Acyclic Preference Systems in P2P Networks

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Acyclic Preference Systems in P2P Networks

Anh-Tuan Gai, Dmitry Lebedev, Fabien Mathieu, Fabien de Montgolfier, Julien Reynier, Laurent Viennot

To cite this version:

Anh-Tuan Gai, Dmitry Lebedev, Fabien Mathieu, Fabien de Montgolfier, Julien Reynier, et al..

Acyclic Preference Systems in P2P Networks. [Research Report] RR-6174, INRIA. 2007, pp.9. �inria- 00143790v2�

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inria-00143790, version 2 - 2 May 2007

a p p o r t

d e r e c h e r c h e

ISSN0249-6399ISRNINRIA/RR--6174--FR+ENG

Thème COM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Acyclic preference systems in P2P networks

Anh-Tuan Gai — Dmitry Lebedev — Fabien Mathieu — Fabien de Montgolfier — Julien Reynier — Laurent Viennot

N° 6174

April 2007

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Unité de recherche INRIA Rocquencourt

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

Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30

Anh-Tuan Gai

∗

, DmitryLebedev

†

, Fabien Mathieu

†

, Fabien de Montgoler

‡

, Julien

Reynier

§

, Laurent Viennot

∗

ThèmeCOM Systèmesommuniants

ProjetsGyroweb

Rapportdereherhe n°6174April20079pages

Abstrat: Inthis work westudy preferenesystems naturalfor thePeer-to-Peer paradigm. Mostof them

fall in threeategories: global,symmetriand omplementary. Allthese systemsshareanayliityproperty.

As aonsequene,theyadmitastable(orPareto eient)onguration,where nopartiipantanollaborate

withbetterpartnersthantheirurrentones.

We analyze the representation of the suh preferene systems and show that any ayli system an be

represented with a symmetri mark matrix. This gives a method to merge ayli preferene systems and

retain theayliity. Wealsoonsider suh propertiesof theorrespondingollaborationgraph,aslustering

oeientanddiameter. Inpartiular,studyingtheexampleofpreferenesbasedonreallatenymeasurements,

weobservethatitsstableongurationisasmall-worldgraph.

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

network,mathings, graphtheory

CRCMARDI

∗

INRIA,domainedeVolueaux,78153LeChesnayedex,Frane

†

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

‡

LIAFA,175,rueduChevaleret,75013ParisFrane

§

MotorolaLabs,ParlesAlgorithmes,SaintAubin,91193GifsurYvetteCedex,Frane

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Résumé : Cet artileest onsaré àdes systèmes de préférenesnaturels dans lesréseaux pair-à-pair. La

plupartdeessystèmesappartiennentàl'unedestroislassesdepréférenessuivantes: globale,symétriqueet

omplémentaire. Cespréférenesontlapartiularitéd'êtreayliques. Enonséquene,ilspossèdentuneunique

ongurationstable(ausensdePareto),oùauunpartiipantnepeutollaboreravedemeilleurspartenaires

quesespartenairesourants.

En analysant leur représentation, nous montrons que tout système de préférenes ayliques peut être

représentéparunematriedenotessymétriques. Nousobtenonsainsiuneméthodepourfusionnerdessystèmes

de préférenes ayliquesen onservant lapropriété d'ayliité. Nous étudions également des propriétés du

graphe de ollaborationde la ongurationstable, omme le diamètre ou leoeient de lustering. Sur un

exampleréeldepréférenesbaséessurdesmesuresdelatenes,nousobservonsquelaongurationstableades

aratéristiquespetit-monde.

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

ouplages,graphes

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1 Introdution

Motivation Inmosturrentpeer-to-peer (P2P)solutionspartiipantsareenouraged toooperatewith eah

other. Sine ollaborationsmay be ostly in terms of network resoures(onnetion establishment, resoure

onsumption, maintenane), the number of onnetions is often bounded by the protool. This onstraint

enourages thelientsto makeaareful hoie amongothers to obtainagood performanefrom thesystem.

The possibility to hoose abetter partner implies that there exists a preferene system, whih desribes the

interestsofeahpeer.

