On the Visualization of Social and other Scale-Free Networks

Transcription

1 On the Visualization of Social and other Scale-Free Networks Yuntao Jia 1, Jared Hoberock 1, Micha Garland 2, and John C. Hart 1 1 University of Illinois at Urbana-Champaign 2 NVIDIA Yuntao Jia yjia3@illinois.edu CS598 Information Visualization 02/18/2010

2 Scale Free Networks Node degree follows power law Few high degree nodes Many low degree nodes Arise often Biology Sociology Networking Physics Layout difficult Highly connected Rary planar y = nodes w/deg. x Example: bo protn interaction graph 1,458 yeast protns 1,948 interactions y = 901 x x = node degree

3 Scale Free Networks Node degrees Highest-degree nodes, hubs Followed by nodes with lower degrees Clustering coefficient Low-degree nodes bong to subgraphs Connected through hubs Common examples Cebrities & fans Big airports & small airports Image courtesy of Wikipedia

4 Betweenness Centrality Proposed by Frean 1977 One of the essential ranking tools Graph G= (V, E), V = n, E = m ( )= -, ( )/, Ries on computing All-Pairs Shortest Paths (APSP) Complexity O(m*n) for unwghted graph [Brandes 01]

5 Betweenness Centrality High-BC nodes Lies on considerable fraction of shortest paths Lies on communication paths between others nodes Connecting subgraphs Low-BC nodes Lies on a few shortest paths Nghbors are wl connected to each other Within subgraphs

6 Layout Alone Not Enough Force-directed approaches [Eades 1984] [Fruchterman and Rngold 1991] [Davidson and Har 1996] [Kamada and Kawai 1989] Other approaches GEM [Frick et al. 1994] GRIP [Gajer & Kobourov 2000] ACE [Koren et al. 2003] FM3 [Hachul & Junger 2004] Topolayout [Archambault et al. 2007] Graph bo laid out using GEM Best planar methods don t work wl on power-law graphs Need to simplify graph to visualize its structure

7 Recent Simplification Methods Node Clustering Brandes et al. 03 (survey) Wu et al. 04 (min-bc, max deg.) Kumar & Garland 06 (stratified) Stochastic Sampling Rafi & Curial 05 (+focus) Leskovec & Faloutsos 06 (walk) Deterministic Filtering Auber et al. 03 (d d weak edges) Lee et al. 06 and Boutin et al. 06 (grew spanning trees) Girvan & Newman 02 and Newman 04 (roved hibc nodes) Us Node/Edge No. 1458/1948 Node/Edge No. 998/1458 Node/Edge No. 1258/1458 Node/Edge No. 348/570 bo Geodesic Clustering [Wu et al. 04] Stochastic Sampling [Rafi & Curial 05] Strength Filtering [Auber et al. 03]

8 Our Approach Filters a graph by roving edges in order of increasing betweenness centrality. Preserve graph connectivity Preserve graph features (e.g. cliques) Node/Edge No. 1458/1948 Node/Edge No. 1458/1458 bo Our result

9 Motivation Betweenness centrality ranks edges How often they appear on shortest paths (SP) High-BC common SP important communication tunns Low-BC less common SP less important tunns Rove low-bc edges Keep important edges/back bone of the graph Rove unimportant edges to facilitate visualizations

10 Why Connectivity? Simply edge filtering is not enough Node/Edge No. 1458/1948 Node/Edge No. 1458/1493 bo Filtering only

11 Motivation Betweenness centrality ranks edges How often they appear on shortest paths (SP) High-BC common SP important communication tunns Low-BC less common SP less important tunns Rove low-bc edges Keep important edges/back bone of the graph Rove unimportant edges to facilitate visualizations Keep connectivity Connectivity itsf is important Keep local graph structures Keep other features

12 Workflow

13 Workflow

14 Compute Edge Metric Graph G= (V, E), V = n, E = m Betweenness centrality [Frean 1977] Ries on computing All-Pairs Shortest Paths Complexity O(m*n) for unwghted graph [Brandes 01] For huge graphs Approximated with random sampling [Jacob et al. 05] O((m+n)*log(n)) with C*log(n) samples where C is a constant For our edge filtering purpose Only rative orders of BC are needed Sect C*log(n) highest degree hub nodes

