The stable configuration in acyclic preference-based systems

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systems

Fabien Mathieu, Gheorghe Postelnicu, Julien Reynier

To cite this version:

Fabien Mathieu, Gheorghe Postelnicu, Julien Reynier. The stable configuration in acyclic preference-

based systems. [Research Report] RR-6628, INRIA. 2008, pp.26. �inria-00318621�

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a p p o r t

d e r e c h e r c h e

N0249-6399ISRNINRIA/RR--6628--FR+ENG

Thème NUM

The stable configuration in acyclic preference-based systems

Fabien Mathieu — Gheorghe Postelnicu — Julien Reynier

N° 6628

September 2008

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Unité de recherche INRIA Rocquencourt

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

Fabien Mathieu

∗

, Gheorghe Postelniu

†

, Julien Reynier

‡

ThèmeNUMSystèmesnumériques

ProjetsGANG

Rapportdereherhe n° 6628September200823pages

Abstrat: Aylipreferenesreentlyappearedasanelegantwayto modelmany distributedsystems. An

ayli instane admits a unique stable onguration, whih an reveal the performane of the system. In

this paper,wegivethestatistial propertiesof thestableongurationforthreelassesofaylipreferenes:

node-basedpreferenes,distane-basedpreferenes,andrandomaylisystems. Usingrandomoverlaygraphs,

we prove using mean-eld and uid-limit tehniques that these systems have an asymptotially ontinuous

independent rank distribution for a proper saling, and the analytial solution is ompared to simulations.

These results provide a theoretial ground for validating the performane of bandwidth-based or proximity-

basedunstruturedsystems.

Key-words: Ayliity,rankdistribution,uidlimit,mean-eld,small-worlds,PDE

SupportedbyCRCMARDIII

SupportedbyANRProjetShaman

∗

OrangeLabs

†

Google

‡

ReehAiMResearh

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Résumé : Les systèmes à préférenes ayliques sont réemment apparus omme une méthode élégante

de modélisation de ertainssystèmes ditribués de typepair-à-pair. Uneinstane ayliqueadmet une unique

onguration stable, auto-stabilisante, qui donne une bonne indiation du omportement du système. Dans

e rapport, nousdonnons ladistributionstatistique dela ongurationstable pourtrois typesdepréférenes

ayliques: lespréférenesglobales(baséessurunordretotaldesn÷uds), lespréférenesdedistane (leplus

proheestpréféré),etlespréférenesayliquesaléatoires. Sousl'hypothèsed'ungraphedeompatibilitéErdös-

Rényi,nousmontronsàl'aidedetehniquesdelimitesuideset dehampmoyenl'existened'unedistribution

limite ontinue. Lapertinenedesrésultatsest vériéeàl'aidedesimulations.

Mots-lés : Systèmesayliques,distribution,limite uid,hampmoyen,petit-mondes,EDP

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Contents

1 Introdution 4

1.1 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Model and notation 5 3 Ayli formulas 5 3.1 Generiformula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Mean-eldapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Node-basedpreferenes 6 4.1 Fluidlimit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.2 Constantdegreesaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.3 Convergenetheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.5 Exatresolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Ayli and distane-based preferenes 9 5.1 Completerankdistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.1.1 Fluidlimit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.1.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5.2 Distanedistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5.3 Aeptablerankdistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 b^-mathinggeneralization 13 6.1 MeanFieldformulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6.2 Fluidlimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6.3 Disussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6.3.1 Stratiationtrade-o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6.3.2 Small-Worldeet ingeometripreferenes . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7 Conlusion 16 A Proofof Theorem1 18 A.1 Uniformonvergene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

A.2 PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

A.3 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

B Exat resolutionofthe node-based stableonguration 21 B.1 Reursiveformula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

B.2 Uniformonvergene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

B.3 PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

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1 Introdution

Mathing problems with preferenes haveappliations in a variety of real-worldsituations, inluding dating

agenies, ollege admissions, roommate attributions, assignment of graduatingmedial students to theirrst

hospitalappointment,orkidneyexhangesprograms[8,9,10,19,20℄.

