Local Graph Partitioning using PageRank Vectors Reid Andersen University of Cali - PDF document

Download presentation
Local Graph Partitioning using PageRank Vectors Reid Andersen University of Cali
Local Graph Partitioning using PageRank Vectors Reid Andersen University of Cali

Tags

this paper

Embed / Share - Local Graph Partitioning using PageRank Vectors Reid Andersen University of Cali


Presentation on theme: "Local Graph Partitioning using PageRank Vectors Reid Andersen University of Cali"— Presentation transcript


Inthispaper,wepresentalocalgraphpartitioningalgorithmthatusespersonalizedPageRankvectorstoproducecuts.BecauseaPageRankvectorisde nedrecursively(aswewilldescribeinsection2),wecanconsiderasinglePageRankvectorinplaceofasequenceofrandomwalkvectors,whichsimpli estheprocessof ndingcutsandallowsgreater exibilitywhencomputingapproximations.WeshowdirectlythatasweepoverasingleapproximatePageRankvectorcanproducecutswithsmallconductance.Incontrast,SpielmanandTengshowthatwhenagoodcutcanbefoundfromaseriesofwalkdistributions,asimilarcutcanbefoundfromaseriesofapproximatewalkdistributions.Ourmethodofanalysisallowsusto ndcutsusingapproximationswithlargeramountsoferror,whichimprovestherunningtime.Theanalysisofouralgorithmisbasedonthefollowingresults:WegiveanimprovedalgorithmforcomputingapproximatePageRankvectors.WeuseatechniqueintroducedbyJeh-Widom[7],andfurtherdevelopedbyBerkhininhisBookmarkColoringAlgorithm[1].ThealgorithmsofJeh-WidomandBerkhincomputemanypersonal-izedPageRankvectorssimultaneously,morequicklythantheycouldbecomputedindividu-ally.OuralgorithmcomputesasingleapproximatePageRankvectormorequicklythanthealgorithmsofJeh-WidomandBerkhinbyafactoroflogn.WeproveamixingresultforPageRankvectorsthatissimilartotheLovasz-Simonovitsmixingresultforrandomwalks.Usingthismixingresult,weshowthatifasweepoveraPageRankvectordoesnotproduceacutwithsmallconductance,thenthatPageRankvectorisclosetothestationarydistribution.WethenshowthatforanysetCwithsmallconductance,andformanystartingverticescontainedinC,theresultingPageRankvectorisnotclosetothestationarydistribution,becauseithassigni cantlymoreprobabilitywithinC.CombiningtheseresultsyieldsalocalversionoftheCheegerinequalityforPageRankvectors:ifCisasetwithconductance(C)f(),thenasweepoveraPageRankvectorpr( ;v) ndsasetwithconductanceatmost,providedthat issetcorrectlydependingon,andthatvisoneofasigni cantnumberofgoodstartingverticeswithinC.Thisholdsforafunctionf()thatsatis esf()= (2=logm).Usingtheresultsdescribedabove,weproducealocalpartitioningalgorithmPageRank-NibblewhichimprovesboththerunningtimeandapproximationratioofNibble.PageRank-Nibbletakesasinputastartingvertexv,atargetconductance,andanintegerb2[1;logm].WhenvisagoodstartingvertexforasetCwithconductance(C)g(),thereisatleastonevalueofbwherePageRank-NibbleproducesasetSwiththefollowingproperties:theconductanceofSisatmost,thevolumeofSisatleast2b�1andatmost(2=3)vol(G),andtheintersectionofSandCsatis esvol(S\C)2b�2.Thisholdsforafunctiong()thatsatis