Stratification in P2P Networks - Application to BitTorrent

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Stratification in P2P Networks - Application to BitTorrent

Anh-Tuan Gai, Fabien Mathieu, Julien Reynier, Fabien de Montgolfier

To cite this version:

Anh-Tuan Gai, Fabien Mathieu, Julien Reynier, Fabien de Montgolfier. Stratification in P2P Networks - Application to BitTorrent. [Research Report] RR-6081, INRIA. 2006, pp.19. �inria-00121974v2�

inria-00121974, version 2 - 27 Dec 2006

a p p o r t

d e r e c h e r c h e

N0249-6399ISRNINRIA/RR--6081--FR+ENG

Thème COM

Stratification in P2P Networks Application to BitTorrent

Anh-Tuan Gai — Fabien Mathieu — Julien Reynier — Fabien de Montgolfier

N° 6081

December 2006

Unité de recherche INRIA Rocquencourt

Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)

Anh-Tuan Gai

∗

, Fabien Mathieu

†

, Julien Reynier

‡

, Fabien de Montgoler

§

ThèmeCOM Systèmesommuniants

ProjetsGyroweb

Rapportdereherhe n° 6081Deember200619pages

Abstrat: Weintrodueamodelfordeentralizednetworkswithollaboratingpeers. Themodelisbasedon

thestablemathingtheorywhihisappliedtosystemswithaglobalrankingutilityfuntion. Weonsider the

dynamis of peers searhing for eient ollaboratorsand weprovethat auniquestable solution exists. We

provethatthesystemonvergestowardsthestablesolutionandanalyzeitsspeedofonvergene.Wealsostudy

the stratiation properties of the model, bothwhen all ollaborationsare possible and for random possible

ollaborations. Wepresenttheorrespondinguidlimitonthehoieofollaboratorsintherandomase.

Asapratialexample,westudytheBitTorrentTit-for-Tatpoliy. Forthissystem,ourmodelprovidesan

interestinginsightonpeerdownloadratesandapossiblewaytooptimizepeerstrategy.

Key-words: P2P, stable marriagetheory, rationalhoietheory, ollaborativesystems, BitTorrent,overlay

network,mathings, graphtheory

CRCMARDI

∗

INRIA,domainedeVolueaux,78153LeChesnayedex,Frane

†

FraneTeleomR&D,3840,ruedugénéralLeler,92130IssylesMoulineaux,Frane

‡

LIENS,45rued'Ulm,75230ParisCedex05,Frane

§

LIAFA,175,rueduChevaleret,75013ParisFrane

Appliation à BitTorrent

Résumé : Cet artilevise àintroduire unnouveaumodèle d'analyse desréseauxdéentralisésbasés surdes

ollaborations entre pairs. Ce modèle repose sur la théorie des mariages stables appliquée à des systèmes

possédantune fontiond'utilitéglobale. Nousétudionsladynamiqueinduiteparlareherhepourhaundes

meilleurspartenairespossiblesetmontronsunthéorèmed'existeneuniité. Nousobservonsunerapidevitesse

de onvergeneet étudionslephénomènedestratiation danslasolutionstable, dansleasoùlegraphedes

ollaborationsréalisablesestompletetdanseluioùilestaléatoire. Pourleasaléatoire,nousprésentonsune

limite uidedelasolution.

Commeexemplepratique,nousétudionslapolitiquedonnantdonnant employéedanslelogiieldepartage

BitTorrent. Pouresystème,notremodèlefournitdesintuitionspertinentessurlesvitessesdetéléhargements

ainsiquesurlespossibilitésd'optimisationdesparamètres.

Mots-lés: pair-à-pair,mariagesstables,hoixrationnels,systèmesollaboratifs,BitTorrent,réseauxoverlay,

ouplages,graphes

1 Introdution

Motivation Collaboration-baseddistributed appliations aresuessfully applied to large salesystems. A

systemissaidtobeollaborativewhenpartiipatingpeersollaborateinordertoreahtheirowngoal(inluding

beingaltruisti). Apartfrom well-knownontentdistribution appliations[4, 5℄, ollaboratinganbeapplied

to numerousappliations suh asdistributedomputing, onlinegaming,orooperativebakup. The ommon

propertyof suh systems isthat partiipating peer exhange resoures. The underlying mehanism provided

byprotoolsforsuhappliationsonsists in seletingwhihpeersto ollaboratewithto maximizeone'speer

benet with regards to its personal interest. This mehanism generally uses a utility funtion taking loal

information as input. One an ask if this approah an provide desirable properties of ollaboration-based

ontentdistributionprotoolslikesalabilityandreliability.

