JEGA: a joint estimation and gossip averaging algorithm for sensor network applications

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for sensor network applications

Nicolas Maréchal, Jean-Benoit Pierrot, Jean-Marie Gorce

To cite this version:

Nicolas Maréchal, Jean-Benoit Pierrot, Jean-Marie Gorce. JEGA: a joint estimation and gossip

av-eraging algorithm for sensor network applications. [Research Report] RR-6597, INRIA. 2008, pp.25.

�inria-00307512�

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Thème COM

JEGA: a joint estimation and gossip averaging

algorithm for sensor network applications

Nicolas Maréchal — Jean-Benoît Pierrot — Jean-Marie Gorce

N° 6597

Unité de recherche INRIA Rhône-Alpes

Ni olas Maré hal

∗

, Jean-BenoîtPierrot

†

,Jean-Marie Gor e

‡

ThèmeCOMSystèmes ommuni ants ProjetARES

Rapportdere her he n°659725juillet200825pages

Abstra t: Distributed onsensusalgorithmsarewidelyusedintheareaof sen-sornetworks.Usually,theyaredesignedtobeextremelylightweightatthepri e of omputationtime. Theyrelyonsimplelo alintera tionrulesbetween neigh-bor nodesandare oftenusedto performthe omputationof spatial statisti al parameters(average,varian e,regression). Inthis paper,we onsider the ase of aparameterestimation from input data streamsat ea h node. An average onsensusalgorithmisusedtoperformaspatialregularizationoftheparameter estimations. Atwosteppro edure ouldbeused: ea h noderstestimatesits ownparameter,andthenthenetworkappliesaspatialregularizationstep. Itis howevermu hmorepowerfultodesignajointestimation/regularizationpro ess. Previouswork hasbeen donefor solvingthis problem but under very restri -tivehypothesesin termsof ommuni ationsyn hroni ity,estimator hoi e and sampling rates. In thispaper,westudy amodiedgossipaveragingalgorithm whi h fullls the sensor networks requirements: simpli ity, lowmemory/CPU usage and asyn hroni ity. By the sameway, we provethat the intuitive idea ofmass onservationprin ipleforgossipaveragingis stableand asympoti ally veried under feedba k orre tions even in presen e of heavily orrupted and orrelatedmeasures.

Key-words: distributed algorithms, gossipalgorithms, epidemi algorithms, averaging,average onsensus,estimation, spa e-timediusion,sensornetworks

∗

CEA-LETIMINATEC,Grenoble,F-38054Fran e(email: ni olas.mare hal ea.fr)

†

CEA-LETIMINATEC,Grenoble,F-38054,Fran e(email: jean-benoit.pierrot ea.fr)

‡

Université de Lyon, INRIA, INSA-Lyon, CITI, F-69621, Fran e (email: jean-marie.gor einsa-lyon.fr)

apteurs

Résumé : Les algorithmes distribués de onsensus de moyenne (average onsensus)sont ourrament utilisés dansle domaine des réseauxde apteurs. Conçuspourêtreextrêmementlégersauprixd'untempsde onvergen ea ru, ilsreposentsurdesintera tionslo alesentre noeudsvoisins etsont prin ipale-ment utilisés pour le al ul de paramètres statistiques empiriques (moyenne, varian e,...). Dans etarti le,nousnousplaçons dansle adredel'estimation sur haquenoeud- apteur d'unparamètreàpartird'unotd'é hantillons. Un algorithme de onsensus de moyenne est utilisé an de régulariser spatialle-mentlesparamètresestimés. Ilseraitpossibled'utiliserunepro édureendeux étapes: haque noeud al ule tout d'abord une première estimation de son paramètre,puisleréseauuniformise esestimationsdansundeuxièmetemps. Il est ependantplusintéressantd'utiliseruns hémad'estimation/régularisation onjointes. Dans depré édents travaux, une solution aété proposée pour ré-soudre eproblèmemaissousdeshypothèsestroprestri tivesentermesde syn- hroniedes ommuni ations,de hoixd'estimateuretde aden ed'é hantillonage. Dans erapport,nousétudionsuneversionmodiéed'unalgorithmede moyen-nage pair-à-pair qui répond à la problématique pré édemment itée tout en respe tant lesspé i itésdes réseauxde apteurs: simpli ité, faible usagedes ressour es(pro esseur,mémoire,...) et asyn hronie.