The study of suh preferene systems is the subjet of b^-mathing ^theory. ^It ^has ^started ^forty-ve ^years

ago with the seminal work of Gale and Shapley on stable marriages [5℄. Although the original paperhad a

ertain rereational mathematis avor, the model turned out to be espeially valuable both in theory and

pratie. Today, b^-mathing's appliations are notlimitedto datingagenies, but inlude ollege admissions, roommates attributions, assignment of graduating medial students to their rst hospital appointments, or

kidney exhanges programs [5, 6,7, 14℄. The goalof the present paper is to expand b^-mathing ^appliation

domainto P2Pnetworksbyusingittomodeltheinterationsbetweenthelientsofsuhnetworks.

Previouswork In[10℄weoveredgeneralaspetsoftheb^-mathing^theory^appliation^to^the^dynamis^of^the

nodeinterations. Weonsideredpreferene systemsnaturalfortheP2P paradigm,andshowedthat mostof

them fall into three ategories: global, symmetri,and omplementary. Wedemonstrated that these systems

share the sameproperty: ayliity. Weproved existeneand uniqueness ofastable ongurationfor ayli

preferenesystems.

Contribution Inthisartile,weanalyzethelinksbetweenpropertiesofloalmarksandtheprefereneliststhat

are generatedwith thosemarks. Weshowthat allaylisystemsanbe reatedwith symmetrimarks. We

provideamethodtomergeanytwoaylipreferenesystemsandretaintheayliproperty. Andnallyour

simulationsshowthat reallatenymarksreateollaborationgraphswithsmall-worldsproperties,in ontrast

withrandomsymmetriorglobalmarks.

Roadmap InSetion 2wedenethe global,symmetri, omplementary, andaylipreferene systems,and

provide a formal desription of our model. In Setion 3 we demonstrate that all ayli preferenes an be

representedusingsymmetripreferenes. WeonsideromplementarypreferenesinSetion4,andtheresults

areextended toanylinearombinationofglobalorsymmetrisystems. Setion 5disussesthepropertiesofa

stablesolutionprovidinganexamplebasedonMeridianprojetmeasurements[13℄. InSetion6wedisussthe

impatofourresults,and Setion7onludes.

2 Denition and appliations of P2P preferene systems

2.1 Denitions and general modeling assumptions

Weformalizehereab^-mathing^model^for^ommon^P2P^preferene^systems.

Aeptane graph Peers may have a partial knowledge of the network and are not neessarily aware of all

other partiipatingnodes. Peers may also wantto avoid ollaborationwith ertain others. Suh riteriaare

represented byan aeptane graphG(V, E)^. ^Neighbors ôf â ^peer p∈ V âre ^the^nodes ^that ^mayôllaborate

withp^. ÂôngurationCîsâ^subsetC˜⊂E ôf^theêxistingollaborationsatagiventime.

Marks We assumepeers use somerealmarks (likelateny, bandwidth,...) to ranktheirneighbors. This is

representedbyavaluedmatrixofmarksm={m(i, j)}^. Â^peer pûsesm(p, i)ândm(p, j)^toômpareiândj^.

Withoutlossofgenerality,weassumethat0îs^the^best^markândm(p, i)< m(p, j)îfândônlyîfp^prefersi^toj^.

Ifpîs^notâ^neighborôfq^,^thenm(p, q) =∞^. ^Weâssume^forônvenieneâ^peerp^hasâ^dierent^mark^forêahôf

itsneighbors.Itimpliesthatapeeranalwaysomparetwoneighborsanddeidewhihonesuitsbettertohim.

Preferenesystem AmarkmatrixM ^reatesânînstaneLôfâ^preferene^system. L(p)îsâ^preferene^list^that

indiateshowapeerp^ranks^its^neighbors. ^The^relation^whenp^prefersq1^toq2^is^denoted^byL(p, q1)< L(p, q2)^.

Note thatdierentmarkmatriesan produethesamepreferenesystem.

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Global preferenes A preferenesystemisglobalifitanbededuedfromglobal marks(m(i, p) =m(j, p) = m(p)^).

Symmetri preferenes A preferenes system is symmetri if it an be dedued from symmetri marks

(m(i, j) =m(j, i)^for^alli, j^).

