anisotropic radial layout for visualizing centrality and ...anisotropic radial layout for...
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AnisotropicRadialLayoutforVisualizingCentralityandStructureinGraphs
MukundRajandRossT.Whitaker
25thInternaBonalSymposiumonGraphDrawingandNetworkVisualizaBonBoston,September2017
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Node-linkDiagram• Goals– ImproveaestheBcs– ReducevisualcluOer– Conveyfeaturesofinterest
• Structuralfeaturesbasedoninternodedistances• Importanceofnodes
Nodelinkdiagramofarandomgraph
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VisualizingStructure• Preserveinternodedistancesinthedrawing
• Method:– DimensionalityreducBontechniques
• Mul+dimensionalScaling(MDS)• t-SNE(stochasBcneighborembedding)
Nodelinkdiagram(MDSlayout)
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MulBdimensionalScaling(MDS)• Conveyssimilaritybetweenobjects• MinimizesenergyfuncBon(stress)
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ImportanceofNodes• Usinggraphstructure– Nodecentrality
• Degreecentrality:numberofneighbors• Closenesscentrality:reciprocalofsum
ofdistances• Betweennesscentrality:numberof
shortestpathspassingthroughthenode• Manymore…
Degreecentrality Closenesscentrality
Betweennesscentrality
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VisualizingCentrality
• Nodesizeencodescentrality• ConflictbetweenposiBon
andsizechannels
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VisualizingCentrality• Radiallayout[Yee2001]
Distancefromcenterencodesbetweennesscentrality.
*FiguregeneratedusingSocNetV.
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VisualizingCentralityandStructure• (Approximate)Distancepreservingradiallayout[Brandes2009,Baingana
2014]
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ObjecBves
?MDSlayoutpreservesstructure
• Conveycentrality• AvoidconflicBngperceptualcues• Reducestructural(distance)distorBon
RadiallayouthighlightsnodecentralityHighlightcentralityandminimizedistancedistorBon
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AnisotropicRadialLayouts• Relaxcircularconstraint• Usestarshapedcurves
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RadialMonotonicity• StrictlydecreasingalongallcenteroutwarddirecBons• Guaranteesstarshapedcontours
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Step1:IniBalizaBon
Nodesizesencodecentrality.
• IniBalizeposiBonsusingMDS.
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Step2:InterpolaBon• SmoothinterpolaBonofthecentrality(e.g.usingthinplatespline)
1
3
2
4
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Step3:MonotonizaBon• Monotonizingalongradialaxis
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Step3:MonotonizaBon• Monotonizinga1DfuncBon[DeOeetal2006]
• Steps1. ConstructdensityesBmate2. Computecdf3. Invertcdf
• Output:smoothmonotonicapproximaBon
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Step3:MonotonizaBon
InterpolaBonfieldonaCartesiangrid
Resampleonaregularpolargrid
InterpolatedandresampledMonotonicfield‘M’
Monotonizealongeachangle
• Radiallymonotonizinga2Dfield
• Ifinputfieldissmooth,independent1DmonotonizaBonresultsinasmooth2Dfield[DeOeetal2006]
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Step4:OpBmizaBonobjecBve• CombineMDSwithcentrality• AddpenaltyfordeviaBonofnodecentralityfromassociatedfieldvalue
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Step5:OpBmizaBon• Usinggradientdescent
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Results
MDSlayout Radiallayout Anisotropicradiallayout
• RandomgraphgeneratedusingtheBarabasi-Albertmodel.
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Results• RandomgraphgeneratedusingtheBarabasi-Albertmodel.
MDSlayout Radiallayout Anisotropicradiallayout
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Results
MDSlayout Radiallayout Anisotropicradiallayout
• Zachary’skarateclubnetwork[Zachary1977]
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Results
MDSlayout Radiallayout Anisotropicradiallayout
• TerroristnetworkfromtheMadridtrainbombingincidentin2004[Rodriguez2005]
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Results
MDSlayout Radiallayout Anisotropicradiallayout
• LesMiserablescharacterassociaBons
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Conclusions• Strategyforpreservingcentralityandstructure• LayoutAlgorithm• Suitableforrealworldnetworks• Futurework– AutomaBcparameteresBmaBon– BeOeropBmizaBonmethod– Experimentwithlargegraph(highernodecount)
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Thanks• ThisworkwassupportedbyNaBonalScienceFoundaBon(NSF)grantIIS-1212806.