whereas a Euler dia-gram does not impose such restriction. Thus a Venn diagram becomes visuallycluttered very quickly as the number of sets increases [2]. Many algorithms existfor drawing Euler diagrams. They differ from each other by their definitions ofEuler diagram and drawable instances of hypergraphs. Flower and Howse [13]define concrete Euler diagram and propose an algorithm to generate concretediagrams automatically up to three sets. An extended definition of Euler dia-gram is given in [30], less restrictive than [13] and the problem of generatingdiagrams up to eight sets is addressed. It is worth to mention that, hypergraphdrawings, in particular, the Subset based drawings differ from Euler diagramsin that the former does not impose regional constraints (e.g. not allowing empty

zones, allowing exactly two points of intersections between contours representingsets) as the latter [27].

Recently, Simonetto et al. [28] propose an approach for generating Euler-like diagrams which are essentially Euler diagrams with no regional restrictions.The sets are drawn as polygons or closed bezier curves under certain conditions.However, with increasing number of intersecting sets, their approach becomesdifficult to interpret. Riche et al. [26] solve this issue by simplifying intersectingsets, either by drawing them as hierarchically arranged rectangular shapes oravoiding intersections all-together by duplicate drawing of elements belongingto multiple sets. Though these methods improve readability of the intersectingsets, they spatially reposition the sets making them ineffective in cases wherethe semantics of the layout is important (e.g. scatter plots, geographical maps).Bubble Sets [7], LineSets [1], Kelp diagrams [8] and KelpFusion [23] are proposedto draw sets encompassing the cases where the spatial position of set elementshave to be preserved. Bubble sets use concave or convex closed isocontours todraw sets. LineSets draw sets by connecting the set elements by simple contin-uous curves to lessen visual clutter. Kelp diagrams envelop the set elements bycolored circles (nesting multiple colors in cases of membership of an element tomultiple sets), construct minimum cost paths connecting the set elements anddraw the paths as thick curves. KelpFusion is a hybrid visualization techniquesimilar to the Kelp diagram in principle and combines line based drawing tech-niques such as LineSets and hull based drawing techniques such as Bubble setsand Euler diagrams.

Subset based drawings such as the one presented in this paper are closelyrelated to Euler-like drawings such as the one proposed by Simonetto et al. [28].Both of these methods draw sets spatially repositioning the set elements andthus do not cater to the cases of grouping objects in maps or scatter plots. [28]applies a force-directed algorithm that preserves edge-crossing properties on theintersection graph of the hypergraph whereas our algorithm is able to apply anyforce-directed algorithm on a class of graphs derived from the hypergraph. [28]approximates the set boundaries by computing polygons whereas our algorithmcomputes convex polygons of the set of points. Thus the resulting drawings by[28] might be concave violating one of the aesthetics we propose in this paper.

3 Aesthetics of hypergraph Drawing

Makinen [22] proposed a set of desirable properties of a subset based hypergraphdrawing. However, empirical study of hypergraph drawings based on their pro-posed set of properties is still lacking in the hypergraph drawing literature. Wepropose the following characteristics of a good hypergraph drawing in subsetstandard.

Firstly, The Concavity metric refers to the number of non-convex hyperedgedrawings drawn by an algorithm. Since convex shapes are visually simpler, min-imizing non-convex shape results in good hypergraph drawing. It is to be noted

that, minimizing non-convex shapes i.e minimizing the Concavity metric helpsensure the first criterion as well.

Secondly, The Planarity metric is defined as the number of non-adjacenthyperedge crossings of a drawing. Two hyperedges Ei and Ej are adjacent ifEi ∩Ej is non-empty. Drawing of a hypergraph, ideally, should have no crossingbetween any pair of non-adjacent hyperedges. In practice, however, having acrossing-free drawing of a hypergraph is rather difficult. Thus it is desirable tohave as little non-adjacent hyperedge crossing (Planarity) as possible. Drawingswith better Planarity help avoid cluttering and thus misinterpretation of therelations being represented.

