Using Multilevel Force-Directed Algorithm to Draw Large Clustered Graph 研究生: 何明彥 指導老師:顏嗣鈞 教授 2018/12/4 NTUEE

OUTLINE Introduction Multilevel Paradigm Experiment result Conclusion Coarsening Phase Uncoarsening Phase Experiment result Conclusion Reference 2018/12/4 NTUEE

Introduction(1/7) (a) network topology (b) social network (c) E-R diagram (d) data flow chart 2018/12/4 NTUEE

Introduction(2/7) Graph is widely-used in computer science recently. entity Vertex relationship between entities Edge ex: 3 4 2 1 0-1,2,3,4 1-0,2,3,4 2-0,1,3,4 3-0,1,2,4 4-0,1,2,3 2018/12/4 NTUEE

Introduction(3/7) Good? Aesthetic? crossing symmetric angular resolution ……. Graph Drawing Algorithm 2018/12/4 NTUEE

Introduction(4/7) Graph Drawing Algorithm Circular Drawing Orthogonal Drawing Straight-Line Drawing (a) Circular Drawing (b) Orthogonal Drawing (c) Straight-Line Drawing 2018/12/4 NTUEE

Introduction(5/7) Straight-Line Drawing—Force-Directed Method Vertex Charged ring by Newton’s Law, repulsive force between vertices Edge Spring by Hook’s Law, attractive Force to make vertices closer 2018/12/4 NTUEE

Introduction(6/7) Clustering 2018/12/4 NTUEE

Introduction(7/7) Usually, Circular drawing can express the aesthetic goal of embedded cluster. Embedded cluster is important. ex: hosts in which LAN for network topology; people in which group for social network try to using Straight-Line Drawing to achieve it. 2018/12/4 NTUEE

Multilevel Paradigm(1/4) Two parts: Graph Coarsening: Partition vertices to form cluster and define a new graph Graph Uncoarsening: To redraw original graph 2018/12/4 NTUEE

Multilevel Paradigm(2/4) Graph Coarsening given graph GL Level-L graph partitioning method Coarser graph GL+1 Level-L+1 2018/12/4 NTUEE

Multilevel Paradigm(3/4) Graph Uncoarsening Level-L+1 given graph GL+1 uncoarsen GL+1; use spring algorithm to draw GL Level-L drawing graph GL 2018/12/4 NTUEE

Multilevel Paradigm(4/4) (a) original graph (b) group vertices (c) coarser graph (d) uncoarsening stage (e) final result 2018/12/4 NTUEE

Coarsening Phase(1/12) Partition vertices to form cluster and define a new graph K-way partitioning: The number of edges of E whose incident vertices belong to different subsets is minimized. 2018/12/4 NTUEE

Coarsening Phase(2/12) Graph Partitioning The number of edges of E whose incident vertices belong to different subsets is so-called edge-cut and should be minimized. b f a g c d h e a a f b d b g c f h c g e e d h Edge-cut=5 Edge-cut=1 2018/12/4 NTUEE

Coarsening Phase(3/12) Two stage Matching stage Contraction stage Using random matching Contraction stage A coarser graph is created by contracting former matching vertex Using Kernighan-Lin algorithm 2018/12/4 NTUEE

Algorithm: Random matching (Go) Input:G0 Output: Match [u] and Map [u]; Match [u] store the vertex which is matched by u Map [u] store the label of v in the coarser graph begin visit vertex in random order If vertex u has not been matched yet then random select one of its unmatched adjacent vertices v If v exist then put v in match [u] map [u]=map [v] else match [u]=u end 2018/12/4 NTUEE

Coarsening Phase(5/12) 2018/12/4 NTUEE

Properties of K-L Algo. 2018/12/4 NTUEE

A B a x b a b A B b x a B A 2018/12/4 NTUEE

if Gj>0 then Algorithm: Kernighan-Lin Algorithm (G) Input: G=(V,E) Output: partition A and B with small edge-cut begin partition G into A and B repeat compute Dv, for i=1 t0 n/2 do find an unmarked pair (ai, bi), that maximizing gab mark ai, bi update Dv for all unmarked find j, such that Gj= is maximized if Gj>0 then move a1,…, aj from A to B move b1,…,bj from B to A until end 2018/12/4 NTUEE

a d b c e f a b c d e f 1 4 3 2 Initial edge cut=22 Random matching For “a”: a d b c e f 1 2 4 3 2018/12/4 NTUEE

a d b c e f f b a c d e f b a c d e f b e a d 2018/12/4 NTUEE

f b Iteration 1 f b a c d e Iteration 2 2018/12/4 NTUEE

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Uncoarsening Phase(1/4) Two stage Mapping stage Put nodes in a region Refinement stage Using FR force-directed method Using Virtual spring for drawing cluster embedded clustered graph 2018/12/4 NTUEE

Algorithm: Uncoarening Input: coarser graph Gi Output: original graph G0 Begin using Force-directed method to draw Gi For each vertex u in Gi if (vertex v in Gi-1 is coarsened into u) place v in a region of u set u as the virtual node again using spring method to draw original graph Go end 2018/12/4 NTUEE

Virtual-Spring to make nodes in cluster close together internal-spring: a spring force between a pair node which belong to a cluster a b external-spring: a spring force between a pair node which not belong to a cluster c d virtual node: each cluster has a virtual node Create virtual spring e g f virtual spring: a spring force between a pair node along the virtual edge h 2018/12/4 NTUEE

Experiment Result(1/6) |V|=25; |E|=60 Original: 2018/12/4 NTUEE

Experiment Result(2/6) Spring Algorithm result: 2018/12/4 NTUEE

Experiment Result(3/6) |Cluster|=5 2018/12/4 NTUEE

Experiment Result(4/6) |V|=42; |E|=75 Original: 2018/12/4 NTUEE

Experiment Result(5/6) Spring Algorithm result: 2018/12/4 NTUEE

Experiment Result(6/6) |Cluster|=6 2018/12/4 NTUEE

Conclusion It is suitable for drawing large graph Multilevel algorithm is fast and no local minimum. K-L algorithm works well for graph partitioning. Embedded cluster graph is available. 2018/12/4 NTUEE

Reference Chris Walshaw, ”A multilevel algorithm for Force-Directed Graph Drawing”. Journal of Graph Algorithm and Applications, vol7,no.3,pp253-285 (2003) J. Chuang, C.Lin, and H. Yen, "Drawing Graphs with Nonuniform Nodes Using Potential Fields," in the Proceedings of the 11th International Symposium on Graph Drawing 2003,(LNCS 2912) pp. 460-465, Perugia, Italy, Sept. 21-24, 2003 Peter Eades, Mao Lin Huang, “Navigating Cluster Graphs using Force-Directed Methods “, Journal of Graph Algorithm and Applications(2000) George Karypis and Vipin Kumar ,”A Fast and High Quality Multilevel Scheme for Partitioning Irregular” .SIAM of Journal on Scientific Computing J.Fruchterman, M.Reingold, “Graph Drawing by Force-Directed Placement”, 1991 B.W.Kernighan, S.Lin, “An efficient heuristic procedure for partitioning graphs.” The Bell System Technical Journal S.Hachul and M.Junger, ”Drawing Large graphs with a potential-field-based multilevel algorithm” GD2004 2018/12/4 NTUEE