Mots- lés: algorithmesdistribués,algorithmesgossip,algorithmesépidémiques, onsensusdemoyenne,estimation, diusionspatio-temporelle,réseauxde ap-teurs

1 Introdu tion

Sensornetworks onsistofagreatamountofsmallentities, allednodes,equipped with low ost hardware in order to balan e the totalnetwork ost. They are ommonly usedfor monitoringphysi al phenomena onwide areassu h as hy-drometry, landslides or res, but also for tra king purposes and in military warfare([1℄,[2℄,[3℄,[4℄,[5℄). Thedire t drawba ksof low ost hardware are nu-merous: severe energy onstraints (battery lifetime), poor CPU and storage abilities, lowtransmissionratesand small ommuni ationrange. Fa ingthese limitationsandobje tives,wirelesssensornetworks(WSN)havetoself-organize theirex hangesandthemaneersensornodesmusta hievetheirmission. Asin most ases, omputations in a entralized fashion be omeuntra table and/or inadaptated, robust distributed algorithms have to be designed. The biggest partof algorithmdesign forWSN isdedi ated to improvingtheperforman es whilepreservingenergy onsumption. Forexample,severaldatafusions hemes developped in order to providea good and ompa t representationof the ob-servedphenomenon an be madeonthebasisof ahigh numberoflowquality measures [6℄ andsimplelo alintera tionsbetweenneighbornodes(gossiping). The parti ular lass of distributed onsensus algorithms is of great interest: they provide arobust way of homogenizing parametersamong network nodes [7℄. Morespe i ally, average onsensus algorithmsseem to be agood hoi e wheneverthestabilityand thequalityofthe onsensuspointisa riti alissue [8℄,andextendtoawidepanelofdatafusiontoolssu hasestimatorsfor statis-ti almoments,linearregressionandpolynomialmaptting([6℄,[9℄). However, a problem o urs when data to be averaged are subje t to u tations. This is often the asewhen astatisti al parameteris estimated from time series of noisydata samples. As thenumberofavailable samplesin reases,the estima-tionpro essnaturallya tsasatemporalregularizations hemeandu tuations are redu ed: one would ask to adjust the urrent state of average onsensus in order to a ount for informations with higher pre ision. For an additive zero-meanstationarymeasurementnoisepro ess,thequalityofthisestimation in reaseswiththenumberofmeasuresand,asa orollary,withtime. Neverthe-less,gossip averagingalgorithmsare veryslowto onvergein omparison with entralized algorithms: gossip-based onsensusalgorithms onverge asymptoti- ally,rarelyinnitetime. Assensornetworkssuerfromheavy onstraintson their resour es,time andenergy must be saved, it thus be omesne essaryto runthe gossip averagingalgorithm whilethe estimation is still in progressby using orre tion me hanisms. This double pro ess should be understood asa spatio-temporalregularizations heme: ea hnodeperformsindividuallyalo al regularization of extra ted features from sampled data (estimation) , while a spatialregularization(averaging)isperformedinordertoextra taglobal har-a teristi . Previousworkhasbeendoneonthistopi ,andanalternativeversion of thealgorithm introdu ed in [9℄is des ribedin this arti le. As explainedin thispaper,theoriginalityofourwork onsistsmainlyinthefullasyn hroni ity that is assumedfor bothdata ex hanges and estimation pro esses,and in the wide rangeof estimators overedby thehypotheses. Moreover,our algorithm an nd many appli ation ontexts: as an example, a lo k syn hronization s heme forwireless sensormakesuse ofit[10℄. Thispaperisorganizedas fol-lows: se tion 2providesashort overviewofgossip-based onsensusalgorithms , theirprin iplesand someknownresults. Inse tion3,asolutionisproposed,

addressingtheproblem oftheparallel estimationandaveragingwhile respe t-ingthephilosophyandparadigmofsensornetworks. Furtherthe onvergen eis proved. Afterthesetheoreti al onsiderations,simulationresultsareprovidedin se tion4inordertogiveaqualitativestudyofitsbehaviourw.r.t. parameters s ale. Inadditionto itsapparantmeaning,this work provesthatprin iples of mass onservation arerespe ted and stablethroughour algorithm,evenwhen measurementsarehighly orruptedand orrelated.