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Fig. 1. Two drawings of the hyperedges {a, b, c}, {b, f}, {d, e, f}. The drawing on theleft has a concave shape, crossing between a non-adjacent hyperedge pair, poor Cov-erage and non-uniform distribution of vertices (clutter). The drawing on the right isaesthetically superior to the one on the left as it has no concavity, no crossing, com-paratively better Coverage and Regularity.

Thirdly, Coverage refers to the ratio of the ‘mean area per vertex’ of thedrawing to the ‘mean area per vertex’ of the entire drawing canvas. The ‘meanarea per vertex’ of a drawing of a hypergraph H(X,E), Mean-APVdrawing(H)is defined as

Mean-APVdrawing(H) =

∑|E|i=1

Area(Ei)|Ei|

|E|(1)

where Area(Ei) is the area of the shape representing Ei and |Ei| is the number ofvertices in the hyperedge Ei (the cardinality of Ei). The ‘Mean area per vertex’of the drawing canvas, Mean-APVcanvas is computed as -

Mean-APVcanvas(H) =Areacanvas|X|

(2)

where, Areacanvas is the area of the drawing canvas. Thus, the Coverage of thehypergraph H(X,E) is defined as

Coverage(H) =Mean-APVdrawing(H)

Mean-APVcanvas(H). (3)

Maximizing the Coverage metric implies that the drawing canvas is utilizedproperly by the drawing, in other words, the drawing is sparse in some way. ACoverage value close to 1 implies almost 100% utilization of the drawing area.

The Coverage as computed above does not take hyperedge crossings intoconsideration. However, since in reality, it is difficult to have drawings which arecrossing-free, Coverage is often more than 1 for larger hypergraphs. However,the fact that a drawing algorithm with higher Coverage is more desirable thanthe one with lower Coverage is still valid.

Fourthly, The Regularity metric is a measure of uniformity of the vertices overthe drawing canvas. Maximizing the Regularity criterion helps ensure less clutterof the vertices over the drawing canvas. Since the Coverage criteria itself failsto capture uniformity of vertices in cases of drawings with crossings, Regularitygives insights into the distribution of vertices over the drawing area in thosecases.

Fig. 1 illustrates how the proposed metrics encapsulate the aesthetics ofa drawing. Note that, some of the properties of a good hypergraph drawingmentioned above are conflicting and thus achieving them simultaneously arerather challenging.

4 Algorithms

Briefly, the family of algorithms we propose transforms the input hypergraphinto a graph, termed as the associated graph of the input hypergraph, draw thegraph using a graph layout algorithm and enclose the vertices of the hyperedgesinside a closed contour.

4.1 From Hypergraph to Graph

We propose four different ways of constructing associated graphs of a hypergraph-namely, complete associated graph, star associated graph, cycle associated graphand wheel associated graph. Each of these constructions gives rise to an algo-rithm.

Consider a hypergraph H denoted by the tuple (X,E = {E1, E2, . . . , En}).For practical purposes, also consider xi.pos denotes the position vector of an ar-bitrary vertex xi ∈ X on the 2D drawing canvas. Furthermore, {x1, x2, . . . , x|Ei|}denotes the set of vertices in the arbitrary hyperedge Ei.

A complete associated graph C(H) of the hypergraph H is a graph whoseset of vertices is X and for each hyperedge Ei, all pair of distinct vertices in Ei

are connected by a unique edge in C(H). To illustrate, the complete associatedgraph of the hypergraph H1 = ({a, b, c, d}, {{a, b, c, d}, {c, d}, {a}}) consists of{a, b, c, d} as the set of vertices and {(a, b), (b, c), (c, d), (a, c), (b, d), (a, d)} as theset of edges. The drawing algorithm which transforms a hypergraph into itscomplete associated graph is termed as the Complete algorithm. The motivationbehind the Complete algorithm stems explicitly from the underlying principleof the force-directed graph layout algorithms and implicitly from the Planar

characteristic of good drawing. Intuitively, fewer crossings are expected to occuramong a set of hyperedges if the constituent vertices of a hyperedge are spatiallycloser to each other than to the vertices of the other hyperedges.

