Comparison of centrality based planarisation for 2 5 d graph drawing

Comparison of Centrality-Based Planarisation for 2.5D Graph Drawing Quan H Nguyen ∗ Seok-Hee Hong † Abstract Planarisation of graphs is a well studied research problem in Graph Theory and has important applications in Graph Drawing to draw non-planar graphs with as few crossings as possible However, the problem of finding the maximum vertex (or edge) induced planar subgraph NP-hard Centrality index is one of the important concepts in Network Analysis for analysing the importance of actors in the social network In this paper, we present centrality-based planarisation algorithms with four variations: vertex addition, vertex deletion, edge addition and edge deletion Our research is inspired by domain specific problem 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 to investigate the impact of different vertex (or edge) orderings on the resulting maximal induced planar subgraphs We present extensive experimental results using both the standard Graph Drawing test suites and popular network models We further extend our centrality-based planarisation algorithm to compute the thickness (both vertex thickness and edge thickness) of graphs Introduction Planarisation of graphs is a well studied research problem in both Graph Theory and Graph Algorithms and has important applications in Graph Drawing to draw non-planar graphs with as few crossings as possible 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 NPhard Nevertheless, there are many heuristics and approximation algorithms available, in particular for computing a maximal edge induced planar subgraph However, the problem of finding a 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 little literature about 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 not aware of any work investigating the impact of different vertex (or edge) orderings on the resulting maximal induced planar subgraphs 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 analysing the importance of actors in the social network There are many different centrality measures available based on the definition of the importance in the specific applications More specifically, we present centrality-based planarisation algorithms with four variations: vertex addition, vertex deletion, edge addition and edge deletion In particular, we investigate the impact of different 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 2.5 dimensions which emphasizes important vertices by placing on the top layer Figure shows an example In this paper, we present extensive experimental results with four variations of planarisation methods and five different centralities (degree, closeness, eccentricity, stress and shortest path betweenness) using five test suites (Rome library: undirected and directed, AT&T graphs, scale-free networks and random networks) In each case, we compare centralities including random ordering Overall, the experimental ∗ University † NICTA of Sydney, Australia, hngu4424@mail.usyd.edu.au (National ICT Australia) and the University of Sydney, Australia, shhong@it.usyd.edu.au 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 attractive solution for planarisation We further extend our centrality-based planarisation algorithm to compute the thickness (both vertex thickness and edge thickness) of graphs, another classical NP-hard problem of partitioning non-planar graphs into several layers of planar subgraphs To our best knowledge, our study is the first to investigate the impacts of orderings (both vertex and edge) in greedy algorithms (both deletion and addition) and to conduct comparison with different centrality measures Further, we use vertex deletion approach for planarisation, as in our application of constructing a 2.5D drawing, this operation is not drastic (i.e we draw the deleted vertices on the top layer) This paper is organised as follows In the next section, we review background Then we present our centrality-based planarisation algorithms with four variations in Section 3, followed by centrality-based thickness-computing heuristics in Section We present extensive experimental results in Section Section concludes 2.1 Background Planarisation of Graphs Graph planarisation, referred to a problem of extracting a maximum planar subgraph from a non-planar graph, has many applications in circuit layout, facility layout