Reently,mathingproblemsalsoappearedasanelegantwaytomodelmanydistributedsystems,inluding

ad-hoandpeer-to-peernetworks[12,6,16,15,7℄. Indistributedsystems,thepreferenesgenerallyomefrom

diret measurements. Thosemeasurementsanbenode-related(CPU,upload/downloadbandwidths,storage,

battery, uptime), oredge-related (Round-Trip Time, physial/virtual distanes, link apaity, o-uptime). In

mostases,theresultingpreferenesareayli: thereannotexistayleofmorethantwonodessuhthateah

nodeprefersitssuessortoitspredeessor. Asaonsequene,therealwaysexistsauniquestableonguration,

whihisself-stabilizing[6,1℄. Thismakesthingsmuheasierthaninothermathing problems,wherending,

ountingand omparingthestable ongurationsaresomeofthemainissues[8,21,17,20℄.

Modeling distributedsystemswith aylipreferenes allowsus to predittheeetiveollaborationsthat

will our,whih, inturn, allowsusto infertheperformaneof agivensystem. Foronveniene, thestudyof

anaylidistributedsystemisoftensplitintotwomain problems:

Howfastisthestabilizationproess? Beausedistributedsystemsareoftenhighlydynami,withonstant

hurnandpreferenealteration,thespeedofonvergeneanbeusedtodeterminehowfartheeetive

ongurationsarefromthetime-evolvingstableonguration.

Whatarethepropertiesofthe stableonguration? Ifthestabilizationproessisfastenough,theeetive

and stable ongurationswill be lose. Analyzing the latteran then givevaluable information on the

former.

Inapreviouswork,Mathieuinvestigatedtherstquestion[16,15℄. Heprovedthat eveniftheonvergene

an be prohibitive under an adversary sheduler, it is fast for realisti senarios. The seond question has

been answeredfor spei aylipreferenes: for real-world lateny-based preferenes, the stable ongura-

tion shows, for b^-mathing^(several ^mates ^per ^nodes ^allowed), small-worldproperties (low diameter and high lusteringoeient)[6℄;fornode-relatedpreferenes,thestableongurationtendstopairnodeswithsimilar

values[7℄: thisisthestratiationeet,whihallows,forinstane,tounderstandupload/downloadorrelations

in inentivenetworkslikeBitTorrent[4℄.

1.1 Contribution

Thestudiesproposedin[7℄and[6℄gaveonlypartial,mostlyempirial,answersaboutthelinkdistributioninthe

stableonguration,andproposedsomeonjetures. Thegoalofthispaperistoompleteandgivetheoretial

proofsontheshapeofthestableonguration.

Weextendtheseminalresultsthatweregivenin[7℄fornode-basedpreferenes: forb= 1^(simple^mathing

ase),weprovetheexisteneofalimitontinuousdistribution andsolvetheorrespondingPartialDierential

Equation(PDE).Thenweapplyasimilarmethodfordistane-basedandrandom-aylipreferenes,andalso

givetheexpliitsolutionoftheorrespondingPDE.

Lastly,weextendtheresultsforb >1^(multiple ^mathings). În^thatâse,^thereîs^no^simpleêxpression^that

givestheexatsolutionofthePDEs system,but disreteequationsareusedtoobserveasymptotialbehavior

ofthedistribution. Fornode-basedpreferenes,theexponentialbehaviorvalidatesthestratiationeet (the

probabilitytobemathedwithadistantpeer dereasesexponentiallywiththedistane),while thepowerlaw

obtainedforthetwootherasesindiatesthatthesmallworldeetobservedin[6℄forlatenyisinfatommon

to alldistane-basedpreferenes 1

.

1.2 Roadmap

InSetion 2wedenethe modeland notationforpreferene-basedsystems. Setion 3givesthegenerimean

eld methodusedin thispapertosolvethesimplemathingase. Theaseofnode-basedpreferenesissolved

in Setion 4,then the resultsare adapted to thedistane-based andrandom-ayli preferenesin Setion 5.

Setion 6 extend the formulas to multiple mathings, and asymptotial properties of the distributions are

desribed. Lastly,Setion7onludes.

1

Lateniesannotbeonsideredasrealdistanes,mainlybeausethetriangularinequalityisnotalwaysveried. However,they

formaninframetri,whihisnotoofarfromarealmetri[13℄.

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2 Model and notation

A preferene-based systemisaset V ôfN ^nodes, ^whose^possibleinterations aredesribedbyan aeptane graphG^,â^mark^matrixmândâ^quota^vetorb^.

