Comparison of centrality based planarisation for 2 5 d graph drawing

In this paper, we present extensive experimental results with four variations of planarisation methodsand five different centralities degree, closeness, eccentricity, stress and shortest

Comparison of Centrality-Based Planarisation for 2.5D Graph

in the social network

In this paper, we present centrality-based planarisation algorithms with four variations: vertexaddition, vertex deletion, edge addition and edge deletion Our research is inspired by domain specificproblem of drawing a social network in 2.5 dimensions which emphasizes important actors by placing

on the top layer More specifically, we compare five different vertex (or edge) centralities in order toinvestigate the impact of different vertex (or edge) orderings on the resulting maximal induced planarsubgraphs We present extensive experimental results using both the standard Graph Drawing testsuites and popular network models We further extend our centrality-based planarisation algorithm

to compute the thickness (both vertex thickness and edge thickness) of graphs

1 Introduction

and has important applications in Graph Drawing to draw non-planar graphs with as few crossings aspossible There are two different approaches for planarising graphs: vertex deletion and edge deletion.Unfortunately, the problem of finding the maximum vertex (or edge) induced planar subgraph NP-hard Nevertheless, there are many heuristics and approximation algorithms available, in particular forcomputing a maximal edge induced planar subgraph

However, the problem of finding a maximal vertex induced planar subgraph is not well investigated sofar, as the vertex deletion operation is too drastic As pointed out in [3], there is relatively little literatureabout it and they are mostly theoretical results Further, the well known simple greedy algorithm, that

is either deleting a vertex or an edge until the graph becomes planar, we do not aware of any workinvestigating the impact of different vertex (or edge) orderings on the resulting maximal induced planarsubgraphs

In this paper, we relate the classical problem of planarisation in graph theory with centrality measures

in network analysis Centrality index is one of the important concepts in Network Analysis for analysingthe importance of actors in the social network There are many different centrality measures availablebased on the definition of the importance in the specific applications

More specifically, we present centrality-based planarisation algorithms with four variations: vertexaddition, vertex deletion, edge addition and edge deletion In particular, we investigate the impact ofdifferent orderings based on centrality for planarisation of graphs with greedy algorithms

Our research motivation was inspired by domain specific problem, i.e drawing a social network in 2.5dimensions which emphasizes important vertices by placing on the top layer Figure 1 shows an example

In this paper, we present extensive experimental results with four variations of planarisation methodsand five different centralities (degree, closeness, eccentricity, stress and shortest path betweenness) usingfive test suites (Rome library: undirected and directed, AT&T graphs, scale-free networks and randomnetworks) In each case, we compare centralities including random ordering Overall, the experimental

Figure 1: Displaying Centrality of a Network in 2.5 Dimensions

results suggest that some of centrality measures such as degree and betweenness, can be an attractivesolution for planarisation

We further extend our centrality-based planarisation algorithm to compute the thickness (both vertexthickness and edge thickness) of graphs, another classical NP-hard problem of partitioning non-planargraphs into several layers of planar subgraphs

To our best knowledge, our study is the first to investigate the impacts of orderings (both vertexand edge) in greedy algorithms (both deletion and addition) and to conduct comparison with differentcentrality measures Further, we use vertex deletion approach for planarisation, as in our application ofconstructing a 2.5D drawing, this operation is not drastic (i.e we draw the deleted vertices on the toplayer)

This paper is organised as follows In the next section, we review background Then we present ourcentrality-based planarisation algorithms with four variations in Section 3, followed by centrality-basedthickness-computing heuristics in Section 4 We present extensive experimental results in Section 5.Section 6 concludes

2 Background

2.1 Planarisation of Graphs

Graph planarisation, referred to a problem of extracting a maximum planar subgraph from a non-planargraph, has many applications in circuit layout, facility layout and in particular Graph Drawing[2] Thereare two approaches for planarisation: vertex deletion and edge deletion

The problem of finding the maximum planar subgraph (i.e the largest induced planar subgraph)

is NP-hard[3] There are a number of algorithms for finding maximal planar subgraph of a non-planargraphs including heuristics and approximation algorithms For details, we refer the extensive survey andbibliography on planarisation [3]

The problem of computing maximal edge-induced planar subgraph has been intensively studied [3].There is a large literature about finding a maximal planar subgraph, including theoretical results andalgorithmic results

However, the problem of computing maximal vertex-induced planar subgraph is not well investigated

so far, as the vertex deletion operation is too drastic As pointed out in [3], there is relatively littleliterature about it and they are mostly theoretical results