esg()= (2=log2m).TherunningtimeofPageRank-NibbleisO(2blog3m=2),whichisnearlylinearinthevolumeofS.Incomparison,theNibblealgorithmrequiresthatChaveconductanceO(3=log2m),andrunsintimeO(2blog4m=5).PageRank-NibblecanbeusedinterchangeablywithNibble,leadingimmediatelytofasteralgorithmswithimprovedapproximationratiosinseveralapplications.Inparticular,weobtainanalgorithmPageRank-Partitionthat ndscutswithsmallconductanceandapproximatelyoptimalbalance:ifthereexistsasetCsatisfying(C)g()andvol(C)1 2vol(G),thenthealgorithm ndsasetSsuchthat(S)and1 2vol(C)vol(S)5 6vol(G),intimeO(mlog4m=3).Thisholdsforafunctiong()thatsatis esg()= (2=log2m).2 Weremarkthatvol(V)=2m,andwewillsometimeswritevol(G)inplaceofvol(V).Theedgeboundaryofasetisde nedtobe@(S)=ffx;yg2Ejx2S;y62Sg;andtheconductanceofasetis(S)=j@(S)j min(vol(S);2m�vol(S)):2.3.DistributionsTwodistributionswewillusefrequentlyarethestationarydistribution, S(x)=(d(x) vol(S)ifx2S0otherwise:andtheindicatorfunction,v(x)=1ifx=v0otherwise:TheamountofprobabilityfromadistributionponasetSofverticesiswrittenp(S)=Xx2Sp(x):Wewillsometimesrefertothequantityp(S)asanamountofprobabilityevenifp(V)isnotequalto1.Asanexampleofthisnotation,thePageRankvectorwithteleportationconstant andpreferencevectorviswrittenpr( ;v),andtheamountofprobabilityfromthisdistributiononasetSiswritten[pr( ;v)](S).ThesupportofadistributionisSupp(p)=fvjp(v)6=0g:2.4.SweepsAsweepisanecienttechniqueforproducingcutsfromanembeddingofagraph,andisoftenusedinspectralpartitioning[11,14].Wewillusethefollowingdegree-normalizedversionofasweep.Givenadistributionp,withsupportsizeNp=jSupp(p)j,letv1;:::;vNpbeanorderingoftheverticessuchthatp(vi) d(vi)p(vi+1) d(vi+1).Thisproducesacollectionofsets,Spj=fv1;:::;vjgforeachj2f0;:::;Npg,whichwecallsweepsets.Welet(p)=minj2[1;Np](Spj)bethesmallestconductanceofanyofthesweepsets.Acutwithconductance(p)canbefoundbysortingpandcomputingtheconductanceofeachsweepset,whichcanbedoneintimeO(vol(Supp(p))logn).2.5.MeasuringthespreadofadistributionWemeasurehowwelladistributionpisspreadinthegraphusingafunctionp[k]de nedforallintegersk2[0;2m].Thisfunctionisdeterminedbysettingp[k]=pSpj;forthosevaluesofkwherek=vol(Spj),andtheremainingvaluesaresetbyde ningp[k]tobepiecewiselinearbetweenthesepoints.Inotherwords,foranyintegerk2[0;2m],ifjistheuniquevertexsuchthatvol(Spj)kvol(Spj+1),thenp[k]=pSpj+k�vol(Spj) d(vj)p(vj+1):Thisimpliesthatp[k]isanincreasingfunctionofk,andaconcavefunctionofk.Itisnothardtoseethatp[k]isanupperboundontheamountofprobabilityfromponanysetwithvolumek;foranysetS,wehavep(S)p[vol(S)]:4 whichensuresthatpisanapproximatePageRankvectorforpr( ;v)afteranysequenceofpushoperations.Wenowformallyde nepushu,whichperformsthispushoperationonthedistributionspandratachosenvertexu. pushu(p;r):1.Letp0=pandr0=r,exceptforthefollowingchanges:(a)p0(u)=p(u)+ r(u).(b)r0(u)=(1� )r(u)=2.(c)Foreachvsuchthat(u;v)2E:r0(v)=r(v)+(1� )r(u)=(2d(u)).2.Return(p0;r0). Lemma1.Letp0andr0betheresultoftheoperationpushuonpandr.Thenp0+pr( ;r0)=p+pr( ;r):TheproofofLemma1canbefoundintheAppendix.Duringeachpush,someprobabilityismovedfromrtop,whereitremains,andaftersucientlymanypushesrcanbemadesmall.Wecanboundthenumberofpushesrequiredbythefollowingalgorithm. ApproximatePageRank(v; ;):1.Letp=~0,andr=v.2.Whilemaxu2Vr(u) d(u):(a)Chooseanyvertexuwherer(u) d(u).