2. A New Force-Directed Graph Drawing Method Based on Edge- Edge Repulsion Chun-Cheng Lin and Hsu-Chen Yen Department of Electrical Engineering, National Taiwan University, Taiwan, R.O.C.

3. Outline Introduction and motivations Model of edge-edge repulsion Implementation and experimental results Conclusion

4. Force-directed graph drawing method An energy model is associated with the graph Low energy states correspond to nice layouts Hence, the drawing algorithm is an optimization process Initial (random) layoutFinal (nice) layoutIteration 1:Iteration 2:Iteration 3:Iteration 4:Iteration 5:Iteration 6:Iteration 7:Iteration 8:Iteration 9: Aesthetical properties Proximity preservation: similar nodes are drawn closely Symmetry preservation: isomorphic sub-graphs are drawn identically No external influences: “ Let the graph speak for itself ” ( This slide comes from Yehuda Harel )

5. Motivations (1/2) The force-directed methods can be categorized according to the types of repulsion vertexedge vertex edge X ( vertex-vertex repulsion ) 1.1. Eades, 1984 1.2. Fruchterman & Reingold, 1991 X ( vertex-edge repulsion ) 2.1. Davidson & Harel, 1996 2.2. Bertault, 1999 the same ( edge-vertex repulsion ) edge-edge repulsion

6. 1.1. Vertex-Vertex Repulsion ( Eades, 1984 ) Model of Vertex-Vertex Repulsion –Nodes → charged rings → repulsive force –Edges → springs → attractive force Properties –The drawing has a high degree of symmetry and uniform edge length let go!

7. 1.2. Vertex-Vertex Repulsion ( ) 1.2. Vertex-Vertex Repulsion ( Fruchterman & Reingold, 1991) Eades’ force formulas –Hook’s law for spring forces f a ( x ) = K  ( x – d )  Modified formula –Newtonian law for repulsive forces f r ( x ) = K / r 2 FR proposed a more efficient version – –Locally consider vertex-vertex repulsion log ( x ) x C2C2

8. 2.1. Vertex-Edge Repulsion ( Davidson & Harel, 1996 ) Drawing graphs using simulated annealing approach –Using the paradigm of simulated annealing (Kirkpatrick, Gelatt, and Vecchi, 1983) –Their approach tries to find an optimal configuration according to a cost function as follows: Penalize the vertex and edge that are too close to each other. ( vertex-edge repulsion ) Penalize the distance between two vertices. ( vertex-vertex repulsion )

9. 2.2. Vertex-Edge Repulsion ( Bertault, 1999 ) A force-directed approach that preserves edge crossing properties –Theorem. Two edges cross in the final drawing iff these edges crossed on the initial layout #(crossings) = 5

10. Motivations (2/2) Def. Angular resolution –the smallest angle formed by two neighboring edges incident to the common vertex in straight line drawing –guaranteeing that arbitrarily small angles cannot be formed by adjacent edges The Zero Angular Resolution Problem –There exist at least two of the edges incident to the common vertex overlapping –Resulting in a bad drawing with edge-edge and vertex-edge crossings simultaneously –Classical force-directed method cannot guarantee the absence of any overlapping of edges incident to the common vertex Edge-edge overlapping vertex-edge overlapping initial final

11. Model of Edge-Edge Repulsion In each iteration: Step1. For each edge (spring), compute spring forces as Eades’, i.e. compute spring forces as Eades’, i.e. Step2. For each pair of neighboring edges, compute repulsive forces according to our model compute repulsive forces according to our model Step3. Adjust the position of every vertex according to the force acting on the vertex acting on the vertex C A B Every edge is replaced by a changed spring.

12. Intuition The magnitudes of the repulsive force due to the two edges and are –Positively correlated with the edge lengths –Negatively correlated with the included angle  C A B f1f1 f 2 = - f 1 f 11 f 12 

13. Potential Field Method Motion planning or Path planning ( Chuang and Ahuja, 1998) ( Chuang and Ahuja, 1998) ++ ++++ + +++++ ++ + + + + + + + + + + + + + + + ++++ S G C A B General formula of the repulsive force between two charged edges

14. Reasons to simplify the repulsive force formula We don’t use the general formulas of edge-edge repulsion because: –The formulas derived from potential fields appears to be a bit complicated and consequently require special care when implementing such a method. From a practical viewpoint, such a complication may not be needed for the purpose of drawing graphs. Therefore, by observing some characteristics of edge- edge repulsion and experimental results of potential fields method, we are able to derive a simplified version of repulsive forces.

15. Simulation on magnitude of force Force magnitude vs. edge length Force magnitude vs. the included angle 

16. Simulation on orientation of force the included angle  the included angle 

17. Simplified force formula The force can be calculated as: Note that f 1 has the advantage that f 1 is determined only by three parameters, facilitating a simple implementation of our drawing algorithm based upon edge-edge repulsion

18. Experimental Results (i) Larger angular resolution Preserve a high degree of symmetry and uniform edge length as the classical method

19. Experimental Results (ii) Hypercubes Because the model of edge-edge repulsion allows at least two vertices coinciding (although such drawings are improper), our approach may produce drawings with more symmetries.

20. Statistics The following is the statistics of the above experimental results with comparison of the classical method and our approach. Observing the column StdDev / AvgLen, i.e. the normalized standard deviation of edge lengths, our approach without costing more running time seems to have equal or more uniform edge length than the classical method. Observing the column `Angular resolution’, the classical method may have the problem of zero or few angular resolution, while our approach normally has larger angular resolution than the classical.

21. Local minimal problem Force-directed method suffers from the local minimal problem, in which the spring force may be too weak to spread the graph Both the classical method and our approach may not handle local minimal problem in some cases, so the following strategies can be applied: –Two-phase method First using the classical method and then using our method –Hybrid strategy Simultaneously using the classical method and our method –Adjusting parameters Parameters play important roles ( The figures shown in this slide comes from Yehuda Harel )

22. Conclusion A new force-directed method based on edge-edge repulsion for generating a straight-line drawing not only preserving the original properties of a high degree of symmetry and uniform edge length but also without zero angular resolution has been proposed and implemented Future work –To handle the local minimal problem Try to use multilevel techniques or optimal heuristics, such as simulated annealing, genetic algorithm, etc. –More experimental results on graphs of huge size and theoretical results on the power of the model of edge-edge repulsion

23. The End Thank you for your attention.