Thequotavetorb ^limits^theollaborations: apeeri^annot^have^moreb(i)simultaneous mates.

The aeptane graphG = (V, E) îs ân ûndireted, non-reexivegraph. It desribes allowed mathings:

a node i ând â^node j ân ^be ^mated ^(we ^say ^that i îs âeptable ^for j^, ând ^vie ^versa) îf, ând ônly îf ⁽ⁱ⁾ {i, j} ∈E^. ^Forînstane,ⁱⁿ peer-to-peernetworks,anodeannotbediretlyonnetedtoallotherpeersofthe system, beauseof salability, andpeers that are notdiretly onneted annot be mated. Inthis paper,we

onsider Erdös-RényigraphsG(N, p)^(eah^possible^edge^exists ^withprobabilitypindependentlyoftheothers;

henetheexpeteddegreeis d=p(N−1)^).

Themarkmatrixmîsûsed^to ônstrut^the^peers'preferenes: giventwonodesj ând kâeptable^fori^,i

ranksj ^better^thankⁱm_i,j < m_i,k ^(the^sign^isarbitrary). Thefollowingmarksareonsidered inthispaper:

Node-based m(., i)îsônstant^(nodes^haveîntrinsi^values). ^These ^preferenesâre^suited^to^modeling^peer-

relatedperformane,likeaess bandwidth,storage,CPU,uptime...

Geometri the nodes are assoiated to N ^points ^piked ûniformly ât ^random ôn â n-dimensional torus (n≥1^). ^The^marksâre ^the^distanes ^between^those ^points. ^These ^preferenes âllowâ^theoretial ânalysisôf

proximity-basedperformane.

Meridian latenies we onsidered random subsets of N ^nodes ^taken ^from ^the 2500 ^nodes ^dataset ^of ^the

Meridian Projet [18℄. The marks are the (symmetri) latenies between those nodes. We do not perform

analysis forthosemarks,but usethemin6.3.2forvalidating thegeometriapproah.

Randomayli eahedgereeivesarandomuniformlydistributed value. Thenameisjustiedbeauseall

aylipreferenesanbedesribedbymarksontheedges(whihisequivalenttoassumethatm^is^symmetri).

Hene uniformly distributed (symmetri) randommarksare aonvenientwayto perform auniformsampling

oftheaylipreferenes[6,1℄.

Alltheonsideredmarksareayli,andthereforea(G, m, b)^systemâdmitsâûnique^stable ônguration C ∈E^, ^whih îs self-stabilizing[12, 16℄. The neighbors of i ⁱⁿ C âre^the ^stable ^mates ôf i^, ând ^the ^notation i↔j îsûsed^toêxpress^that iândj âre^stable^mates.

Weassumeforsimpliitythatmîsômpleteând^not^limited^to^theêdgesôfG^. ^Forâllônsidered^preferenes

butrandomayli,theompletionisstraightforward.Forrandomaylipreferenes,weassumethatdummy

randomvaluesareassignedtonon-aeptableedges.

The preferenes are denoted like follows: ifj îs âeptable ^for i^, r_i(j) ^denotes ^the^rank ôf j ⁱⁿ i^'s ^list ⁽1

beingthe best). r_i îsâlled ^the âeptable ^rankingôf i^. Îf i ^has^more^than k âeptable^neighbors, r_i⁻¹(k)îs

thek^th^node ⁱⁿi^'sâeptable^ranking. ^Similarly,^forj 6=i^, Ri(j)^denotes^the^rankôfj ⁱⁿ ^theômplete ^graph

(the aeptabilityonditionisomitted). Ri îsâlled^theômplete ^rankingôfi ^FôrK < N^,R⁻¹_i (K)îs^theK^th

nodeini^'s^omplete^ranking.

AllstablematingprobabilitiesthataredisussedinthisartilearedesignedbyD^. ^Subsripts^and^arguments

areusedto preisethemeaningofD ^whenever^needed. ^For^instane:

DRi(K)îs^theprobabilitythati ^hasâ^stable^mate^withômplete^rankK^.

DN,d(i, j) îs ^theprobabilitythat i↔ j^, ^knowing ^there îsN ^nodes ând ^that ^theêxpeted ^degree ôf ^the

aeptanegraphisd^.

forc≤b(i,)^,Dri,c(k)^is^theprobabilitythat thec^th^stable^mate ^ofi^has^relative^rankk^.