Complementary preferenes A preferenes systemis omplementary ifit anbededued from marks of the

form m(i, j) =v(j)−c(i, j)^,^wherev(j)^values^the^resoures^possessed^byjândc(i, j)^the^resoures^thatiând j ^haveⁱⁿômmon¹^.

Ayli preferenes A preferenessystemis ayliifitontainsnopreferene yle. Apreferene yleis a

yleofat leastthreepeers suhthat eaupeer stritlyprefersitssuessorto itspredeessoralongtheyle.

Quotas Eahpeerp^hasâ^quotab(p)^(possiblyînnite)ôn^the^numberôf^linksîtân^support. Âb^-mathing îs

aongurationC^that^respets^the^quotas. Îf^the^quotasâre^greater^than^the^numberôf^possibleollaborations, thensimplyC=E ^would^beânôptimal^solution^forâll.

Blokingpairs Weassumethatthenodesaimtoimprovetheirsituation,i.e. tolinktomostpreferredneighbors.

A pairofneighborspândq îsâ^bloking ^pair ôfâôngurationC îf{p, q} ∈E\C ând^both^prefer^to ^hange

theongurationC ând^to^link^with^theêahôther. ^Weâssume^that ^systemêvolves^by^disrete^steps. Âtâny

step twonodes anbelinkedtogether if and only iftheyform ablokingpair. Those nodes may droptheir

worstperforminglinks tostaywithintheirquotas. AongurationC îs^stable îf^no^bloking^pairsêxist.

Loving pair Peersp, q^formâ^loving^pairîfp^prefersq^toâllîtsôther^neighborsândq^,ⁱⁿîts^turn,^prefersp^to

allotherneighbors.It impliesastronglink whihannotbedestroyedinthegivenpreferenesystem.

2.2 Preferene systems and appliation design

DependingontheP2Pappliation,severalimportantriterionanbeusedbyanodetohooseitsollaborators.

Weintroduethefollowingthree typesasrepresentativeofmostsituations:

Proximity: distanesinthephysialnetwork,inavirtualspaeorsimilaritiesaordingtosomeharateristis,

Capaity related: networkbandwidth,omputingapaity,storageapaity,

Distintion: omplementaryharaterofresouresownedbydierentpeers.

Notiethatthesestypesorrespondrespetivelytothedenitionsofsymmetri,globalandomplementary

preferenesystemategories.

Examples of symmetri preferenes are P2P appliations whih optimize latenies. A lassial approah

for distributed hash-table lists of ontatsis seletingthe ontats with the smallest round trip time (RTT)

in thephysial network. InPastry [15℄, a node will alwaysprefer ontats withsmallest RTT among all the

ontatsthat ant into agivenroutingtable entry. Moregenerally,building alowlateny overlaynetwork

with bounded degreerequires toselet neighbors withsmall RTTs. Optimizinglatenies betweenplayersan

alsoberuial,forinstane,foronlinereal-timegamingappliations[11℄. Suhpreferenesaresymmetrisine

themarkapeerp^gives^to^some^peerqîs^the^sameâs^the^markq^gives^top^(the ^RTT^betweenpândq^).

Similarly,massivelymultiplayeronlinegames(MMOG)requireonnetingplayerswithnearbyoordinates

inavirtualspae[9,8℄. Againthisanbemodeledbysymmetripreferenesbasedonthedistaneinthevirtual

spae. Someauthorsalsoproposeto onnetpartiipantsofalesharingsystemaordingtothesimilarityof

theirinterests[3,16℄, whihis alsoasymmetrirelation.

BitTorrent[2℄isan exampleof aP2Pappliationthat usesaapaityrelated preferenesystem. Inbrief,

a BitTorrent peer uploads to peers it has most downloaded from during the last ten seonds. This is an

implementation of the well known Tit-for-Tat strategy. Themark of a peer an thus be seen hasits upload

apaitydividedbyitsollaborationquota.

ThisglobalpreferenenatureofBitTorrentshouldbetemperedbythefatthatonlypeerswithomplemen-

tarypartsoftheleareseleted. Pushingforwardthisrequirementwouldleadtoanotherseletionriterionfor

BitTorrent: prefereneforthepeerspossessingthemostomplementarysetoflepiees. Inotherwords,eah

1

Ofourse,inthisasethepreferredneighborhasalargermark.