15 Workflow

16 Graph Feature detection Graph features Cliques NP-Complete probl Fast approximation O(m*n) [Chiricota et al. 03] User defined features GS MER PRU JNS PFG GE sp Edge No Our result Edge No. 703

17 Workflow

18 Edge Filtering Edges BC Metric Threshold t = 1.25 eh

19 Edge Filtering Edges BC Metric Edges BC Metric Threshold t = eh Sort eh

20 Edge Filtering Edges BC Metric Edges BC Metric Threshold t = eh Sort eh

21 Edge Filtering Edges BC Metric Edges BC Metric Threshold t = eh Sort eh

22 Edge Filtering Edges BC Metric Edges BC Metric Threshold t = eh Sort eh

23 Edge Filtering Edges BC Metric Edges BC Metric Threshold t = eh Sort eh

24 Edge Filtering Edges BC Metric Edges BC Metric Threshold t = eh Sort eh

25 Edge Filtering Edges BC Metric Edges BC Metric Threshold t = eh Sort eh

26 Workflow

27 Recover connectivity Roved Edges BC Metric stack connectivity test

28 Recover connectivity Roved Edges BC Metric stack

29 Recover connectivity Roved Edges BC Metric stack

30 Recover connectivity Roved Edges BC Metric stack

31 Recover connectivity Roved Edges BC Metric stack

32 Workflow

33 Workflow

34 Result SIGGRAPH 07 Dani Cohen-Or Dani Cohen-Or Maneesh Agrawala Szymon Rusinkiewicz Wojciech Matusik John Barnwl Maneesh Agrawala Fredo Durand Szymon Rusinkiewicz Markus Gross Fredo Durand Wojciech Matusik Markus Gross John Barnwl siggraph-07 Edge No. 441 Our results Edge No. 129

35 Result stock corration GS MER PRU JNS PFG GE Edge No Edge No. 703 sp Our result Graph stratification [Kumar and Garland 06]

36 Result VIS researchers Node/Edge No. 618/1218 Node/Edge No. 618/523 vis-community Our result

37 Results autonomous syst (AS) AT&T WorldNet Services Sprint UUNET Lev Technologies, 3 Communications, Inc. Inc. Cogent Communications UUNET Technologies, Inc. Cogent Communications AT&T WorldNet Services Sprint Lev 3 Communications, Inc. Edge No Edge No as.r Our result Dataset in courtesy of CAIDA s ranking at

38 Empirical Estimate Verification 7 x SSE of Distance Matrix Random edge sampling Edge filtering with betweenness centrality (BC) Edge filtering with approximated BC Error Raining Edges RMSE of Betweenness Centrality Random Edge Sampling Our Exact Edge filtering Our Approximated Edge filtering Percentage Error Rative error of BC approximation Raining Edges

39 Performance Graph Nodes Edges Timing siggraph s sp s bo s cg_web s as-r s hep-th s flickr s Time 10 5 Performance on G(V,E) ( E + V )*log( V )

40 Discussion Doesn t work wl on planar and other non-power-law graphs But interesting neverthess Planar graph Our result

41 Discussion Doesn t visualize the entire dataset Hierarchical Edge Bundles [Holten 06] Adding back roved edges as straight lines Adding back roved edges as B-splines

42 Hierarchical Edge Bundles for General Graphs, Jia et al.

43 Conclusion A new graph simplification approach for scale-free graphs Filter edges to rove distracting edge crossings Effectivy improves the visualization of scale-free graphs Efficient approximation of betweenness centrality computation

44 Questions? Thanks for your attention! Acknowledgent NSF grant IIS Thanks reviewers for thr valuable suggestions Thanks David for sharing the flickr dataset NVIDIA research Implentation based on Tulip graph visualization syst Implentation based on The Visualization ToolKit (VTK)