A cycle associated graph Cy(H) of the hypergraph H is a graph whose setof vertices is X and for each hyperedge Ei, a cycle is formed by adding edges(x1, x2), (x2, x3), . . . , (x|Ei|, x1) in Cy(H). The cycle formed is unique if we con-sider the vertices in Ei sorted clockwise according to their position in the draw-ing. The drawing algorithm which transforms a hypergraph into its cycle asso-ciated graph is named the Cycle algorithm. To illustrate, the cycle associatedgraph of the hypergraph H1 mentioned before consists of {a, b, c, d} as the set ofvertices and {(a, b), (b, c), (c, d), (a, d)} as the set of edges. Sometimes attractiveforces between the vertices in the complete subgraphs of C(H) are too strongwhich in turn results in a cluttered drawing. The desire to have a sparse drawingwith good Coverage and Regularity is the driving force of the Cycle algorithmsince Cy(H) is a subgraph of C(H).

If we allow vertices other than X into our associated graph, transformationsof other kinds emerge. Given the position vectors of the vertices in X and anarbitrary hyperedge Ei in the hypergraph H, the barycenter of the vertices inEi denoted as bi is the unique vertex located at the position

∑ki=1

xi.posk . The

set of barycenters of H denoted as B is {b1, b2, . . . , bn}.

Fig. 2. Ahypergraphwith 4hyperedges.

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Fig. 3. C(H)of the hyper-graph in Fig 2.

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Fig. 4. Cy(H)of the hyper-graph in Fig 2.

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Fig. 5. S(H)of the hyper-graph in Fig 2.

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Fig. 6. W (H)of the hyper-graph in Fig 2.

A star associated graph S(H) of a hypergraph H = (X,E) is a graph whoseset of vertices is H ∪ B and for each hyperedge Ei and its barycenter bi, a staris formed by adding edges (bi, x1), (bi, x2), . . . , (bi, x|Ei|). To illustrate, S(H1)of the hypergraph H1 mentioned before consists of {a, b, c, d, b1, b2} as the setof vertices and {(b1, a), (b1, b), (b1, c), (b1, d), (b2, c), (b2, d)} as the set of edges.The drawing algorithm which transforms a hypergraph into its star associatedgraph is named the Star algorithm. The design principle of the star algorithmis the fact that, spatial nearness among the vertices from the same hyperedgeand remoteness among the vertices from distinct hyperedge can be achieved ifall the vertices feel the attractive force towards the barycenter.

A wheel associated graph W (H) of the hypergraph H is a graph whose set ofvertices is H ∪ B and for each hyperedge Ei and its barycenter bi, a wheelis formed by adding edges (bi, x1), (bi, x2), . . . , (bi, x|Ei|), (x1, x2), (x2, x3), . . . ,(x|Ei|, x1). To illustrate, W (H1) of the hypergraph H1 mentioned above con-sists of ({a, b, c, d, b1, b2} as the set of vertices and {(b1, a), (b1, b), (b1, c), (b1, d),(b2, c), (b2, d), (a, b), (b, c), (c, d), (a, d)}) as the set of edges. The drawing algo-rithm which transforms a hypergraph into its wheel associated graph is namedthe Wheel algorithm. Note that, the set of hyperedges in the wheel associatedgraph is the union of the set of hyperedges of the cycle and the star associatedgraphs i.e W (H) = S(H) ∪ Cy(H).

Fig. 3 - 6 are the associated graphs constructed from the hypergraph in Fig.2.

4.2 The Algorithm

The generalized form of the proposed family of algorithms are as follows -

Algorithm 1 Force-directed hypergraph drawing algorithm

1: Given a hypergraph H(X,E), initialize the position xi.pos of each vertex xi ∈ Xrandomly over the drawing canvas.