and in particular Graph Drawing[2] There are 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-planar graphs including heuristics and approximation algorithms For details, we refer the extensive survey and bibliography 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 and algorithmic 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 little literature about 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 not aware of any work investigating the impact of different vertex (or edge) orderings on the resulting maximal induced planar subgraphs 2.2 Thickness of Graphs The thickness of a graph G, denoted θ(G), is the minimum number of planar subgraphs of G whose union is G Note that computing thickness of a given graph G is NP-complete There are few algorithmic results about computing thickness of graphs, however there are many structural results about thickness in topological graph theory [3] Nevertheless, there are some heuristics with experimental results available For details, we refer the survey paper on thickness [3, 4] 2.3 Centrality in Social Network Analysis In this section, we briefly review the definition of each centralities For details, see [1] 2.3.1 Degree Degree centrality cD (v) of a vertex v is defined as the degree d(v) of v for undirected graphs For directed graphs, there are two variants of the degree centrality: the in-degree centrality ciD (v) = d− (v) and the out-degree centrality coD (v) = d+ (v) In this paper, we use cD (v) = ciD (v) + coD (v) as a measurement of degree centrality of directed graphs 2.3.2 Eccentricity 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 as follows: 1 = (1) cE (u) = e(u) max{d(u, v) : v ∈ V } 2.3.3 Closeness We denote the sum of the distances from a vertex u ∈ V to any other vertex in a graph G = (V, E) as the total distance v∈V d(u, v) In social network analysis, a person with small total distance maybe more important than a person with high total distance Centrality value based on this concept is called closeness Closeness centrality of a vertex u is formally defined as follows: cC (u) = d(u, v) (2) v∈V 2.3.4 Stress Centrality Stress centrality is defined based on the set of shortest paths in the graph Formal definition of stress centrality of a vertex u is as follows: cS (u) = σst (u) (3) s=v∈V t=v∈V where σst (u) denotes the number of shortest paths containing u The same definition can be applied for edges as follows: cS (e) = σst (e) (4) s∈V t∈V 2.3.5 Shortest path Betweenness Centrality Let δst (v) denote the fraction of shortest paths between s and t that contain vertex v δst (v) = σst (v) σst where σst denotes the number of all shortest paths between s and t (5) Shortest path betweenness centrality of a vertex v is defined as follows: cB (v) = δst (v) (6) s=v∈V t=v∈V 2.4 Derived Edge Centralities Derived edge centralities such as degree, closeness and eccentricity are computed by using edge graph The edge graph of G = (V, E) is G′ = (E, K) where K is the set of all edges e = ((x, y), (y, z)) where (x, y), (y, z) ∈ E That is, two edges have a connection if they are adjacent to the same vertex y (with the first one in- and the second outbound for directed graphs) For computing stress and shortest path betweenness edge centralities, we use incidence graph The incidence graph of G = (V, E) is G′′ = (V ∪ E, {(v, e) | ∃w : e = (v, w) ∈ E} ∪ {(e, w) | ∃v : e = (v, w) ∈ E} That is a real vertex and an edge vertex become linked if they are incident in the original graph Figure shows an example of edge graph and incidence graph respectively a a a c c b b b c Figure 2: Example of an edge graph and an incidence graph Centrality-Based Planarisation In this section, we describe centrality-based planarisation methods with four variations: vertex deletion, vertex addition, edge deletion and edge addition 3.1 Vertex Deletion The simple greedy algorithm deletes vertices of a graph G one by one (together with its incident edges) until the remaining graph is planar Here we use centrality indices from the highest to the lowest 3.2 Vertex Addition Here we start with an empty graph, and then add one vertex at a time (from the lowest to the highest centrality index), testing planarity, until all the vertices are considered VertexAdditionPlanarisation () INPUT: G = (V, E) OUTPUT: planar graph G′ = (V ′ , E ′ ) V2 ← sorted list of V based on centrality index from the lowest to