2 Distributed onsensus algorithms

2.1 Prin iples of gossip-based onsensus algorithms

Distributed onsensusalgorithms/proto ols aimat agreeing all network nodes with a ommon value or ade ision in a de entralized fashion. From a signal pro essing ontext,this an beunderstood asaspatial regularizationpro ess. When data ex hanges onsist of lo al, asyn hronous and simple intera tions betweenneighbornodes, su halgorithms referedto asgossip-based. The par-ti ularsub lassof gossip-basedaverage onsensusalgorithmsdoesnotlimitto the omputation of averages, but extends to the extra tion of a large variety of aggregated and statisti al quantities like sums/produ ts, max/min values, varian es, quantiles and rank statisti s ([11℄ and [12℄). More dire t appli a-tionslikeleast-squares regressionof modelparametershavealsobeenadapted to thisalgorithms ([6℄,[9℄). All these spe i ities makegossip-based onsensus algorithmsgood andidatesforsensornetworksappli ations,wherebandwidth, energy onsumptionandCPU/memoryusageareenduringseverelimitationsfor thesakeof nodes lifetime and size. Despite their suboptimality

1

, pure gossip onsensusalgorithm anbeusedasapreludetomoresophisti atedalgorithms byhomogenizingparametersupongroupsofnodes(forexample,theredu tion of arrierosetfor redu ing timedrift of TDMA s hemes). The performan e analysis of su h algorithms relies essentially on diusion speed statisti s and is then loselyrelated to performan es of ooding/multi asting pro essesand mixing time of Markov hains: someasymptoti bounds on onvergen e time aregivenin[12℄and[11℄.

2.2 Gossip averaging

Inpra ti alappli ations, onsensusalgorithmsareoftenusedinordertoeasily homogenize some parameters. However, one should distinguish situations in whi h the agreedvalue is riti al. Forexample, agreeing on a meeting point is not as riti al as dete ting the position of a sniper. Obtaining an average is in general mu h slower than any uniformization algorithm based on ood-ing/broad astingte hniques,but ensuresagoodqualitityof onsensus.

Gossip-based average onsensus algorithms (gossip averaging) have been widelystudiedin literature([13℄,[14℄,[11℄,...). As suggestedbytheir denomina-tion,theyaimat omputingaglobal averageof lo alvalues. This paperfo us onaversionbasedonanasyn hronouspeer-to-peer ommuni ationsmodel as presented in [6℄. Su h a model frequently o urs in sensor networks appli a-tionswherefullsyn hroni ityisneitherguaranteednoreasilytra table. Under

1

thisassumption,anintera tion onsistsin hoosingapairofneighbornodesat ea hiterationandmakingalo alaveragingbetweenthem. A gossipaveraging algorithmisthendes ribedbyalineardieren eequationoftheform:

X

k+1

= W

k

X

k

(1)

where

X

k

isthe ve torof nodes' valuesat iteration

k

(

X

0

ontainsthe initial valuestobeaveraged)and

W

k

isa

n

-by-

n

matrixwhi hdes ribesinstantaneous pairwise intera tions. In fa t,

X

k

anbe seenasa stateve tor, andits om-ponents as estimates of the global average of intial values. Following Boyd's notations,

W

k

ouldbewritten:

W

k

= I −

(e

i

− e

j

)(e

i

− e

j

)

T

2

∆

= W

ij

(2) where

e

i

= [0 . . . 0 1 0 . . . 0]

T

is a

n

-dimensional ve torwith

i

th

entryequal to

1

(here, double index

ij

meansthat node

i

onta ts node

j

). This learly orrespondstothefollowingrule(timeintobra kets,node'sIDasindex):

x

i

(k + 1) = x

j

(k + 1) =

x

i

(k) + x

j

(k)

2

(3)

For onvenien e, the

W

k

are onsideredtobei.i.d. randommatri es. Ea h

W

ij

is hosenwithprobability

p

ij

,i.e. a ordingto theprobabilitythat node

i

initiates aniterationinvolvingnode

j

.