2: Transform hypergraph H = (X,E) into a graph G.3: Run any force-directed algorithm to draw the graph G. The position of the ver-

tices in H is set to the position of the vertices in G returned by the graph layoutalgorithm.

4: for each vertex xi ∈ X do5: xi is drawn as a circle centered at point xi.pos.6: end for7: for each hyperedge Ei ⊂ X in H do8: Find the Convex hull of the set of points denoted by the centers of the circles

representing the vertices of Ei. Let, Ei ⊆ Ei be the vertices in the boundary of theconvex hull.

9: Find the pairwise outtangents of the circles representing the vertices in Ei.10: Compute the pairwise intersections of the out-tangents.11: Draw the Catmull-Rom spline [6] defined by the intersecting points as the

control points.12: end for13: Return the splines as the drawing of the hyperedges and circles as the vertices.

Line 1 initializes the position of the vertices randomly. It is well-known thatthe initial position of vertices largely affects the performance of the force-basedgraph layout algorithm [16]. To this end, we optionally suggest two alternativeways of initializing the vertices.In ‘circular initialization’ the position xi.pos of each vertex xi is initialized to thecoordinate of a randomly generated point on a circle of radius k|Ei| where k ∈ Nand |Ei| is the size of the hyperedge containing xi. The heuristic of circular initial

positioning stems from the desire to achieve spatial nearness of the vertices ofthe hyperedge. To avoid clutter and allow more space for the larger hyperedges,the radius of the circle is made proportional to the hyperedge cardinality.

In ‘Grid based initialization’ drawing canvas is divided into m×m grid with

m =√

Areacanvas

|X| , the vertices are placed in each grid sequentially and xi.pos is

initialized to a random point inside the grid xi has been placed in. The valueof m chosen is the finest granularity of the canvas where each grid can containonly one vertex. The heuristic of grid based initial positioning originates fromthe desire of having a uniform vertex position after the graph-drawing layout.

Line 2 essentially transforms the hypergraph drawing problem into a graphdrawing problem. The motivation of transforming the hypergraph to a graph isto leverage the graph layout algorithm (especially but not limited to Force-basedgraph layout algorithms) to find the placement of the vertices of the hypergraph.

A Graph drawing algorithm is used in Line 3 to find the embedding of theassociated graph and line 4-6 draws the physical realization of the vertices of thehypergraph based on that embedding.

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Fig. 7. The ver-tices of a hyper-edge {a, b, c, d}after drawingits Completeassociated graph.

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Fig. 8. Convexhull of the hy-peredge (line 8)and its borderingpoints (a,b,c).

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Fig. 9. Pair-wiseouttangents ofthe borderingvertices (line9) and pointsx,y,z as theirintersections.

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Fig. 10. The hy-peredge drawn asa closed curvewith x,y,z as con-trol points (line10).

Line 8-11 draws each hyperedge as closed spline. Fig. 7-10 illustrate how ahyperedge is drawn from its associated graph (line 8-11).

Enveloping the vertices of the hyperedges inside a closed curve is essentialfor visual clarity and distinction from one hyperedge from the other. Envelopingcan as well be done in other ways as mentioned in [4] and [8]. However, splinesappeared to us as a simple, elegant and visually pleasant choice.

Note that, the algorithm follows some of the characteristics of good drawingby design. For instance, it rarely draws any non-convex shape since only thevertices in the boundary of the convex hulls are used to draw the contour. Thusthe convexity of the drawn closed contours is induced by the convex hulls.