the highest G′ ← empty graph while V2 has next element v ← next vertex in V2 Add v into G′ if G′ is not planar then Remove v from G′ endif endwhile VertexDeletionPlanarisation () INPUT: G = (V, E) OUTPUT: planar graph G′ = (V ′ , E ′ ) Sort V V1 ← sorted list of V based on centrality index from the highest to the lowest while G is not planar v ← next vertex in V1 Remove v and its incident edges from G endwhile Figure 3: Centrality-Based Planarisation Algorithms with Vertex Deletion and Vertex Addition 3.3 Edge Deletion This is similar to the Vertex Deletion method Here we delete edges based on edge centrality index, from the highest to the lowest 3.4 Edge Addition This is similar to the Vertex Addition method Here we add edges based on edge centrality index, from the lowest to the highest EdgeAdditionPlanarisation () EdgeDeletionPlanarisation () INPUT: G = (V, E) INPUT: G = (V, E) OUTPUT: planar graph G′ = (V, E ′ ) OUTPUT: planar graph G′ = (V, E ′ ) Build incidence or edge graph Ge = (Ve , Ee ) of G Build incidence or edge graph Ge = (Ve , Ee ) of G Map each e in E with ve in Ve Map each e in E with ve in Ve E4 ← sorted list of edges E based on edge E3 ← sorted list of edges E based on edge centralities on G from the lowest to the highest centralities on G from the highest to the lowest G′ ← empty graph while G is not planar while V4 has next element e ← next edge in E3 e ← next edge in V2 Remove e from G Add e into G′ endwhile if G′ is not planar then Remove e from G′ endif endwhile Figure 4: Centrality-Based Planarisation Algorithms with Edge Deletion and Edge Addition Centrality-Based Thickness We propose heuristics for computing thickness (both vertex and edge thickness) of the graph based on the 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 G thickness = while G is NOT planar thickness = thickness + Perform centrality-based planarisation by removing a vertex (or edge) set S G←G−S endwhile Figure 5: Computing Thickness using Centrality-based Planarisation 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 Maximum Insight, written in Java3D) For test data, we use standard Graph Drawing test suites including Rome library (both directed and undirected: ALF/CD and ALF/CU, in GML format) and AT&T graphs, as well as scale-free networks and random graphs Each of Rome library data sets contains more than 8000 graphs The AT&T graphs have 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, edge deletion, 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 40 test cases In each test case, we compare the impact of the different ordering with five centralities and random ordering We compute minimum, maximum and average ratios between the number of deleted nodes (or edges) and the number of nodes (or edges) in original graph We also compared the minimum, maximum and average for thickness 5.1 5.1.1 Planarisation Vertex Deletion For Rome directed graphs, degree centrality performs very well on average and also in worst case Other centralities 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 shortest path betweenness are also good High ratios of deleted nodes in both closeness and eccentricity indicate that 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 Degree centrality 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, stress and shortest path betweenness are the best In summary, for all data sets, degree centrality is the best in overall, and stress and shortest path betweenness follow 5.1.2 Vertex Addition For Rome directed graphs, degree centrality performs very well on average and in worst case Stress and shortest 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 were deleted 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, only about 0.1 of nodes were deleted from original graphs, as shown in Figure 23 Degree, stress and shortest path betweenness are the best For random networks, Figure 24 shows that degree, stress and shortest path betweenness again are the best Closeness, eccentricity random ordering have slightly higher ratios than the others In summary, for all data sets, degree, stress and shortest path betweenness centrality are the best overall Random ordering, in contrast, is the worst in all cases Note that vertex addition is twice better than vertex deletion for planarisation in overall 5.1.3 Edge Deletion For Rome directed graphs, degree centrality performs very well on average