W = E [W

k

] = E [W

0

] =

1

n

X

i,j

p

ij

W

ij

(4)

where

n

standsforthenumberofnetworknodes. In[14℄, thefollowingstatementisproven:

Proposition1([14℄). If

W

isadoubly-sto hasti ergodi matrix, then

X

k

−−−−→

k→∞

¯

θ1

where

θ =

¯

1

n

P

n

i=1

x

i

(0)

,and

1 = [1, . . . 1]

T

∈ R

n

.

Thismeansthat onvergen etothetrueaveragevalueisensuredifandonly if the largesteigenvalue(in modulus) of

W

is equalto 1, with multipli ity

1

. Inotherwords,

W

isthetransitionmatrixofaMarkov hainwhoseunderlying (weighted) transitiongraph

G

is strongly onne ted, i.e. forea h pair

(i, j)

of verti esof

G

,apathfrom

i

to

j

andapathfrom

j

to

i

areexisting. In[15℄,the authorsprovedthatproposition1isequivalenttothefollowingproposition.

Proposition2([15℄). Let

G = (V, E)

be thegraphsu hthat: ˆ theset ofverti es

V

isthe setof networknodes.

ˆ thereis an edge between twoverti es only if the orresponding nodes are intera ting innitelymany times.

Then,

X

k

onvergesto

θ

¯

if

G

is onne ted.

Anotherwaytodene

E

withrespe ttotheset

E

k

ofa tivelinks(edges)at timeinstant

k

isthefollowing:

E =

∞

\

n=0

∞

[

i=n

E

k

= lim sup

n

E

n

2.3 Performan e analysis of noiseless gossip averaging

The onvergen e speed of thestandard gossip averagingalgorithm isstrongly relatedtothese ondlargesteigenvalue(inmodulus),

λ

2

of

W

(see[15℄). Given linkprobabilities,performan esarequantiedintermsof

ǫ

-averagingtime,i.e. theminimaltimethatguaranteesarelativeerroroforder

ǫ

withprobabilityat least

1 − ǫ

:

T

ave

(ǫ) = sup

X

0

∈R

n

inf

(

k : Pr

"

X

k

− ¯

θ1

2

kX

0

k

2

≥ ǫ

#

≤ ǫ

)

(5)

Boydandal. derivedupperandlowerboundsfor

T

ave

(ǫ)

basedonthese ond largesteigenvalueofthe orrespondingmatrix

W

:

0.5 log ǫ

−1

log λ

2

(W )

−1

≤ T

ave

(ǫ) ≤

3 log ǫ

−1

log λ

2

(W )

−1

(6)

Theeigenvalue

λ

2

isfun tion ofthelink probabilitiesbetweenpairofneighbor nodes. Thus,one antrytoredu e

λ

2

bya tingonneighborlinks,intwoways:

ˆ link reation/deletiona ordingtotopology-basedheuristi s.

ˆ neighborsele tionuniformlyandrandomly hosen,ornot: potential heuris-ti s.

Somepapersdes ribed solutionsforthe (distributed)optimization of

λ

2

for a given network using semidenite programming[16℄. The hoi e of algorithms withsu h omplexityisquestionableforsensornetwork appli ations: their op-erationshould remainreasonablyfeasibleonthe hosenar hite ture, and per-forman egainsmust beimportantenoughto ompensatefortime andenergy lostduringoptimization. However,thiss hemeisofgreatinterestina topology-stablenetwork.

2.4 Gossip averaging of noisy measures

Insomeappli ations,theparameterstobeaveragedareestimatedfromsampled streamsofdata. Re ently,atremendeous workhasbeenpublishedto enhan e performan es and robustness of onsensusalgorithms ([17℄,[18℄) and to dene proto ols for thems. However, one of the most important pra ti al problem is thepresen e of additivemeasurementnoise. When oupling noises orrupt ex hanges, Boyd et al provedthat the onsensuspoint deviatesfrom the true averagevalue,andtakestheformofarandomwalkover