(b)Applypushuatvertexu,updatingpandr.3.Returnp,whichsatis esp=apr( ;v;r)withmaxu2Vr(u) d(u). Lemma2.LetTbethetotalnumberofpushoperationsperformedbyApproximatePageRank,andletdibethedegreeofthevertexuusedintheithpush.ThenTXi=1di1  :Proof.Theamountofprobabilityonthevertexpushedattimeiisatleastdi,thereforejrj1decreasesbyatleast diduringtheithpush.Sincejrj1=1initially,wehave PTi=1di1,andtheresultfollows. ToimplementApproximatePageRank,wedeterminewhichvertextopushateachstepbymain-tainingaqueuecontainingthoseverticesuwithr(u)=d(u).Ateachstep,pushoperationsareperformedonthe rstvertexinthequeueuntilr(u)=d(u)forthatvertex,whichisthenremovedfromthequeue.Ifapushoperationraisesthevalueofr(x)=d(x)aboveforsomevertexx,thatvertexisaddedtothebackofthequeue.Thiscontinuesuntilthequeueisempty,atwhichpointeveryvertexhasr(u)=d(u).WewillshowthatthisalgorithmhasthepropertiespromisedinTheorem1.TheproofiscontainedintheAppendix.6 Whenalazywalkstepisappliedtothedistributionp,theamountofprobabilitythatmovesfromutovis1 2p(u;v).ForanysetSofvertices,wehavethesetofdirectededgesintoS,andthesetofdirectededgesoutofS,de nedbyin(S)=f(u;v)2Eju2Sg;andout(S)=f(u;v)2Ejv2Sg;respectively.Lemma4.Foranydistributionp,andanysetSofvertices,pW(S)1 2(p(in(S)[out(S))+p(in(S)\out(S))):TheproofofLemma4canbefoundintheAppendix.WenowcombinethisresultwiththeinequalityfromLemma3torelateapr( ;s;r)toitself.Incontrast,theproofofLovaszandSimonovits[9,10]relatesthewalkdistributionsp(t)andp(t+1),wherep(t+1)=p(t)W,andp(0)=s.Lemma5.Ifp=apr( ;s;r)isanapproximatePageRankvector,thenforanysetSofvertices,p(S) s(S)+(1� )1 2(p(in(S)[out(S))+p(in(S)\out(S))):Furthermore,foreachj2[1;n�1],phvol(Spj)i shvol(Spj)i+(1� )1 2phvol(Spj)�j@(Spj)ji+phvol(Spj)+j@(Spj)ji:TheproofofLemma5isincludedintheAppendix.ThefollowinglemmausestheresultfromLemma5toplaceanupperboundonapr( ;s;r)[k].Moreprecisely,itshowsthatifacertainupperboundonapr( ;s;r)[k]�k 2mdoesnothold,thenoneofthesweepsetsfromapr( ;s;r)hasbothsmallconductanceandasigni cantamountofprobabilityfromapr( ;s;r).ThislowerboundonprobabilitywillbeusedinSection6tocontrolthevolumeoftheresultingsweepset.Theorem3.Letp=apr( ;s;r)beanapproximatePageRankvectorwithjsj11.Letand beanyconstantsin[0;1].Eitherthefollowingboundholdsforanyintegertandanyk2[0;2m]:p[k]�k 2m + t+p min(k;2m�k)1�2 8t;orelsethereexistsasweepcutSpjwiththefollowingproperties:1.