Toahievethese properties, thefamous protool BitTorrent[4℄ implements aTit-for-Tat (TFT)exhange

poliy. More preisely, eah node knows a subsetof all other nodes of the system and ollaborates with the

bestonesfromitspointofview: ituploadstotheontatsithasmostdownloadedfrominthelast10seonds.

In other words,the utility of peer p ^{for node} q ^{is equalto the quantity of data peer} q ^{hasdownloadedfrom} p ^{(in the last} measurementperiod). The main interest in using the TFT poliy is the resultinginentive to ooperate. Thenatureoftheutilityfuntionthenleadstoalusteringproesswhihgatherpeerswithsimilar

uploadperformanestogether,alledstratiation.

Reently,muhresearhhasbeendevotedtothestudyofthephenomenon. Sofar,however,whileithasbeen

measuredandobservedbysimulations,ithasnotbeenformallyproved. Understandingstratiation isarst

steptowardsabetteromprehensionoftheimpatof theutilityfuntion onasystembehavior. Atheoretial

frameworktoanalyzeandomparedierentutilityfuntionisneeded: hoosingautilityfuntionthatbestsuits

agivenappliationisquitediult. Moreimportantly,itisnotlearwhethertheutilityfuntionsimplemented

lead to desirable properties. We introdue a generi framework that allows an instantiation of (known and

novel) utilityfuntions that model ollaboration. We further present athorough analysis of alass ofutility

funtions basedonglobalrankingagreements,suhasthatofBitTorrentTFTpoliy. Thisframeworkalsots

gossip-basedprotoolsusedbyapeertodisoveritsrank[8℄.

Contribution First,weproposeamodel basedonthestablemathing theory. Thismodel desribesdeen-

tralizednetworkswherepeersrankeahothersandtrytoollaboratewiththebest peersforthem.

Seond,wefousonsystemswithaglobalrankingutilityfuntion (eahpeerhasanintrinsivalue)in the

framework of stable mathing. We provethat suh asystem alwaysadmits a uniquestable solution towards

whih itonverges. Weverifythroughsimulationsthespeedof onvergene withoutandwith hurn (arrivals

ansdepartures).

Third,westudystratiation inatoymodel offullyonnetednetworkswhere everypeer anollaborate

with all other peers. If every peer tries to ollaborate with the same number of peers, we observe disjoint

lustering. Butwithavariablenumberofollaborationsperpeer,lusteringturns intostrongstratiation.

Fourth,wedesribestratiationinrandomgraphs. ForErdös-Rényigraphs,thedistributionofollaborat-

ingpeershasauidlimit. Thislimitingdistributionshowsthatstratiationisasalableresult.

Lastly, wepropose apratialappliation of ourresultsto the BitTorrentTFT poliy. Assumingontent

availabilityisnotabottlenekinaBitTorrentswarm,ourmodelleadstoaninterestingharaterizationofthe

downloadrateapeer anexpet asafuntion of itsuploadrate. This desriptionleadsto possiblestrategies

foroptimizingthedownloadforagivenuploadrate.

Roadmap InSetion2wedeneourmodel. Setion3presentsastudyontheproblemdynamis. Setion4

desribesstratiationinaompleteneighborhoodgraphandSetion5,inrandomgraphs. Setion6disusses

theappliationofourresultsto BitTorrentandSetion 7onludesthepaper.

2 Model

P2Pnetworksareformedbyestablishinganoverlaynetworkbetweenpeers. Apeeratsbothasaserveranda

lient. Eahpeerp^{hasaboundednumber}b(p)^ofollaborationslots. Asthenetworkevolves,peersontinuously searhafternew(orbetter)partners.Eah protoolhasitsownapproahtohandlingthese dynamihanges.