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peershouldtrytoexhangewithpeerspossessingalargenumberofbloksitneeds. Weallthisaomplemen-

tarypreferenesystem. Note,thatthiskindofprefereneshangesontinuouslyasnewpieesaredownloaded.

However,thepeerswiththemostomplementarysetofbloksarethose,whoenablelongestexhangesessions.

In its more generalform, the seletion of partners for ooperative le downloadan be seen as a mix of

severalglobal,symmetri,andomplementarypreferenesystems.

3 Ayli preferenes equivalene

In [10℄, we showed that global, symmetri and omplementary preferenes are ayli, and that any ayli

b^-mathing^prefereneînstane^hasâûnique^stableonguration. Andaylisystemsalwaysonvergetoward theirstableonguration. However,sineayliityisnotdened byonstrution,oneanaskifotherkindof

aylipreferenesexist. Thissetionisdevotedtoanswerthisquestion.

Theorem1. LetP ^beâ^set ôfn ^pêers,A^be^the ^setôf âll^possible âyli^preferene înstanesônP^,S ^be^the

set of all possible symmetri preferene instanes on P^,G ^be^the ^setôf âll ^pôssible ^global ^preferene înstanes,

then

G$A=S

Therestofthissetiononsistsoftheproofoftheorem1: wewillrstshowS ⊂ A^andG ⊂ A^,^thenA ⊂ S^,

whihwillbefollowedbyG 6=A^.

Lemma1. Global andsymmetri preferenesystems areayli

Proof. from [10 ℄ Let usassumetheontrary, andassumethat there isairular listof peers p1, . . . , pk ^(with

k ≥3^),^suh ^that êah ^peerôf ^the^list ^stritly ^prefers îts^suessor^to îtspredeessor. Written in theform of marksitmeansthatm(pⁱ, pi+1)< m(pⁱ, pi−1)^forâlli^modulok^. ^Takingâ^sum^forâll^possiblei^,^we^get

k

X

i=1

m(pⁱ, pi+1)<

k

X

i=1

m(pⁱ, pⁱ₋1)^.

Ifmarksareglobal,thisanberewritten

P^k

i=1m(pⁱ)<P^k

i=1m(pⁱ)^,ândîf^theyâre^symmetri,P^k

i=1m(pⁱ, pi+1)<

P^k

i=1m(pⁱ, pⁱ+1)^. ^Both^areimpossible,thusglobalandsymmetrimarksreateayliinstanes.

Thenextpart,A ⊂ S^,ûses^the^loving^pairs^desribedⁱⁿ ^2.1. ^We^rst^prove^theêxisteneôf^loving^pairsⁱⁿ

Lemma 2.

Lemma2. A nontrivialayliprefereneinstanealways admits atleast onelovingpair.

Proof. Aformalproofwaspresentedin[10℄. Inshort,ifthere isnolovingpair,oneanonstrutapreferene

ylebyonsideringasequeneofrsthoiesofpeers.

Algorithm1: Construtionof asymmetrinotematrixm ^givenânâyli^preferenes înstaneL ônn

peers

N := 0

forallpândq^,m(p, q) = +∞^(by^default, ^peers^do^notâeptêahôther)

whilethereexistsaloving pair {a, b}^do m(a, b) :=m(b, a) :=N

Removea^from^the^preferene^listL(b)^andb ^fromL(a) N :=N+ 1

Lemma 3. Let L ^be â ^preferene înstane. Âlgorithm ¹ ônstruts â ^symmetri ^mark ^matrix ⁱⁿ O(n²) ^time

that produesL^.

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Proof. Thematrixoutput islearly symmetri. Neighboringpeers getnite marks,whileothers haveinnite

marks. If aninstane ontainsalovingpair {a, b} ^then m(a, b) = m(b, a)^an^be^the^best ^mark^sinea^and b

mutuallyprefertoanyotherpeers. AordingtoLemma 2suhalovingpairalwaysexistsinayliase. By

removingthepeersaândb^from^their^preferenes^lists,^weâre^lead^toâ^smallerâyliînstane^with^the^same

preferene listsexept aândb âre^nowûnaeptable^to êah ôther. ^The^proessôntinuesûntilâll^preferene

lists are eventually empty. The marks are given in inreasing order, therefore when m(p, q) ^and m(p, r) ^are

nite, m(p, q)< m(p, r)ⁱ^the^loving^pair{p, q} ^is^formed ^before ^the^loving^pair{p, r}ⁱp^prefersq ^tor^.