Further, the well known simple greedy algorithm, that is either deleting a vertex or an edge untilthe graph becomes planar, we do not aware of any work investigating the impact of different vertex (oredge) orderings on the resulting maximal induced planar subgraphs

2.3 Centrality in Social Network Analysis

In this section, we briefly review the definition of each centralities For details, see [1]

Degree centrality cD(v) of a vertex v is defined as the degree d(v) of v for undirected graphs For directed

of degree centrality of directed graphs

Eccentricity e(u) of a vertex u denote the maximum distance from u to a random vertex v in the graph,where e(u) = max{d(u, v) : v ∈ V } More formally, eccentricity centrality of a vertex u is defined asfollows:

We denote the sum of the distances from a vertex u ∈ V to any other vertex in a graph G = (V, E) as

more important than a person with high total distance Centrality value based on this concept is calledcloseness Closeness centrality of a vertex u is formally defined as follows:

where σst(u) denotes the number of shortest paths containing u

The same definition can be applied for edges as follows:

s∈V

Xt∈V

Let δst(v) denote the fraction of shortest paths between s and t that contain vertex v

Shortest path betweenness centrality of a vertex v is defined as follows:

2.4 Derived Edge Centralities

Derived edge centralities such as degree, closeness and eccentricity are computed by using edge graph

(x, y), (y, z) ∈ E That is, two edges have a connection if they are adjacent to the same vertex y (withthe first one in- and the second outbound for directed graphs)

For computing stress and shortest path betweenness edge centralities, we use incidence graph The

E} That is a real vertex and an edge vertex become linked if they are incident in the original graph.Figure 2 shows an example of edge graph and incidence graph respectively

V1← sorted list of V based on

centrality index from the highest to the lowest

whileGis not planar do

v ← next vertex in V1

Remove v and its incident edges from G

endwhile

VertexAdditionPlanarisation()INPUT: G = (V, E)

OUTPUT: planar graph G′= (V′

Figure 3: Centrality-Based Planarisation Algorithms with Vertex Deletion and Vertex Addition

OUTPUT: planar graph G′= (V, E′)

Build incidence or edge graph Ge= (Ve, Ee) of G

Map each e in E with vein Ve

E3← sorted list of edges E based on edge

centralities on G from the highest to the lowest

whileGis not planar do

e ← next edge in E3

Remove e from G

endwhile

EdgeAdditionPlanarisation()INPUT: G = (V, E)

OUTPUT: planar graph G′= (V, E′)Build incidence or edge graph Ge= (Ve, Ee) of GMap each e in E with ve in Ve

E4← sorted list of edges E based on edgecentralities on G from the lowest to the highest

G′← empty graphwhile V4has next element do

Figure 4: Centrality-Based Planarisation Algorithms with Edge Deletion and Edge Addition

4 Centrality-Based Thickness

We propose heuristics for computing thickness (both vertex and edge thickness) of the graph based onthe centrality-based planarisation methods More specifically, we repeat the vertex (or edge) deletion(or addition) planarisation algorithm until the remaining induced subgraph becomes planar

Thickness()INPUT: G = (V, E)OUTPUT: thickness of Gthickness= 1

whileGis NOT planar dothickness= thickness + 1Perform centrality-based planarisation by removing a vertex (or edge) set S

G ← G − Sendwhile

Figure 5: Computing Thickness using Centrality-based Planarisation

5 Implementation and Experimental Results

For implementation, we use JUNG (Java Universal Network/Graph Framework) to compute centralities,and LEDA (written in C++) to test planarity For visualisation, we use GEOMI (Geometry for MaximumInsight, written in Java3D)

For test data, we use standard Graph Drawing test suites including Rome library (both directed andundirected: ALF/CD and ALF/CU, in GML format) and AT&T graphs, as well as scale-free networksand random graphs Each of Rome library data sets contains more than 8000 graphs The AT&T graphshave more than 1200 test cases in GraphML format We also generated about 10000 random graphs and

10000 scale-free networks using Erdos-Renyi and Barabasi-Albert generators in JUNG

We experiment with each of the four greedy algorithms (i.e vertex deletion, vertex addition, edgedeletion, and edge addition) for planarisation and thickness Thus, in total we have eight test cases.Further, in each case, we run experiments with five data sets Thus, the whole experiments have 40test cases In each test case, we compare the impact of the different ordering with five centralities andrandom ordering We compute minimum, maximum and average ratios between the number of deletednodes (or edges) and the number of nodes (or edges) in original graph We also compared the minimum,maximum and average for thickness.