Forexample,aprotoollikeeDonkey[5, 1℄ optimizesindependently twopreferenelists ontheserverandon

the lientsides. More reent protools, like BitTorrent [4℄, make ause ofa gametheoreti approah, where

eahpeertriestoimproveitsownpayo. Itresultsinkeepingone preferenelistpernode.

Letus suppose that eahpeer p^{hasaglobal mark} S(p)^{, whih mayrepresentits available bandwidth,its}

omputational apaities, or its sharedstorage apaity. Eah peer wantsto ollaborate with best partners

who havehighest marksS(p)^{. This models manynetworks preferenes systems,albeitnot allnetworks have}

suhranking. Forinstane, inhessplaying,playershaveanintrinsivalue(ELOrating),although theydon't

generallywantto engagepeople farbetterorworsethanthem.

Somepeersmightnotbewilling toooperatewithsomeothers. Forinstane, peersthat haveno ommon

interest orareunaware ofeah other. We introdue anaeptanegraph torepresentompatibilities. A pair

(p, q)^{belongsto theaeptanegraphif,and onlyif(i)bothpeersare interestedin} ollaboration. Without lossofgenerality, wean supposeaeptabilityis asymmetrirelation: ifp^{is unaeptablefor}q^,q ^willnever

be able to ollaborate with p^{so we anassume} q ^{is also unaeptablefor} p^{. We denote by onguration or}

mathing thesubgraphoftheaeptanegraphthatrepresentstheeetiveollaborationbetweenpeers. The

degreeofapeerp^{inaongurationisboundedby}b(p)^.

A bloking pair foragivenongurationis aset of twopeers unmathed togetherwishing to bemathed

together(evenifitmeansdropping oneoftheirurrentollaborations). Aongurationwithoutblokingpair

issaidtobestable. Inastableonguration,asinglepeerannotimproveitssituation:itisaNashequilibrium.

If a number of olloborations is limitedto 1^{, the problem is known asthe stable roommates problem [7℄.}

It is an extension of the famous stable marriage problem introdued by Gale and Shapley in 1962 [6℄. If we

assumeeahpeerp^{wantstoollaboratewithupto}b(p)^{otherpeers,theframeworkisalled stable}b^-mathing

problem 1

[3℄.

As it holds for all theories of stable mathings, the existene of a stable onguration depends on the

preferenerulesusedtorankpartiipantandontheaeptanegraph. Inthisworkwestudytheimpatofthe

rulesderivedfromaglobalrankingonapeer-to-peernetworkbehavior. Inpartiular,wendthepropertiesof

thestableongurations.

3 Existene and onvergene properties of a stable onguration

Global rankingmathingisoneofthesimplestasesofmathingproblems. Tan[13℄hasshownthat existene

and uniqueness ofstable solutionswere relatedto prefereneylesin theutilityfuntion. A prefereneyle

of length k ^{is a set} i1, . . . , ik ^of k ^{distint peers suh that eah peer of the yle prefers its suessor to its}

predeessor. AsprovedbyTan,astableongurationexistsithereisnooddprefereneyleoflengthgreater

than 1^{. He also provedthat if noeven yle of lengthgreater than} 2 ^{exists, then the stable ongurationis}

unique. If peers havean intrinsivalue, nostrit preferenes yleanour(see belowfor ties), so aglobal

rankingmathingproblemhasoneandonlyonestable solution.

ThissolutionisveryeasytoomputeknowingtheglobalrankingS^,b^{andtheaeptanegraph. Theproess}

is given byAlgorithm 1: eah peer p^startswith b(p)^{available onnetions. First, thebest peer}p1 ^{piks the}

bestb(p1)^{peersfromitsaeptanelist. As}p1^{isthebest,thehosenpeersgladlyaept(realltheaeptane}

graph is symmetri) and the resulting ollaborations are stable (no bloking pairan unmath them). Note

thatifthereisnotenoughaeptablepeers,p1^{maynotsatisfyallitsonnetions. Peershosenby}p1^haveone

lessonnetion available. Then seond best peer p2 ^{doesthe same,andso on...Byimmediatereurrene,all}

onnetionsmadearestable. Whentheproessreahesthelastpeer,theonnetionsarethestableonguration

for the problem. As it was saidbefore, allonnetions are not neessarilysatised. Forinstane, if the last

peerstillhasavailableonnetionswhenitsturn omes,hisonnetionswillnotbefethed, asallpeersabove

him havebyonstrutionspentall theironnetions. Thisis, of ourse, aentralizedalgorithm,but weshall

see belowthatdeentralizedalgorithmsworkaswell.