and in worst case Closeness and 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 all the 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 higher values 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 worst with 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 betweenness centrality is slightly better than the others in worst case and on average See Figure 30 For Rome undirected 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 quite bad in worst case Random ordering is the best with maximum ratio of about 0.33 See Figure 33 For random networks, shortest path betweenness centrality is the best on average and in worst case See Figure 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 we will see below 5.2 5.2.1 Thickness Vertex Deletion 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 For Rome undirected graphs, closeness and eccentricity again perform not quite well Average of the thickness values 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 on average Closeness and eccentricity perform quite well For random networks, Figure 39 shows that closeness 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 performs well in most data sets, except for ALF/CD data set Closeness and eccentricity, however, seem to be not appropriate for computing thickness using vertex deletion 5.2.2 Vertex Addition For Rome directed graphs, all of the centralities and random ordering have approximately same average thickness value Maximum thickness of stress centrality is 3, while the others have maximum value of See Figure 40 For Rome undirected graphs, as shown in Figure 41, degree and random ordering are the best For AT&T graph, the centralities work nearly the same in worst case and on average Random ordering 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 random ordering 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 vertex addition Eccentricity is nearly as good as random ordering in most data sets, except for ALF/CU data set Overall, vertex addition is better than vertex deletion for thickness except scale-free networks 5.2.3 Edge Deletion Figure 45 shows that for Rome directed graphs, stress is the best on average It is interesting that random ordering is the best with maximum thickness of 3, and is the second best with average thickness of 2.4 For Rome undirected graphs, closeness is the worst, and random ordering is the best, as shown in Figure 46 For AT&T graph, Figure 47 shows that eccentricity is the best, and random ordering is the second best 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 random networks, 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 edge deletion 5.2.4 Edge Addition For Rome directed graphs, Figure 50 shows that all of them perform nearly the same However, stress centrality is slightly better than the others For Rome undirected graphs, Figure 51 shows that stress centrality is best in worst case, and the second best on average Random ordering is the best in worst case For AT&T graph, degree centrality is the worst in worst case, with maximum thickness of 5, while the others have maximum thickness of Other centralities and random ordering perform similar in worst 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 For random networks, Figure 54 shows clearly that stress centrality is the best, both in worst case and on average In contrast, degree and closeness are 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 and random ordering Overall, edge addition is also better than edge deletion for computing thickness in all cases 5.3 Centrality-Based 2.5D Graph Drawing Figure shows a directed graph in Rome data set (ALF/CD/graphwith17nodes/ug.15.17) with 18 vertices and 36 edges It is drawn into layers using vertex deletion Figure 14 shows a drawing of a bigger graph from AT&T graphs (g.94.2.graphml) with 94 vertices and 174 edges It has thickness of 5, which is computed using vertex deletion with eccentricity centrality 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 and betweenness, may be an attractive choice for planarisation Figures 6, 7, 8, 9, 10, 11, 12, 13 summarise 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 of Graphs, Prentice-Hall, 1998 [3] A Liebers, Planarizing Graphs - A Survey and Annotated Bibliography, JGAA, vol 5, no 1, pp - 74, 2001 [4] P Mutzel, T Odenthal, M Scharbrodt, The Thickness of Graphs: A Survey, Graphs and