5 Experiments, Results, and Analysis

5.1 Dataset

We test the scalability and effectiveness of our proposed algorithm in light ofthe metric proposed in Section 3 on both synthetic and real dataset. We gen-erate random 3-uniform hypergraphs with 2000 vertices and varying number ofhyperedges and use them as our synthetic dataset. A 3-uniform hypergraph isa hypergraph where every hyperedge has three vertices. We use the DBLP 1

co-authorship network as our real dataset. We trimmed off attributes other thanauthors from the dataset. Thus our dataset is a list of authors to be representedas a list of hyperedges. Though the algorithm can be run on arbitrarily large datawe restricted ourselves to hypergraphs up to 2500 hyperedges for both datasets.

5.2 Experimental Setup

All programs are run on a MacBook Pro with quad-core 2.7 GHz Intel R© Core i5processor, 8GB memory running OSX 10.11.6. The layout algorithms have beenimplemented and rendered using Processing language2. The rendering engineused is Processings P2D renderer which is built on top of OPENGL. All draw-ings have been rendered on a canvas of size 800x800 throughout the experiment.The Cycle algorithm of our implementation sorts the vertices of a hyperedgein clockwise order. Also, k in the initialization parameter of the circular ini-tialization has been set to 5 heuristically. The temperature parameter of theFruchterman-Reingold’s algorithm has been set to 40, the number of iterationparameter has been set to 40.

5.3 Implementation of the metrics

The shape representing each hyperedge has been approximated by their respec-tive convex hulls computed in algorithm 1. Hence, the Concavity of a hypergraphdrawing becomes equal to the number of non-convex polygons. The Planarity isapproximated by the number intersecting convex hull pairs where each pair ofhull corresponds to each pair of non-adjacent hyperedge. Equation 3 is used tocompute Coverage.

In order to measure Regularity, we divide the drawing canvas into a collec-

tion of square grid each of size m =√

Areacanvas

|X| at the finest granularity and√Areacanvas at the coarsest granularity. For each granularity, we generate |X|

uniformly distributed points into the grids and compute the frequency distri-bution of the uniformly distributed points over the drawing area. We comparethis distribution with the frequency distribution of the positions of the verticesin the drawing canvas by running Kolmogorov-Smirnov test (K–S test) and thep-value of the test is used as the measure of Regularity.

1 http://dblp.uni-trier.de/xml/2 https://processing.org/

5.4 Aesthetics of the drawings

The proposed algorithms have been used to draw random 3-uniform hypergraphsas well as partial DBLP co-authorship network. Some drawings drawn by theWheel algorithm are shown in Fig. 11.

(a) A synthetic 3-uniform hy-pergraph with 5 hyperedge.

(b) A hypergraph with 100 hyperedges sampled fromthe DBLP dataset drawn over a circular drawing area

Fig. 11. Example of some drawings by the Wheel algorithm

5.5 Experiments regarding Effectiveness and Scalability

For each family of algorithms, we experiment with three different ways of ini-tializing the position of the vertices - ‘Random initialization’, ‘Circular initial-ization’ and ‘Grid based initialization’. In order to evaluate the effectivenessof the proposed family of algorithms, we construct 5 hypergraphs by randomlysampling 500, 1000, 1500, 2000, 2500 hyperedges from the entire dataset. Werefer to this dataset as the ‘randomly sampled dataset’. In order to evaluatethe Scalability of the proposed family of algorithms, we construct a hypergraphby randomly sampling 2500 hyperedges from the entire dataset and the rest bydownsampling 2000, 1500, 1000, 500 hyperedges each from the previously sam-pled partial-hypergraph in sequence. We refer to this dataset as the ‘subsampleddataset’. After we construct the sequence of hypergraphs, we run each algorithm10 times on each of the hypergraphs and evaluate the first 3 metrics (Concav-ity, Planarity and Coverage) after each run of the algorithms. We compute themean and the standard deviation of those metrics over different runs. To eval-uate Regularity, we compute the mean of the vertex positions over the differentruns, use the mean to compute the frequency distribution and perform K-S test

for each granularity of the drawing area, starting from the finest granularity ofd|X|e × d|X|e till the coarsest granularity of 1× 1, each time decrementing it by

half. In other words, we increment the grid size by 2 starting from√

Areacanvas

|X|

till√Areacanvas. Since our drawing algorithms enforces convexity in the closed

contour representation of each hyperedge by construction, the Concavity hasbeen found to be zero in all the experiments. Thus we abstain ourselves fromproviding the results with repetition here.