R

n

: themeansquared normof lo al deviationsin reaselinearlywith time. In[19℄, asolutionis pro-posed to redu etherate ofdeviation, whilein [9℄theeortisdoneto provide simultaneousaveragingandestimation. Nevertheless,theworkdonein[9℄relies onex hangessyn hroni ityandonlyfo usonthe aseofleastmeansquare es-timation. Theproblemisthento ndasolutiontodesyn hronizingex hanges whilepreserving the onvergen ein aseof moregeneralestimates. Themain di ultyin thisstudy remainsthe onvergen erateof theestimationpro ess. Forexample,thevarian eoftheoptimalestimatorfortheaverageof normally distributed values onverges inversely proportionaly to the number of values. In[9℄,theproofofse ond-order onvergen eisbasedonthe onditionthatthe varian e

σ

t

hasa nite norm

l

2

_(N)

. In general, the varian eof an estimator doesnotfulll thisrequirement.

3 Joint estimation and gossip averaging

Similarlyto the initial algorithmpresentedin [9℄, we propose to run an asyn- hronousspatialregularization(averaging)pro essworkingjointlywiththe lo- alestimationoftheparameters(temporalregularization). Themaindieren e standsin thefullasyn hronoi ityofours heme,the hosenweightsfor intera -tions,andtheretroa tionpro ess. Moreover,the onvergen eofouralgorithm is proovedin the following fora widerange of lo al estimatorsunder the sin-gle assumptionthat theirspatio-temporal ovarian esde reaseto 0with time, withoutanyassumptionneitheron:

ˆ their onvergen eratew.r.t. tothenumberofsamples.

ˆ datasamplingrates.

3.1 Measurement pro ess and estimation

Duringthemeasurementpro ess,nodes olle tsamplesrelatedtosomedataof interest. The goal ofthe estimation phase is to extra t someunkown param-eters or hara teristi sof theoriginal datadistribution only from samples. In parti ular, an estimatorof someparameter

θ

issaid to be unbiased, ifat any time,themeanoftheestimatoris

θ

. Inthiswork,aproperestimationpro essis onsidered,i.e. onvergingtotheexpe tedparameterasthenumberofsamples grows(the varian etendstoward

0

withtime): this estimationisreferedtoas temporalregularization. Asanexample,oneshouldbeinterestedinestimating themean

µ

ofsomereal-valueddistribution

D

(ofnite varian e

σ

2

). It iswell knownthat, given i.i.d. samples

v

k

from thedistribution, the sampleaverage onvergestothetruemean

µ

,i.e.:

∀k ∈ N

∗

_{, v}

k

∼ D(µ, σ)

⇒



















E



1

n

P

n

k=1

v

k



= µ

V



1

n

P

n

k=1

v

k



=

σ

2

n

−−−−→

_n→∞

0

Cov



1

m

m

P

k=1

v

k

,

_n

1

n

P

l=1

v

l



=

σ

2

max(m,n)

−−−−−−−−−→

_{max(m,n)→∞}

0

(7)

Inrealexperiments, samplesmaybe orrelatedand theirdistribution may hangethroughtime. However,some onsistan iesinthemeasurementand esti-mationpro essesareassumed: measurements anbetakenfromatime-varying distributionbuttheparameterstoestimatemustremain onstantthroughtime. Forinstan e, interferen esin wireless ommuni ationsare subje t to the net-worka tivitydynami s, andareusually modelled asa entered randomnoise. In other words, their varian es (power) vary with time, but their means are onstantandequalto

0

.

3.2 Des ription of the algorithm

Letus startwithsomenotationsand onventions. Foranynode

i

, the urrent estimationoftheparameter

θ

i

givensamplesavailableatnode

i

uptotime

k

is

denotedby

θ

ˆ

k

i

. These estimatorsaregroupedinthe

n

-dimensionalve tor

Z

k

:

Z

k

=

h

ˆ

θ

₁

(k)

, . . . , ˆ

θ

(k)

n

i

T

(8)

As lo al estimators are assumed unbiased, the ve tor

E

[Z

k

]

is onstant throughtimeandisequaltotheve torofparameterstobeestimated.