Combinatorics, 14, pp 59 - 73, 1998 0.5 0.45 0.4 0.35 Ratios 0.3 ALFCD ALFCU AT&T Barabasi-Albert Erdos-Renyi 0.25 0.2 0.15 0.1 0.05 Degree Closeness Eccentricity Stress SBetweenness Random Figure 6: Vertex Deletion- Planarisation Summary 0.3 0.25 Ratios 0.2 ALFCD ALFCU AT&T Barabasi-Albert Erdos-Renyi 0.15 0.1 0.05 Degree Closeness Eccentricity Stress SBetweenness Random Figure 7: Vertex Addition- Planarisation Summary 0.6 0.5 Ratios 0.4 ALFCD ALFCU AT&T Barabasi-Albert Erdos-Renyi 0.3 0.2 0.1 Degree Closeness Eccentricity Stress SBetweenness Random Figure 8: Edge Deletion- Planarisation Summary 0.3 0.25 Ratios 0.2 ALFCD ALFCU AT&T Barabasi-Albert Erdos-Renyi 0.15 0.1 0.05 Degree Closeness Eccentricity Stress SBetweenness Figure 9: Edge Addition- Planarisation Summary 10 Random 2.5 Ratios ALFCD ALFCU AT&T Barabasi-Albert Erdos-Renyi 1.5 0.5 Degree Closeness Eccentricity Stress SBetweenness Random Figure 10: Vertex Deletion- Thickness Summary 2.5 1.5 Ratios ALFCD ALFCU AT&T Barabasi-Albert Erdos-Renyi 0.5 Degree Closeness Eccentricity Stress SBetweenness Random Figure 11: Vertex Addition- Thickness Summary 11 3.5 2.5 Ratios ALFCD ALFCU AT&T Barabasi-Albert Erdos-Renyi 1.5 0.5 Degree Closeness Eccentricity Stress SBetweenness Random Figure 12: Edge Deletion- Thickness Summary 2.5 Ratios ALFCD ALFCU AT&T Barabasi-Albert Erdos-Renyi 1.5 0.5 Degree Closeness Eccentricity Stress SBetweenness Figure 13: Edge Addition- Thickness Summary 12 Random Appendix Figure 14: Visualisation of Thickness with Vertex Deletion on AT&T graph 13 0.9 0.8 Ratio 0.7 MIN 0.6 MAX 0.5 AVG 0.4 SUM(yi)/SUM(xi) 0.3 0.2 0.1 Degree Closeness Eccentricity Stress SP Betweeness Random Figure 15: Vertex Deletion- Planarisation on ALF/CD data set 0.9 0.8 0.7 MIN 0.6 Ratio MAX 0.5 AVG 0.4 SUM(yi)/SUM(xi) 0.3 0.2 0.1 Degree Closeness Eccentricity Stress SP Betweeness Random Figure 16: Vertex Deletion-Planarisation on ALF/CU data set 0.9 0.8 0.7 MIN Ratio 0.6 MAX 0.5 AVG 0.4 SUM(yi)/SUM(xi) 0.3 0.2 0.1 Degree Closeness Eccentricity Stress SP Betweeness Random Figure 17: Vertex Deletion- Planarisation on AT&T graph 0.9 0.8 MIN 0.7 MAX 0.6 Ratio AVG 0.5 SUM(yi)/SUM(xi) 0.4 0.3 0.2 0.1 Degree Closeness Eccentricity Stress ShBet Random Figure 18: Vertex Deletion- Planarisation on Barabasi-Albert scale-free network 14 0.9 0.8 0.7 MIN 0.6 Ratio MAX 0.5 AVG 0.4 SUM(yi)/SUM(xi) 0.3 0.2 0.1 Degree Closeness Eccentricity Stress ShBet Random Figure 19: Vertex Deletion- Planarisation on Erdos-Renyi random network 0.4 0.35 MIN 0.3 MAX 0.25 Ratio AVG 0.2 SUM(yi)/SUM(xi) 0.15 0.1 0.05 Degree Closeness Eccentricity Stress SP Betweeness Random Figure 20: Vertex Addition- Planarisation on ALF/CD data set 0.4 0.35 0.3 MIN MAX 0.25 Ratio AVG 0.2 SUM(yi)/SUM(xi) 0.15 0.1 0.05 Degree Closeness Eccentricity Stress SP Betweeness Random Figure 21: Vertex Addition-Planarisation on ALF/CU data set 0.6 0.5 MIN MAX 0.4 Ratio AVG SUM(yi)/SUM(xi) 0.3 0.2 0.1 Degree Closeness Eccentricity Stress SP Betweeness Random Figure 22: Vertex Addition- Planarisation on AT&T graph 15 0.7 0.6 MIN 0.5 MAX 0.4 Ratio AVG SUM(yi)/SUM(xi) 0.3 0.2 0.1 Degree Closeness Eccentricity Stress ShBet Random Figure 23: Vertex Addition- Planarisation on Barabasi-Albert scale-free network 0.6 0.5 MIN 0.4 Ratio MAX AVG 0.3 SUM(yi)/SUM(xi) 0.2 0.1 Degree Closeness Eccentricity Stress ShBet Random Figure 24: Vertex Addition- Planarisation on Erdos-Renyi random network 16 0.9 0.8 0.7 MIN 0.6 Ratio MAX 0.5 AVG SUM(yi)/SUM(xi) 0.4 0.3 0.2 0.1 Degree Closeness Eccentricity Stress ShBet Random Figure 25: Edge Deletion- Planarisation on ALF/CD data set 0.9 0.8 MIN 0.7 MAX Ratio 0.6 AVG 0.5 SUM(yi)/SUM(xi) 0.4 0.3 0.2 0.1 Degree Closeness Eccentricity Stress ShBet Random Figure 26: Edge Deletion-Planarisation on ALF/CU data set 1.2 MIN 0.8 Ratio MAX AVG 0.6 SUM(yi)/SUM(xi) 0.4 0.2 Degree Closeness Eccentricity Stress ShBet Random Figure 27: Edge Deletion- Planarisation on AT&T graph 17 1.2 MIN 0.8 Ratio MAX AVG 0.6 SUM(yi)/SUM(xi) 0.4 0.2 Degree Closeness Eccentricity Stress ShBet Random Figure 28: Edge Deletion- Planarisation on Barabasi-Albert scale-free network 0.9 Ratio 0.8 0.7 MIN 0.6 MAX AVG 0.5 SUM(yi)/SUM(xi) 0.4 0.3 0.2 0.1 Degree Closeness Eccentricity Stress ShBet Random Figure 29: Edge Deletion- Planarisation on Erdos-Renyi random network 0.45 0.4 0.35 MIN 0.3 Ratio MAX 0.25 AVG SUM(yi)/SUM(xi) 0.2 0.15 0.1 0.05 Degree Closeness