ThealgorithmrunsinO(n²)^time^beauseânîterationôf^the^while^loop^takesO(1)^time. Â^loving^pairân

espeiallybefound in onstanttime bymaintainingalist onalllovingpairs. Thelistis updatedin onstant

timesine,afteraândb^beame^mutuallyunaeptable,eahnewlovingpairontainseitheraândîts^new^rst

hoie,orb ^and^its^new^rst^hoie.

All ayli preferenes are not global preferenes. A simpleounter-exampleuses 4 ^peers p1^, p2^, p3 ^and p4

withthefollowingpreferenelists:

L(p1) : p2, p3, p4 L(p2) : p1, p3, p4 L(p3) : p4, p1, p2 L(p4) : p3, p1, p2

Lîs âyli,^but p1 ^prefersp2 ^to p3 ^whereasp4 ^prefers p3 ^to p2^. p1 ândp4 ^ratep2 ândp3 ^dierently,^thus

theinstaneisnotglobal.

4 Complementary and Composite Preferene Systems

Complementary preferenes appear in systems where peers are equally interested in the resoures they do

not have yet. As said in Setion 2.1, omplementary preferenes an be dedued from marks of the form

m(p, q) =v(q)−c(p, q)⁽ⁱⁿ^thisâse,^marksôf^higher^valuesârepreferred).

Theexpressionofaomplementarymarkmatrixm^showsîtîsâ^linearômbinationôf^previously^disussed

global and symmetri markmatries: m =v−c^, ^where v ^denes â^global ^preferene ^system ând c ^denes â

symmetrisystem.

Theorem2showsthatomplementarymarks,andmoregenerallyanylinearombinationofglobalorsym-

metrimarks,produeaylipreferenes.

Theorem2. Letm1 ândm2 ^be^global ôr^symmetri ^marks. Âny^linearômbinationôfλm1+µm2 îsâyli.

Proof. The proofis pratiallythesameasforLemma 1. Letus suppose that thepreferenesystem indued

bym=λm1+µm2 ôntainsâ^preferene^ylep1, p2, . . . , p^k, p^k+1=p1^,^fork≥3^. ^Weâssume^wlog^that m1îs

global, m2 ^symmetriând^that^marksôf^higher^valuesâre^preferred^form^. ^Then m(pⁱ, pⁱ+1)> m(pⁱ, pⁱ₋1)^for

alli^modulok^. ^Tâking â^sumôverâll^possiblei^,^we^get

k

X

i=1

m(pⁱ, pi+1)>

k

X

i=1

m(pⁱ, pi−1) =

k

X

i=1

(λm1(pⁱ, pi−1) +µm2(pⁱ, pi−1))^,^but

k

X

i=1

(λm1(pⁱ, pⁱ₋1) +µm2(pⁱ, pⁱ₋1)) = λ

k

X

i=1

m1(pⁱ₋1) +µ

k

X

i=1

m2(pⁱ₋1, pⁱ)

= λ

k

X

i=1

m1(pi+1) +µ

k

X

i=1

m2(pⁱ, pi+1)

=

k

X

i=1

m(pi, pi+1)^.

This ontraditionprovestheTheorem.

Theorem 2 leads to the question whether any linearombination of ayli preferenes expressed by any

kindofmarksisalsoayli. Theexamplebellowillustratesthatingeneralitisnottrue:

M1=





0 3 1 2 0 1 3 1 0



 ^,M2 =





0 1 2 1 0 3 1 2 0



 ^,M1+M2=





0 4 3 3 0 4 4 3 0





The preferene instane induedby M1+M2 ^has ^the^yle 1,2,3^, ^while ^bothM1 ând M2 âre âyli^(both

produeglobalpreferenes).