Note on ties Ties in preferene lists make the mathing problems more diult to resolve [11℄ without

bringing moreinsightaboutthestratiationissuesstudiedin thispaper. Simulationshaveshownourresults

hold ifweallowties,but equationsarehardto proveasexisteneof astable mathingannot beguaranteed.

Thusforthesakeofsimpliity,weshallsuppose utilitiesaredistint, thatisS(q)6=S(p)^foranyp6=q^.

Convergene Oneanaskwhat isthepointinstudying astable ongurationin adynamialontextsuh

asP2Psystems,wherepeersarriveanddepartwhenevertheywish,andwhereutilityfuntionsandaeptane

listsanutuate. Wehavenotprovedyetthattheproessofpeerstryingindependentlytoollaboratetothe

bestpeerstheyknowanreahthestable state.

1

inthispaperthewordmathingstandsforb^{-mathing(unlessotherwisestated)}

Algorithm1: Stableongurationin globalranking

Data: An aeptanegraphG^withn^{peers,aglobalranking}S(p)^{,andmaximalnumberofonnetions} b(p)

Result: Theuniquestable ongurationoftheb^{-mathing problem}

Leta^beavetorinitializedwithb

foreahpeer i^{sortedininreasing} S(p)^{(bestpeer rst)do}

foreahpeer j ^{sortedin inreasing} S(p) ^{startingjustafter} i^do

if (i, j)∈G^anda(i)>0 ^anda(j)>0 ^then

onnet(i, j⁾ a(i) =a(i)−1 a(j) =a(j)−1

end

end

end

Weintroduetheoneptofinitiative to modeltheproessbywhih apeermayhange itsmates. Given

aongurationC^{, wesaythat peer}p^{takes the initiative when itproposes toother peersto beits newmate.}

Basially,p^mayproposepartnershipto anyaeptable peer. ButonlyblokingpairsofC ^{representaninter-}

esting newpartnership. Ifp^{anndsuh abloking mate,theinitiativeisalledative beauseitsueedsin}

modifyingtheonguration(bothpeerswillhangetheirsetofmates).

Tondablokingmate,p^{ontatspeersfromitsaeptanelist. Weidentify severalstrategiesdepending}

onhowp^{sansitsaeptanelist:}

best mate whenthepeerseletsthebest(if any)availableblokingmate. This happensifp^knowstherank

ofallitsaeptablepeersandwhether theywillollaborateornot,

deremental when thelist is irularly sannedstartingfrom the last asked peer. This happens ifp^knows

therankofallitsaeptablepeers, butnotiftheywillollaborate,

random whenasinglepeerisseletedatrandom. Thishappensifp^hasnoinformationonitsneighborsuntil itasks.

Ofourse, whenbest mate initiativeispossible,itseemstobethebest strategytomaximizeapeer'sown

prot,but itsupposesagoodknowledgeofthesystemismaintained.

We an now omplete our model with initiatives: starting from any initial onguration, an instane of

our model evolvesbeauseof initiativestaken bypeers. Infat,it anonlyevolvetowardsthe uniquestable

onguration,asshownbyTheorem 1.

Theorem1 The stable solution anbe reahed inB/2initiatives, whereB=P

pb(p)^{isthe maximalnumber}

of onnetions. Moreover, any sequeneof ative initiatives starting from any initial onguration eventually

reahes thestable onguration.

Proof: InAlgorithm1,eahonnetionanbe obtainedbyinitiative. As thestable ongurationpossesses

uptoB/2^{pairings,thisensurestherstpartofthetheorem. Weprovetheonvergenebyshowingasequene}

of ative initiatives an never produe twie the same onguration. There is a nite number of possible

ongurations,soifwekeepalteringtheongurationthroughinitiatives,weeventuallyreahaonguration

that annotbealteredwithanyinitiative: thestableonguration.