5.1 Planarisation

For Rome directed graphs, degree centrality performs very well on average and also in worst case Othercentralities and random ordering, however, have a worst case of around 70% or above See Figure 15.For Rome undirected graphs, degree centrality again performs very well, although stress and shortestpath betweenness are also good High ratios of deleted nodes in both closeness and eccentricity indicatethat these two centralities are not useful for planarisation See Figure 16

For AT&T graph, random ordering works quite bad, as shown in Figure 17 Despite of high values

in worst cases, all the centralities have average ratios of approximately 0.06, which is very good Degreecentrality is again the best among them

For scale-free networks, degree, stress and shortest path betweenness centrality are nearly the same

in worst case and on average See Figure 18 For random networks, Figure 19 shows that degree, stressand shortest path betweenness are the best

In summary, for all data sets, degree centrality is the best in overall, and stress and shortest pathbetweenness follow

For Rome directed graphs, degree centrality performs very well on average and in worst case Stress andshortest path betweenness are also good Random ordering is the worst See Figure 20

For Rome undirected graphs, random ordering and eccentricity are not good on ALF/CU data set,

as shown in Figure 21 The others work quite well, around 0.15 on average

For AT&T graph, all the centralities work quite badly in worst cases, around 0.5 of the nodes weredeleted from original graph, as in Figure 22 Although on average, the ratio is only about 0.04

For scale-free networks, on average all the centralities and random ordering are quite good, onlyabout 0.1 of nodes were deleted from original graphs, as shown in Figure 23 Degree, stress and shortestpath betweenness are the best For random networks, Figure 24 shows that degree, stress and shortestpath betweenness again are the best Closeness, eccentricity random ordering have slightly higher ratiosthan the others

In summary, for all data sets, degree, stress and shortest path betweenness centrality are the bestoverall Random ordering, in contrast, is the worst in all cases Note that vertex addition is twice betterthan vertex deletion for planarisation in overall

For Rome directed graphs, degree centrality performs very well on average and in worst case Closenessand eccentricity are the worst See Figure 25 For Rome undirected graphs, degree centrality is the best,while closeness performs worst See Figure 26

Figure 27 shows that, for AT&T graph, degree, closeness, eccentricity and random ordering are allthe same, with the average of 0.2 Stress centrality does not perform good in this case

Stress and betweenness centralities seem not appropriate for planarising scale-free networks, as shown

in Figure 28 Degree centrality is the best For random networks, Figure 29 shows that stress has highervalues in worst case and on average Degree and eccentricity are the best

In summary, for all data sets, degree centrality is the winner and surprisingly the random ordering

is the second in overall Closeness is the worst with Rome data set, while stress centrality is the worstwith other data sets In general, the ratios in worst cases are quite high

5.1.4 Edge Addition

For Rome directed graphs, all perform approximately the same, however shortest path betweennesscentrality is slightly better than the others in worst case and on average See Figure 30 For Romeundirected graphs, shortest path betweenness centrality is the best on average See Figure 31

For AT&T graph, all perform the same on average Shortest path betweenness centrality is the best

in worst case See Figure 32

For scale-free networks, on average all perform the same Closeness and eccentricity perform quitebad in worst case Random ordering is the best with maximum ratio of about 0.33 See Figure 33 Forrandom networks, shortest path betweenness centrality is the best on average and in worst case SeeFigure 34

In summary, For all data sets, shortest path betweenness centrality is the best and stress centrality

is the second in overall Closeness and eccentricity appear not a good choice for scale-free networks.Surprisingly, degree and random ordering perform badly in other four data sets

Note that again edge addition is twice better than edge deletion for planarisation in worst case and

on average In general, addition is also better than deletion for computing thickness in all cases as wewill see below

5.2 Thickness

For Rome directed graphs, degree and stress centrality are the best Average of thickness of all centralities

is 2.2 Closeness and eccentricity perform quite bad in worst case and on average See Figure 35 ForRome undirected graphs, closeness and eccentricity again perform not quite well Average of the thicknessvalues is around 2.2 See Figure 36

For AT&T graph, Figure 37 shows that the average of thickness is 1.4, which is very good Closeness

is still the worst

For scale-free networks, Figure 38 shows that random ordering is the worst in worst case and onaverage Closeness and eccentricity perform quite well For random networks, Figure 39 shows thatcloseness and eccentricity are the worst The average of thickness is around 2.3

In summary, for all data sets, degree and stress centrality are the best overall Shortest path performswell in most data sets, except for ALF/CD data set Closeness and eccentricity, however, seem to be notappropriate for computing thickness using vertex deletion