¯

Z

= E [Z

∆

k

] = [θ

1

, . . . , θ

n

]

T

(9) Inthispaper,itisusefulto onsiderthedieren ebetween

Z

k

anditsmean, whi hisdenoted by

B

k

:

B

k

∆

= Z

k

− ¯

Z =

h

b

(k)

₁

, . . . , b

(k)

n

i

(10)

Wealsodenethe ovarian eterm

C

kl

ij

whi hmeasurestherelationbetween

omponentsof

Z

k

throughtime:

C

kl

ij

∆

= E



b

θ

i

(k)

− θ

i

 

b

θ

j

(l)

− θ

j



= E

h

b

(k)

i

b

(l)

j

i

(11)

Inthefollowingofthisarti le,theproofofthe onvergen ewillrelyonthe onlyassumptionthat these omponentsareasympoti allyun orrelated,i.e.

∀(i, j) ∈ [1, n]

2

, C

kl

ij

−−−−−−→

(k+l)→∞

0

(12)

Formostofsensornetworkappli ations,thisassumptionisquitenotrestri tive assu ientlyspa ed(temporallyand/orspatially)samplestendstobe un or-related too. Inmany ases, the Stolz-Cesarótheorem and its extensions help inndingsu ient onditionsonsample onvarian esforensuringassumption (12).

Theproposedalgorithmisbasedonthestandardgossipaveragingalgorithm (1),uponwhi hasimplefeedba kisaddedtoa ountforestimatedparameters updates. Thismodiedalgorithmtakestheformofanonhomogeneoussystem ofrst-orderlineardieren eequations,andisdened by:



X

k+1

= W

k

X

k

+ Z

k+1

− Z

k

Z

0

= X

0

= [0 . . . 0]

T

(13)

The proof of the e ien y of this algorithm relies on the onvergen e of every omponentofthestateve tor

X

k

tothespatialaverageoftheestimated parameters

(θ

i

)

i=1..n

whilethoseof

Z

k

should onvergetothelo alparameters:

X

k

−−−−→

k→∞

1

n

n

X

i=1

θ

i

!

1

Z

k

−−−−→

k→∞

¯

Z = [θ

1

, . . . , θ

n

]

T

Insystem(13),the(random)matrix

W

k

fulllsthesame onditionsasinthe lassi gossipavaragingalgorithms: it anbe onstantthroughtimeortakenat

random, but its meanmust be adoubly sto hasti 2

ergodi 3

matrix. The be-haviourofoursystemisveryeasytodes ribequalitatively. Thepara ontra ting matri es(see[20℄)

W

k

homogenizethevaluesof

X

k

whilethe omponentsof

Z

k

arestabilizing. Ifea h

z

i

(k)

is ergodi ,

Z

k+1

− Z

k

tendsto 0as

X

k

tendstoa ve tor olinearto

[1 . . . 1]

. Theterm

Z

k+1

− Z

k

impliesapermanent orre tion of thetotalweightsof

X

k

. One anthen on ludethat after aninnite time, thesumofthe(identi al) omponentsof

X

k

isequalto thesumofthoseof

Z

¯

. Inthenexttwoparts,aformalproofofthisanalysis isgiven.

3.3 First-order moment onvergen e

The rst result asserts that the mean limit of

X

k

is naturally the average of estimatedparameters,i.e.

θ =

¯

1

n

P

n

i=1

θ

i

: Proposition3.

lim

k→∞

E

_[X

_k

_{] = ¯}

_θ1

(14)

Proof. All along the proof, we takebenet of the re ursive form e ien y of updateequation(13):

E

_[X

_k+1

_{] = E [W}

_k

_X

_k

_{+ Z}

_k+1

_{− Z}

_k

_]

(15)

= E [W

k

] E [X

k

] + E [Z

k+1

] − E [Z

k

]

(16)

= W E [X

k

] + ¯

Z − ¯

Z

(17)

Thisrelationistrueex eptfor

k = 0

:

E

[X

1

] = E [Z

1

] = ¯

Z

. Itfollows:

E

_[X

_k+1

_{] = W}

k

E

_[Z

₁

_]

(18)

= W

k

_Z

_¯

(19)

Theergodi ityof

W

statesthat

W

k

onvergesto

11

T

n

as

k

grows. Together withrelation(19),ityeldstotheexpe tedresult:

E

_[X

_k

_{] −−−−→}

k→∞

11

T

n

Z = ¯

¯

θ1

(20)

