232K - views

Local Graph Partitioning using PageRank Vectors Reid Andersen University of Cali

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

Tags : this paper
Embed :
Pdf Download Link

Download Pdf - The PPT/PDF document "Local Graph Partitioning using PageRank ..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.

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