For Rome directed graphs, all of the centralities and random ordering have approximately same averagethickness value Maximum thickness of stress centrality is 3, while the others have maximum value of 2.See Figure 40 For Rome undirected graphs, as shown in Figure 41, degree and random ordering are thebest

For AT&T graph, the centralities work nearly the same in worst case and on average Randomordering works slightly better than the others See Figure 42

For scale-free networks, all the centralities perform nearly the same in worst case and on average.Surprisingly, random ordering works slightly better than the others See Figure 43 For random networks,all the centralities perform nearly the same in worst case and on average Eccentricity and randomordering perform slightly better than the others See Figure 44

In summary, for all data sets, random ordering and degree centrality are the best choices for vertexaddition Eccentricity is nearly as good as random ordering in most data sets, except for ALF/CU dataset Overall, vertex addition is better than vertex deletion for thickness except scale-free networks

For AT&T graph, Figure 47 shows that eccentricity is the best, and random ordering is the secondbest Shortest path betweenness is the worst.

For scale-free networks, the maximum thickness computed by stress and betweenness centrality is 11,which is significantly higher compared with the maximum thickness computed by the others, as shown

in Figure 48 Degree and random ordering are the best Figure 49 is quite interesting, as for randomnetworks, degree and closeness are the worst, while the random ordering is the best

In summary, random ordering is the best and stress centrality is ranked as the second in most cases.With ALF/CU data set, stress centrality is the best, while the random ordering is the second best.Closeness and eccentricity, however, seem to be not appropriate for computing thickness using edgedeletion

For Rome directed graphs, Figure 50 shows that all of them perform nearly the same However, stresscentrality is slightly better than the others For Rome undirected graphs, Figure 51 shows that stresscentrality is best in worst case, and the second best on average Random ordering is the best in worstcase

For AT&T graph, degree centrality is the worst in worst case, with maximum thickness of 5, whilethe others have maximum thickness of 4 Other centralities and random ordering perform similar inworst case and on average See Figure 52

For scale-free networks, Figure 53 shows that all of them perform similar in worst case and on average.Average thickness is about 2.1, while the maximum is 4 For random networks, Figure 54 shows clearlythat stress centrality is the best, both in worst case and on average In contrast, degree and closenessare the worst

In summary, stress centrality is the best and degree centrality, in contrast, performs worst in all cases.With ALF/CU and scale-free networks, there is no significant difference among the five centralities andrandom ordering Overall, edge addition is also better than edge deletion for computing thickness in allcases

5.3 Centrality-Based 2.5D Graph Drawing

Figure 1 shows a directed graph in Rome data set (ALF/CD/graphwith17nodes/ug.15.17) with 18 tices and 36 edges It is drawn into 2 layers using vertex deletion

ver-Figure 14 shows a drawing of a bigger graph from AT&T graphs (g.94.2.graphml) with 94 verticesand 174 edges It has thickness of 5, which is computed using vertex deletion with eccentricity centrality

6 Conclusions

In this paper, we propose simple greedy algorithms for planarisation based on the centrality of vertices

or edges In particular, we investigate the impact of different orderings (both vertex and edge) based

on centrality for planarisation with greedy algorithms (both deletion and addition approach) Further

we extend our approach to another classical problem of computing thickness of graphs In summary,the extensive experimental results suggest that some of centrality measures such as degree, stress andbetweenness, may be an attractive choice for planarisation Figures 6, 7, 8, 9, 10, 11, 12, 13summarise the comparisons for each methods

References

[1] U Brandes and T Erlebach, Network Analysis: methodological foundations, Springer, 2005

[2] G Di Battista, P Eades, R Tamassia and I G Tollis, Graph Drawing: Algorithms for the Visualization ofGraphs, Prentice-Hall, 1998

[3] A Liebers, Planarizing Graphs - A Survey and Annotated Bibliography, JGAA, vol 5, no 1, pp 1 - 74,2001

[4] P Mutzel, T Odenthal, M Scharbrodt, The Thickness of Graphs: A Survey, Graphs and Combinatorics,

14, pp 59 - 73, 1998

Figure 6: Vertex Deletion- Planarisation Summary

Figure 7: Vertex Addition- Planarisation Summary

Figure 8: Edge Deletion- Planarisation Summary

Figure 9: Edge Addition- Planarisation Summary

Figure 10: Vertex Deletion- Thickness Summary

Figure 11: Vertex Addition- Thickness Summary

Figure 12: Edge Deletion- Thickness Summary

Figure 13: Edge Addition- Thickness Summary