AnisotropicRadialLayoutforVisualizingCentralityandStructureinGraphs

MukundRajandRossT.Whitaker

25thInternaBonalSymposiumonGraphDrawingandNetworkVisualizaBonBoston,September2017

Node-linkDiagram•  Goals–  ImproveaestheBcs– ReducevisualcluOer– Conveyfeaturesofinterest

•  Structuralfeaturesbasedoninternodedistances•  Importanceofnodes

Nodelinkdiagramofarandomgraph

VisualizingStructure•  Preserveinternodedistancesinthedrawing

•  Method:– DimensionalityreducBontechniques

•  Mul+dimensionalScaling(MDS)•  t-SNE(stochasBcneighborembedding)

Nodelinkdiagram(MDSlayout)

MulBdimensionalScaling(MDS)•  Conveyssimilaritybetweenobjects•  MinimizesenergyfuncBon(stress)

ImportanceofNodes•  Usinggraphstructure–  Nodecentrality

•  Degreecentrality:numberofneighbors•  Closenesscentrality:reciprocalofsum

ofdistances•  Betweennesscentrality:numberof

shortestpathspassingthroughthenode•  Manymore…

Degreecentrality Closenesscentrality

Betweennesscentrality

VisualizingCentrality

•  Nodesizeencodescentrality•  ConflictbetweenposiBon

andsizechannels

VisualizingCentrality•  Radiallayout[Yee2001]

Distancefromcenterencodesbetweennesscentrality.

*FiguregeneratedusingSocNetV.

VisualizingCentralityandStructure•  (Approximate)Distancepreservingradiallayout[Brandes2009,Baingana

2014]

ObjecBves

?MDSlayoutpreservesstructure

•  Conveycentrality•  AvoidconflicBngperceptualcues•  Reducestructural(distance)distorBon

RadiallayouthighlightsnodecentralityHighlightcentralityandminimizedistancedistorBon

AnisotropicRadialLayouts•  Relaxcircularconstraint•  Usestarshapedcurves

RadialMonotonicity•  StrictlydecreasingalongallcenteroutwarddirecBons•  Guaranteesstarshapedcontours

Step1:IniBalizaBon

Nodesizesencodecentrality.

•  IniBalizeposiBonsusingMDS.

Step2:InterpolaBon•  SmoothinterpolaBonofthecentrality(e.g.usingthinplatespline)

1

3

2

4

Step3:MonotonizaBon•  Monotonizingalongradialaxis

Step3:MonotonizaBon•  Monotonizinga1DfuncBon[DeOeetal2006]

•  Steps1.  ConstructdensityesBmate2.  Computecdf3.  Invertcdf

•  Output:smoothmonotonicapproximaBon

Step3:MonotonizaBon

InterpolaBonfieldonaCartesiangrid

Resampleonaregularpolargrid

InterpolatedandresampledMonotonicfield‘M’

Monotonizealongeachangle

•  Radiallymonotonizinga2Dfield

•  Ifinputfieldissmooth,independent1DmonotonizaBonresultsinasmooth2Dfield[DeOeetal2006]

Step4:OpBmizaBonobjecBve•  CombineMDSwithcentrality•  AddpenaltyfordeviaBonofnodecentralityfromassociatedfieldvalue

Step5:OpBmizaBon•  Usinggradientdescent

Results

MDSlayout Radiallayout Anisotropicradiallayout

•  RandomgraphgeneratedusingtheBarabasi-Albertmodel.

Results•  RandomgraphgeneratedusingtheBarabasi-Albertmodel.

MDSlayout Radiallayout Anisotropicradiallayout

Results

MDSlayout Radiallayout Anisotropicradiallayout

•  Zachary’skarateclubnetwork[Zachary1977]

Results

MDSlayout Radiallayout Anisotropicradiallayout

•  TerroristnetworkfromtheMadridtrainbombingincidentin2004[Rodriguez2005]

Results

MDSlayout Radiallayout Anisotropicradiallayout

•  LesMiserablescharacterassociaBons

Conclusions•  Strategyforpreservingcentralityandstructure•  LayoutAlgorithm•  Suitableforrealworldnetworks•  Futurework– AutomaBcparameteresBmaBon–  BeOeropBmizaBonmethod–  Experimentwithlargegraph(highernodecount)

Thanks•  ThisworkwassupportedbyNaBonalScienceFoundaBon(NSF)grantIIS-1212806.