...

The omplementary umulativedistributionfuntion (CCDF) ofD îs ^denoted S^, ând^the^saled ^versionôfD

and S^are ^denotedD^andS^.

3 Ayli formulas

We rst onsider the ase b = 1 ^(simple ^mathing) ^(the ^results ^will ^be ^extended ^to ^multiple ^mathings ⁱⁿ

Setion 6). Wegiveageneriformulathat desribestheompleterankofthemate C(i)^of^a^peeri^.

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3.1 Generi formula

Let DRi(K) ^be ^the probability that Ri(C(i)) = K ^(the probability that the mate of i^, ^if ^any, ^has ^rank K^).

The CCDF of D ^is SRi(K) := 1−PK−1

L=1^, ^whih ^is ^the probability that i^'s^mate ^has^a ^rank^greater ^than K

(Ri(C(i))≥K⁾ôr^has^no ^mate ^(short ^notation: Ri(C(i)) ≮K^). ^Fôllowing ^theâpproah ^proposed ⁱⁿ ^[7℄, ^we

rstgiveageneriexatformulathat desribesDRi^,^then^we^propose^a^simplied ^mean-eldapproximation.

InordertosolveDRi(K)^,ôneân ôbserve^thatiîs^mated^withîtsK^th^peerj=R⁻¹_i (K)î:

{i, j}îsânêdgeôf^theâeptane^graph;^this^happens^withprobabilitypâsGîs^supposed^to^beâG(N, p)

graph.

i^is^not^mated^with^a^node^better^thanj ⁽R_i(C(i))≮K^);

j ^is^not^mated^with^a^node^better^thani⁽R_j(C(j))≮R_j(i)^).

Thisleadsto thefollowingexatformula:

DRi(K) =pP(Ri(C(i))≮K)×

×P(Rj(C(j))≮Rj(i)|Ri(C(i))≮K)

=pSRi(K)P(Rj(C(j))≮Rj(i)|Ri(C(i))≮K)

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3.2 Mean-eld approximation

Solving (1) is diult to handle, mainly beause of possible orrelations between Rj(C(j)) ≮ Rj(i) ând R_i(C(i))≮K^. ^The^solutionîs^toâdoptâ^meanêld assumption:

Assumption 1 Theevents nodeiîs^not^withâ^node^better^thanj ând ^nodej îs^not^withâ^node^better^than i âreindependent.

Thisassumptionhasbeenproposedin[7℄tosolve(1)intheaseofnode-basedpreferenes. Itisreasonable

whenN îs^largeândpîs^small. ^Then⁽¹⁾ân^beapproximatedby

D_R_i(K) =pS_R_i(K)SRj(Rj(i))^. ⁽²⁾

Now,inthenexttwosetions,weproposeto solveEquation2forspeipreferenes.

4 Node-based preferenes

We assumeherethat the preferenesomes from markson nodes. This is equivalentto assume atotalorder

among thenodes. Thereforewedonotneed to expliitthemarkmatrix m^, ând^weânûseân ôrdered^node

labelinginstead. Wearbitraryhoose1, . . . , N âs^labels,1 ^been^the^best^(if 1îs ^ranked^rst^forâll^nodes^that

aept1^,^and^so^on.^.^.^).

Beausethenodes'labelexpresstheirompleteranks,weandiretlyonsiderD(i, j)^,^theprobabilitythat nodeiîs^mated^with^nodej^. ^Nodej ^has^rankj ^foriîfj < i^,ândj−1 îfj > i^,^beauseâ^rank^does^not^rank

itself. ThisgivestherelationbetweenD ^andDR^:

D(i, j) =







DRi(j)^ifj < i^,

0îfj=i^(matingîs^not^reexive), D_R_i(j−1)îfj > i^.

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UsingtheCCDFS(i, j) := 1−Pj−1

k=1D(i, k)^,^we^get^the^node-based^version^of^Equation^2:

D(i, j) =

0^ifi=j^,

pS(i, j)S(j, i)^otherwise. ⁽⁴⁾

Thisequation,whihwasoriginallyproposedin [7℄,whihalsoshowthatitgivesaverygoodapproximationof

empirialdistribution. It anbenumeriallysolvedbyusingadoubleiteration.