Theproofisindeedsimple. Ifasequeneofinitiativesinduesayleofatleasttwodistintongurations,

then oneanextrat aprefereneyleoflengthgreaterthan3^{: let}p1 ^{beapeerwhosemateshangethrough}

theyle. Callp2 ^thebestpeerp1 ^{isunstablypairedwithduringtheyle,and}p3 ^thebestpeerp2^isunstably

pairedwith duringtheyle. p1 ^isnotp3^and p2 ^prefersp3^to p1^{, otherwisethepair}{p1, p2} ^{wouldnotbreak}

duringtheyle. Iteratingtheproess,webuildasequeneofpeer(pk)^suhthatpk ^preferspk+1^topk−1^,until

wend i < j ^{suh that} pi =pj^{. Theirular list}(pi, pi+1, . . . , pj−1) ^{is apreferene yle. As global ranking}

doesnotallowprefereneyles,thisisnotpossible,soasequeneofativeinitiativesanneverproduetwie

thesameonguration.

Theorem1provesthat instati onditions(nojoin ordeparture, onstantutility funtion), aP2Psystem

will onverge to the stable state. Toprove this stable state is worthstudying, we haveto show onvergene

0 5 10 15 20 25 30 35 40 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Initiatives per peer

Disorder

n=100,d=50 n=1000,d=10 n=1000,d=50

Figure1: Starting fromC_∅^{, onvergenetowardsthestablestatefordierentparameters}

is fast in pratie (Algorithm1is optimal in numberof initiativesbut diult to implement in alargesale

system) andansustainaertainamountofhurn. As aompleteformal proofofthisis beyondthesopeof

this paper,weusedsimulations.

In oursimulations, peers were labeled from 1 ^to n ^{(the numberof peers). These labels dene the global}

ranking,1^{beingthebestpeerand}n^theworst(ifi < j^,peeri^{isbetterthanpeer}j^{). Weuse}Erdös-Renyiloopless symmetrigraphsG(n, d)^{asaeptanegraphs,where}d^{istheexpeteddegree(eahedgeexists}independently withprobability

d

n−1^{). Only}1^{-mathingwasonsidered.}

FormeasuringthedierenebetweentwoongurationsC1 ^andC2 ^{weusethedistane}

D(C1, C2) = Σⁿ_i=1kσ(C1, i)−σ(C2, i)k. 2 n(n+ 1)^,

where σ(C, i)^{denotesthemateof}iⁱⁿ C^{(byonvention,}σ(C, i) =n+ 1 ^ifi^isunmatedinC^).

D ^isnormalized: thedistane betweenaomplete mathingandtheemptyongurationC_∅ ^isequalto 1^.

Thedisorder denotesthedistanebetweentheurrentongurationandthestableonguration.

Ateahstepoftheproesswesimulate,apeerishosenatrandomandperformsabestmateinitiative(the

initiativeanbeativeornot). Toomparesimulationswith dierentnumbern ^{ofpeers, wetakeasequene}

ofn^suessiveinitiativesasabase unit(thatanbeseenasoneexpetedinitiativeper peer).

Arst setof simulationsismadeto provearapidonvergenewhenthe aeptanegraphis stati. Inall

simulations,the disorderquiklydereases, and the stable ongurationis reahedin lessthan ndinitiatives (that is d ^{base unit). Figure 1 shows onvergene starting from the empty onguration for three typial}

parameters: (n, d) = (100,50)^,(n, d) = (1000,10)^,(n, d) = (1000,50)^.

Thenweinvestigatetheimpatofanatomialterationofthesystem. Startingfromthestableonguration,

weremoveapeerfromthesystemandobservetheonvergenetowardsthenewstableonguration. Weobserve

bigvarianesinonvergenepatterns,butonvergenealwaystakeslessthand^{baseunitsanddisorderisalways}

small. Note,that duetoadominoeet,removingagoodpeergenerallyinduesmoredisorderthanremoving

a bad peer. This is shown by Figure 2. We ran the simulations 100 ^{times and seleted four} representative trajetories,aswedidnotwishtoaverageoutinterestingpatterns.

Finally, we investigate ontinuous hurn. A peer an be removed or introdued in the system anytime,

aordingto ahurn rate parameter. Simulations showthat asthehurn rateinreases,the systembeomes

unable to reah the instant stable onguration. However, the disorder is kept under ontrol. That means