(Spj);2.pSpj�vol(Spj) 2m� + t+q min(vol(Spj);2m�vol(Spj))1�2 8t;forsomeintegert,3.j2[1;jSupp(p)j].TheproofcanbefoundintheAppendix.WecanrephrasethesequenceofboundsfromTheorem3toprovethetheorempromisedatthebeginningofthissection.Namely,weshowthatifthereexistsasetofvertices,ofanysize,thatcontainsaconstantamountmoreprobabilityfromapr( ;s;r)thanfromthestationarydistribution,thenthesweepoverapr( ;s;r) ndsacutwithconductanceroughlyp lnm.WeremarkthatthisappliestoanyapproximatePageRankvector,regardlessofthesizeoftheresidualvector:theresidualvectoronlyneedstobesmalltoensurethatapr( ;s;r)islargeenoughthatthetheoremapplies.Theproofisgivenintheappendix.8 6AnalgorithmfornearlylineartimegraphpartitioningInthissection,weextendourlocalpartitioningtechniquesto ndasetwithsmallconductance,whileprovidingmorecontroloverthevolumeofthesetproduced.TheresultisanalgorithmcalledPageRank-Nibblethattakesascalebaspartofitsinput,runsintimeproportionalto2b,andonlyproducesacutwhenit ndsasetwithconductanceandvolumeroughly2b.WeprovethatPageRank-Nibble ndsasetwiththesepropertiesforatleastonevalueofb2[1;dlogme],providedthatvisagoodstartingvertexforasetofconductanceatmostg(),whereg()= (2=log2m). PageRank-Nibble(v;;b):Input:avertexv,aconstant2(0;1],andanintegerb2[1;B],whereB=dlog2me.1.Let =2 225ln(100p m).2.ComputeanapproximatePageRankvectorp=apr( ;v;r)withresidualvectorrsatisfyingmaxu2Vr(u) d(u)2�b1 48B.3.CheckeachsetSpjwithj2[1;jSupp(p)j],toseeifitobeysthefollowingconditions:Conductance:(Spj),Volume:2b�1vol(Spj)2 3vol(G),ProbabilityChange:p2b�p2b�1�1 48B,4.IfsomesetSpjsatis esalloftheseconditions,returnSpj.Otherwise,returnnothing. Theorem7.PageRank-Nibble(v;;b)canbeimplementedwithrunningtimeO(2blog3m 2).Theorem8.LetCbeasetsatisfying(C)2=(22500log2100m)andvol(C)1 2vol(G),andletvbeavertexinC for =2=(225ln(100p m)).Then,thereissomeintegerb2[1;dlog2me]forwhichPageRank-Nibble(v;;b)returnsasetS.AnysetSreturnedbyPageRank-Nibble(v;;b)hasthefollowingproperties:1.(S),2.2b�1vol(S)2 3vol(G),3.vol(S\C)�2b�2:TheproofsofTheorems7and8areincludedintheAppendix.PageRank-NibbleimprovesboththerunningtimeandapproximationratiooftheNibblealgo-rithmofSpielmanandTeng,whichrunsintimeO(2blog4m=5),andrequires(C)=O(3=log2m).PageRank-NibblecanbeusedinterchangeablywithNibbleinseveralimportantapplications.Forexample,bothPageRank-NibbleandNibblecanbeappliedrecursivelytoproducecutswithnearlyoptimalbalance.AnalgorithmPageRank-PartitionwiththefollowingpropertiescanbecreatedinessentiallythesamewayasthealgorithmPartitionin[15],soweomitthedetails.Theorem9.ThealgorithmPageRank-Partitiontakesasinputaparameter,andhasex-pectedrunningtimeO(mlog(1=p)log4m=3).IfthereexistsasetCwithvol(C)1 2vol(G)and(C)2=(1845000log2m),thenwithprobabilityatleast1�p,PageRank-PartitionproducesasetSsatisfying(S)and1 2vol(C)vol(S)5 6vol(G).10 7AppendixTodemonstratetheequivalenceoflazyandstandardPageRankvectors,letrpr( ;s)bethestandardPageRankvector,de nedtobetheuniquesolutionpoftheequationp= s+(1� )pM,whereMistherandomwalktransitionmatrixM=D�1A.Weprovethefollowingproposition.Proposition3.pr( ;s)=rpr(2 1+ ;s).Proof.Wehavethefollowingsequenceofequations.pr( ;s)= s+(1� )pr( ;s)Wpr( ;s)= s+(1� 2)pr( ;s)+(1� 2)pr( ;s)(D�1A)(1+ 2)pr( ;s)= s+(1� 2)pr( ;s)(D�1A)pr( ;s)=(2 1+ )s+(1� 1+ )pr( ;s)mSincepr( ;s)satis estheequationforrpr(2 1+ ;s),andsincethisequationhasauniquesolution,theresultfollows. ProofofProposition1.Theequationp= s+(1� )pWisequivalentto s=p[I�(1� )W].Thematrix(I�(1� )W)isnonsingular,sinceitisstrictlydiagonallydominant,sothisequationhasauniquesolutionp. ProofofProposition2.Thesuminequation(2)thatde nesR isconvergentfor 2(0;1],andthefollowingcomputationshowsthatsR obeysthesteadystateequationforpr( ;s). s+(1� )sR W= s+(1� )s 1Xt=0(1� )tWt!W= s+s 1Xt=1(1� )tWt!