Eccentricity Stress ShBet Random Figure 30: Edge Addition- Planarisation on ALF/CD data set 0.45 0.4 0.35 MIN 0.3 Ratio MAX 0.25 AVG 0.2 SUM(yi)/SUM(xi) 0.15 0.1 0.05 Degree Closeness Eccentricity Stress ShBet Random Figure 31: Edge Addition-Planarisation on ALF/CU data set 18 0.8 0.7 0.6 MIN Ratio 0.5 MAX 0.4 AVG 0.3 SUM(yi)/SUM(xi) 0.2 0.1 Degree Closeness Eccentricity Stress ShBet Random Figure 32: Edge Addition- Planarisation on AT&T graph 0.6 0.5 MIN 0.4 Ratio MAX AVG 0.3 SUM(yi)/SUM(xi) 0.2 0.1 Degree Closeness Eccentricity Stress ShBet Random Figure 33: Edge Addition- Planarisation on Barabasi-Albert scale-free network 0.8 0.7 0.6 MIN 0.5 Ratio MAX AVG 0.4 SUM(yi)/SUM(xi) 0.3 0.2 0.1 Degree Closeness Eccentricity Stress ShBet Random Figure 34: Edge Addition- Planarisation on Erdos-Renyi random network 19 4.5 3.5 Thickness 2.5 max avg 1.5 0.5 Degree Closeness Eccentricity Stress SP Betweeness Random Figure 35: Vertex Deletion- Thickness on ALF/CD data set 4.5 3.5 Thickness 2.5 max avg 1.5 0.5 Degree Closeness Eccentricity Stress SP Betweeness Random Figure 36: Vertex Deletion- Thickness on ALF/CU data set Thickness max avg Degree Closeness Eccentricity Stress SP Betweeness Random Figure 37: Vertex Deletion- Thickness on AT&T graph 20 4.5 3.5 Thickness max 2.5 avg 1.5 0.5 Degree Closeness Eccentricity Stress SP Betweeness Random Figure 38: Vertex Deletion- Thickness on Barabasi-Albert scale-free network Thickness max avg Degree Closeness Eccentricity Stress SP Betweeness Random Figure 39: Vertex Deletion- Thickness on Erdos-Renyi random network 3.5 2.5 Thickness max avg 1.5 0.5 Degree Closeness Eccentricity Stress SP Betweeness Random Figure 40: Vertex Addition- Thickness on ALF/CD data set 3.5 2.5 Thickness max avg 1.5 0.5 Degree Closeness Eccentricity Stress SP Betweeness Random Figure 41: Vertex Addition- Thickness on ALF/CU data set 21 4.5 3.5 Thickness max 2.5 avg 1.5 0.5 Degree Closeness Eccentricity Stress SP Betweeness Random Figure 42: Vertex Addition- Thickness on AT&T graph 4.5 3.5 Thickness max 2.5 avg 1.5 0.5 Degree Closeness Eccentricity Stress SP Betweeness Random Figure 43: Vertex Addition- Thickness on Barabasi- Albert scale-free network 3.5 Thickness 2.5 max avg 1.5 0.5 Degree Closeness Eccentricity Stress ShBet Random Figure 44: Vertex Addition- Thickness on Erdos- Renyi random network 22 4.5 3.5 Thickness max 2.5 avg 1.5 0.5 Degree Closeness Eccentricity Stress SP Betweeness Random Figure 45: Edge Deletion- Thickness on ALF/CD data set Thickness max avg Degree Closeness Eccentricity Stress SP Betweeness Random Figure 46: Edge Deletion- Thickness on ALF/CU data set 12 10 Thickness max avg Degree Closeness Eccentricity Stress SP Betweeness Random Figure 47: Edge Deletion- Thickness on AT&T graph 23 12 10 Thickness max avg Degree Closeness Eccentricity Stress SP Betweeness Random Figure 48: Edge Deletion- Thickness on Barabasi-Albert scale-free network Thickness max avg Degree Closeness Eccentricity Stress SP Betweeness Random Figure 49: Edge Deletion- Thickness on Erdos-Renyi random network 3.5 Thickness 2.5 max avg 1.5 0.5 Degree Closeness Eccentricity Stress SP Betweeness Random Figure 50: Edge Addition- Thickness on ALF/CD data set 3.5 Thickness 2.5 max 1.5 avg 0.5 Degree Closeness Eccentricity Stress SP Betweeness Random Figure 51: Edge Addition- Thickness on ALF/CU data set 24 Thickness max avg Degree Closeness Eccentricity Stress SP Betweeness Random Figure 52: Edge Addition- Thickness on AT&T graph 4.5 Thickness 3.5 2.5 max avg 1.5 0.5 Degree Closeness Eccentricity Stress SP Betweeness Random Figure 53: Edge Addition- Thickness on Barabasi-Albert scale-free network Thickness max avg Degree Closeness Eccentricity Stress SP Betweeness Random Figure 54: Edge Addition- Thickness on Erdos-Renyi random network 25 ... review the definition of each centralities For details, see [1] 2. 3.1 Degree Degree centrality cD (v) of a vertex v is defined as the degree d( v) of v for undirected graphs For directed graphs,... edge deletion for computing thickness in all cases 5. 3 Centrality- Based 2. 5D Graph Drawing Figure shows a directed graph in Rome data set (ALF/CD/graphwith17nodes/ug. 15. 17) with 18 vertices and... Insight, written in Java 3D) For test data, we use standard Graph Drawing test suites including Rome library (both directed and undirected: ALF/CD and ALF/CU, in GML format) and AT&T graphs, as well as