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Fig. 12. Effectiveness of the algorithms(with random initialization) on the randomlysampled dataset

Effectiveness on the DBLP Dataset The effectiveness of the algorithmsis measured using the ‘randomly sampled dataset’, meaning, a good algorithmshould have consistent performance on any set of hyperedges. Fig. 12 shows theresult on the DBLP dataset. Fig. 12(a) shows the performance of the algorithmson the Planarity metric. The sizes of the sampled hypergraphs with hyperedges

of different size are shown along the X-axis and the average number of non-adjacent intersecting hyperedges over different runs of our algorithms are shownin Y-axis. Standard deviations of the number of intersections are shown as errorbars. As we observe from Fig. 12(a), the Complete and the Wheel algorithmresults in the least and the Cycle algorithm gives the most number of hyperedgecrossings. Thus, the Complete and the Wheel algorithms are better than theStar and the Cycle algorithms in terms of Planarity. We also observe that, thestandard deviation of the Planarity metric is higher for larger hypergraphs. Fig.12(b) shows Coverage metric (in Y-axis) of the drawing generated by the differentalgorithms (marked by different color) over hypergraphs with different numberof hyperedges (in X-axis). The Cycle algorithm gives the highest value for cover-age over the others. However, judging from the algorithms poor performance inPlanarity metric in comparison to theWheel and the Complete algorithms, werealize that the latter two are relatively better than the former in Coverage. Thelog of p-values for the K-S test performed for different granularity (in X-axis) ondifferent data samples (in Y-axis) has been shown as a heat map in Fig. 12(c).The darker each cell is colored the higher the p-value is in that granularity forthat particular sample. We observe that in the coarser granularity the colors aredarker and the intensity decreases as the granularity increases. The vertices arerather uniform in coarser granularity. However, for finer granularity, the vertexdistribution over the canvas is highly unlikely to be uniform. Since inherentlythe Fruchterman-Reingold’s algorithm scatters disconnected components awayfrom each other as mentioned by Jacomi et. al. [17], resulting in disconnectedcomponents scattered along the border of the drawing. Complete, Star and theCycle algorithms generate relatively sparser drawing and the uniformity of thedrawings by those three algorithms are closely matched with each other. Moreimportantly, we notice that the Star and the Cycle algorithms are apparentlytrading off Planarity for Regularity.

Fig. 13 shows the effectiveness of the algorithms (each initializing vertices ina circular manner) on the DBLP dataset. Fig. 13(a) shows the performance ofthe algorithms on the Planarity metric. We observe that the Complete algorithmand Wheel algorithms result in the least and the Star algorithm gives the mostnumber of hyperedge crossings. In terms of the Coverage metric, as we canobserve from Fig. 13(b), the Star algorithm gives higher value in coverage overthe others. The Star and the Complete algorithms are relatively better than theCycle algorithm, because of their comparable Coverage but better Planarity.The Fig. 13(c) shows the Regularity metric for the algorithms. The drawingsgenerated by all the algorithms are closely matched with each other in terms ofsparsity. Roughly speaking, the Star and the Complete algorithms are relativelybetter than the Cycle algorithm when the vertices are initialized in a circularmanner.

Fig. 14 shows the effectiveness of the family of algorithms, each initializingvertices uniformly over the drawing area partitioned into grids. Fig. 14(a) showsthat the Star algorithm has slightly more hyperedge crossings than the others.Therefore, we conclude that the Star algorithm is worse than the others in

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Fig. 13. Effectiveness of the algorithms(with circular initialization) on the randomlysampled dataset

terms of Planarity. However, Fig. 14(b) implies that the Star algorithm hasbetter Coverage. All the algorithms except the Star algorithm performs equallywell in terms of Regularity, as shown in Fig. 14(c). Therefore, we conclude thatthe grid based initialization is not favoring any algorithm in particular to gainbetter performance than the others.To summarize, the Complete and the Wheel algorithms are moderately effectiveon the real dataset than the other algorithms.