=s 1Xt=0(1� )tWt!=sR :SincethesolutiontothesteadystateequationisuniquebyProposition1,itfollowsthatpr( ;s)=sR . ProofofLemma1.Afterthepushoperation,wehavep0=p+ r(u)u:r0=r�r(u)u+(1� )r(u)uW:Usingequation(5),p+pr( ;r)=p+pr( ;r�r(u)u)+pr( ;r(u)u)=p+pr( ;r�r(u)u)+[ r(u)u+(1� )pr( ;r(u)uW)]=[p+ r(u)u]+pr( ;[r�r(u)u+(1� )r(u)uW])=p0+pr( ;r0):12 Thisprovesthe rstpartofthelemma.Toprovethesecondpart,recallthatphvol(Spj)i=p(Spj)foranyintegerj2[0;n].Also,foranysetofdirectededgesA,wehavetheboundp(A)p[jAj].Therefore,phvol(Spj)i=p(Spj) s(Spj)+(1� )1 2pin(Spj)[out(Spj)+pin(Spj)\out(Spj) shvol(Spj)i+(1� )1 2ph in(Spj)[out(Spj) i+ph in(Spj)\out(Spj) i:Allthatremainsistoboundthesizesofthesetsintheinequalityabove.Noticethat in(Spj)[out(Spj)j+jin(Spj)\out(Spj) =2vol(Spj);and in(Spj)[out(Spj) � in(Spj)\out(Spj) =2j@(Spj)j:Thisimpliesthat in(Spj)[out(Spj) =vol(Spj)+j@(Spj)j;and in(Spj)\out(Spj) =vol(Spj)�j@(Spj)j:Theresultfollows. ProofofTheorem3.Letkj=vol(Spj),let kj=min(kj;2m�kj),andletft(k)= + t+p min(k;2m�k)1�2 8t:Assumingthattheredoesnotexistasweepcutwithallofthepropertiesstatedinthetheorem,wewillprovebyinductionthatthefollowingholdsforallt0:p[k]�k 2mft(k),foranyk2[0;2m]:(7)Forthebasecase,equation(7)holdsfort=0,withanychoiceof and.Toseethis,noticethatforeachintegerk2[1;2m�1],p[k]�k 2m1p min(k;2m�k)f0(k):Fork=0andk=2mwehavep[k]�k 2m0f0(k).Theclaimfollowsbecausef0isconcave,p[k]islessthanf0foreachintegervalueofk,andp[k]islinearbetweentheseintegervalues.Assumeforthesakeofinductionthatequation(7)holdsfort.Toprovethatequation(7)holdsfort+1,whichwillcompletetheproofofthetheorem,itsucestoshowthatthefollowingequationholdsforeachj2[1;jSupp(p)j]:p[kj]�kj 2mft+1(kj):(8)14 ProofofTheorem2.Let=(apr( ;s;r)).Theorem3impliesapr( ;s;r)(S)�vol(S) vol(G) t+p min(vol(S);2m�vol(S))1�2 8t;foranyintegert0andanyk2[0;2m].Ifwelett=d8 2ln2p m e,thenwehaveapr( ;s;r)(S)�vol(S) vol(G) d8 2ln2p m e+ 2;whichimplies 2 d8 2ln2p m e 9 2lnm:Theresultfollowsbysolvingfor. ProofofLemma6.We rstprovethefollowingmonotonicitypropertyforthePageRankoper-ator:foranystartingdistributions,andanyk2[0;2m],pr( ;s)[k]s[k]:(9)ThisisaconsequenceofLemma5;ifweletp=pr( ;s),thenforeachj2[1;n�1]wehavephvol(Spj)i shvol(Spj)i+(1� )1 2phvol(Spj)�j@(Spj)ji+phvol(Spj)+j@(Spj)ji shvol(Spj)i+(1� )phvol(Spj)i;wherethelastlinefollowsfromtheconcavityofp[k].Thisimpliesthatpr( ;s)[kj]s[kj],wherekj=vol(Spr( ;s)j),foreachj2[1;n�1].Theresultfollows,sinces[k]isconcave,andpr( ;s)[k]islinearbetweenthepointswherek=kj.TheamountofprobabilitythatmovesfromCtoCinthestepfrompr( ; C)topr( ; C)Wisboundedby1 2pr( ; C)[j@(C)j],sincej@(C)jisthenumberofdirectededgesfromCtoC.Bythemonotonicityproperty,pr( ; C)[j@(C)j] C[j@(C)j]=j@(C)j vol(C)=(C):UsingtherecursivepropertyofPageRank,[pr( ; C)]�C=[ C+(1� )pr( ; C)W]�C(1� )[pr( ; C)]�C+1 