Scalability on the Dataset Scalability of the algorithms is measured by evalu-ating their consistent performance in the metrics using the ‘subsampled dataset’,meaning, a good algorithm should have consistent performance in drawing anested sequence of sets of hyperedges. Fig. 15 shows the scalability of the familyof algorithms on the DBLP dataset, each initializing vertices randomly at thebeginning. From Fig. 15(a), we observe that, the Complete algorithm has thebest and Cycle algorithm has the worst Planarity. From Fig. 15(b) we observethat, the Cycle algorithm has higher coverage. Our experiment on the Regular-

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ity metric showed that the algorithms have heat map similar to each other asobserved in Fig. 12(c). Thus, we omit the figure for the sake of brevity. The

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Wheel and the Complete algorithm are relatively better choice than the Cyclealgorithm regarding the proposed metrics.

Fig. 16 shows the scalability of the family of algorithms, each initializingvertices in a circular manner. In Fig. 16(a), we observe that the Star algorithmproduces more hyperedge crossings than the other three algorithms. However,Star algorithm has slightly better coverage than the other algorithms as observedin Fig. 16(b). Judging from both Planarity and Coverage, the Complete andthe Wheel algorithms are better performing than the Star algorithm. Since weobserved no significant difference among the algorithms in terms of Regularitywhich was observed in the effectiveness experiment in Fig. 13(c) as well, wechoose to omit the heat map for Regularity metric for the sake of brevity.

Fig. 17 shows the scalability of the algorithms on the sequentially subsam-pled dataset. Fig. 17(a) shows that Wheel and Complete algorithm has betterPlanarity than the other algorithms. From Fig. 17(b), we observe that the Star

algorithm has better Coverage than the other algorithms. Since we observed nosignificant difference among the algorithms in terms of Regularity as mentionedin the effectiveness experiment in Fig. 14(c), we choose to omit the heat map forRegularity metric for the sake of brevity.

To summarize, the Complete and the Wheel algorithm are more scalable interms of their performance than the other algorithms.

Experiment on the Synthetic Dataset The results we observed on syn-thetic dataset closely resembles our result on the real dataset. We only mentionthe key observations here and omit the demonstration. First, the Coverage forthe Star algorithm is the highest in every synthetic dataset, irrespective of ini-tial positioning of the vertices. Second, the Regularity of the Star algorithm istherefore relatively better than the others. Third, Star algorithm has the worstPlanarity when initialized in a circular manner. Fourth, all the algorithms havesimilar performance in Regularity and Planarity in the grid based initializationof vertices.

6 Conclusion

We propose a family of algorithms for drawing hypergraphs and their variants.We also propose a set of aesthetic and measurable criteria for evaluating theperformance of subset based hypergraph drawing algorithms in general. We em-pirically evaluate the effectiveness and scalability of our proposed algorithmswith different initial positioning of the vertices. The drawings by our algorithmsare not only aesthetically pleasing in a qualitative way but also reasonably wellfollow a set of quantitative criteria of good hypergraph drawing. However, thedrawings generated have lesser uniformity than expected stemming from scat-tering of disconnected components by the underlying force-directed graph layoutalgorithm. In future, we would like to extend Fruchterman-Reingold’s algorithmfrom two-dimensional canvas to higher dimension by modeling the hyperedges aselastic multidimensional manifolds rather than a graph with the hope of havingbetter drawings.

Acknowledgement

This work is supported by the National Research Foundation, Prime Minister’sOffice, Singapore under its Campus for Research Excellence and TechnologicalEnterprise (CREATE) programme.

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