2pr( ; C)[j@(C)j](1� )[pr( ; C)]�C+1 2(C):Thisimplies[pr( ; C)](C)(C) 2 : 16 ProofofTheorem7.AnapproximatePageRankvectorp=apr( ;v;r),withresidualvectorrsatisfyingmaxu2Vr(u) d(u)2�b 48B,canbecomputedintimeO(2blogm )usingApproximatePageRank.ByTheorem1,wehavevol(Supp(p))=O(2blogm ).Itispossibletocheckeachoftheconditionsinstep4,foreverysetSpjwithj2[1;jSupp(p)j],intimeO(vol(Supp(apr( ;v;r)))logn)=O(2blog2m ):Therefore,therunningtimeofPageRank-NibbleisO(2blog2m )=O(2blog3m 2): ProofofTheorem8.ConsiderthePageRankvectorpr( ;v).SincevisinC ,andsince(C) 1 96B,wehavepr( ;v)[vol(C)]�vol(C) 2m(1�(C) )�1 21 2�1 96:Wehaveset sothat t1=25whent=d8 2ln(100p m)e,andwiththischoiceoftwehave t+p min(vol(C);2m�vol(C))1�2 8t1 25+1 100:Since1 2�1 96�5 12+1 25+1 100,thefollowingequationholdswith =5 12.pr( ;v)[vol(C)]�vol(C) 2m� + t+p min(vol(C);2m�vol(C))1�2 8t:(10)LetB=dlog2me.Foreachintegerbin[1;B],let b= (9 10+1 10b B).Considerthesmallestvalueofbin[1;B]forwhichthefollowingequationholdsforsomek2b.pr( ;v)[k]�k 2m� b+ t+p min(k;2m�k)1�2 8t;forsomeintegert0:(11)Equation(10)showsthatthisequationholdswithb=Bandk=m.Letb0bethesmallestvalueofbforwhichthisequationholds,andletk0besomevaluesuchthatk0mandsuchthatthisequationholdswithb=b0andk=k0.Noticethatsb0�1k0sb0,becauseifequation(11)holdsforb=b0andk=k0,italsoholdsforb=b0�1andk0.WhenPageRank-Nibbleisrunwithb=b0,theapproximatePageRankvectorapr( ;v;r)com-putedbyPageRank-Nibblehasonlyasmallamountoferroronasetofvolumek0:theerrorissmall18 Toprovethatthereisasigni cantdi erencebetweenapr( ;v;r)2b0andapr( ;v;r)2b0�1,observethatequation(12)doesnotholdwithb=b0�1andk=2b0�1.Therefore,foreveryintegert0,apr( ;v;r)h2b0�1i�k0 2m b0�1+ t+q min(2b0�1;2m�2b0�1)1�2 8t:(13)Wealsoknowthatforsomeintegert,apr( ;v;r)[k0]�k0 2m�( b0�1+1 48B)+ t+p min(k0;2m�k0)1�2 8t:(14)Since2b0�1k0m,wehavep min(sb0�1;2m�sb0�1)p min(k0;2m�k0).Takinganintegertthatmakesequation(14)true,andpluggingthisvalueoftintoequations(14)and(13),yieldsthefollowinginequality.apr( ;v;r)h2b0i�apr( ;v;r)h2b0�1iapr( ;v;r)[k0]�apr( ;v;r)h2b0�1i�1 48B:WehaveshownthatSjmeetsalltherequirementsofPageRank-Nibble,whichprovesthatthealgorithmoutputssomecutwhenrunwithb=b0.Wenowprovealowerboundonvol(S\C),whichholdsforanycutSoutputbyPageRank-Nibble,withanyvalueofb.Letp0[k]=p[k]�p[k�1].Sincep0[k]isadecreasingfunctionofk,p0h2b�1ip2b�p2b�1 2b�2b�1�1 2(b�1)48B:Itisnothardtoseethatcombiningthislowerboundonp02b�1withtheupperboundp�C(C) givesthefollowingboundonthevolumeoftheintersection.vol(Sj\C)2b�1�p�C p0[2b�1]�2b�1�2b�1(48B(C) ):Sincewehaveassumedthat(C) 1 96B,wehavevol(S\C)�2b�1�2b�2=2b�2: 20

By: trish-goza
Views: 232
Type: Public

Local Graph Partitioning using PageRank Vectors Reid Andersen University of Cali - Description


In this paper we present an algorithm for local graph partitioning using personalized PageRank vectors We develop an improved algorithm for computing approximate PageRank vectors and derive a mixing result for PageRank vectors similar to that for ra ID: 4187 Download Pdf

Related Documents