3.4 Se ond-order moment onvergen e

Nowthat ouralgorithmis provedto onvergeonaveragetowardsthe onstant ve tor

θ1

¯

,thequalityofthis onvergen emustbeanalyzed. Inotherwords, an themean distan ebetween onsensusand trueaverage(statisti ally)bemade arbitrarily small ? A positiveansweris demonstrated below. Te hni al issues stand in the randomnature of matri es appearingin the algorithm, and also in thesimplefa tthat ovarian esarenotassumedto haveanynite

ℓ

p

norm (in anydire tion). Moreinterestingly, the se ondorder onvergen e is usefull tostatethe onvergen ein probabilityasexplainedin thenextse tion.

2

itsrowsand olumnssumto1 3

Proposition 4.

lim

k→∞

E

h

X

_k

_{− ¯}

_θ1

2

i

= 0

(21)

Proof. Theproofofproposition4ismorefastidiousthante hni al. Asthenext equationshows,thedeviationisdire tlyrelatedtothese ond-ordermomentof

X

k

:

E

h

X

_k

_{− ¯}

_θ1

2

i

_{= E}

h

_X

_k

_{− ¯}

_θ1

T

_X

_k

_{− ¯}

_θ1

i

(22)

Theright-handsideofequation(22) anbedeveloppedas:

E



_X

_k

T

_X

_k



_{− 2¯}

_θE



₁

T

_X

_k



_{+ E}

h

¯

_θ1

2

i

(23)

whi hgivestrivially:

E

h

X

_k

_{− ¯}

_θ1

2

i

_{= E}

h

_kX

_k

_k

2

i

_{− n¯}

_θ

2

(24)

Thus,thegoalisnowtoprovethat

E

h

kX

k

k

2

i

onvergesto

n¯

θ

2

as

k

in reases. Werewritethere ursivesystem(13)intoamoree ientway. Forthis,wedene

Ψ(k, i)

and

Φ(k, i)

in

M

n

(R)

4 by:

Ψ(k, i)

∆

=



I

if

i ≥ k

W

k−1

W

k−2

. . . W

i+1

W

i

otherwise (25)

Φ(k, i)

∆

= Ψ(k, i) − Ψ(k, i + 1)

(26) Byputting

Φ(k, i)

and

Ψ(k, i)

intosystem(13),oneobtainsamoree ient formulationfor

X

k

:







X

k

= Z

k

+

k−1

_P

i=1

Φ(k, i)Z

i

Z

0

= X

0

= [0 . . . 0]

T

(27)

Thisnotationhelpsproving onvergen eofthese ondordermomentof

X

k

where lassi al upper bounding (see [14℄) would fail. As one should expe t

from expression (27),

E

h

kX

k

k

2

i

ould be written with se ond-ordermoments

ofmeasures:

E

h

kX

k

k

2

i

= E

h

kZ

k

k

2

i

+ 2E

"

Z

T

k

k−1

X

i=1

Φ(k, i)Z

i

#

+ E





k−1

X

i=1

Φ(k, i)Z

i

2





(28)

Thelimitofthersttermiseasilyderivedas:

Proposition 5.

lim

k→∞

E

h

_kZ

_k

_k

2

i

₌

¯

_Z

2

(29)

4

M

Proof.

E

h

_kZ

_k

_k

2

i

₌

n

X

p=1

E



_θ

(k)

_p



2



(30)

−−−−→

k→∞

n

X

p=1

E

_[θ

_p

_]

2

₌

¯

_Z

2

(31)

Inordertomaketheproof learertothereader,thetwolasttermsof equa-tion(28) anbede omposedbyseparatingnoisyanddeterministi omponents:

E

"

Z

T

k

k−1

_X

i=1

Φ(k, i)Z

i

#

= E

"

¯

Z

T

k−1

_X

i=1

Φ(k, i) ¯

Z

#

+ E

"

B

T

k

k−1

_X

i=1

Φ(k, i)B

i

#

+ E

"

B

k

T

k−1

X

i=1

Φ(k, i) ¯

Z

#

+ E

"

¯

Z

T

k−1

_X

i=1

Φ(k, i)B

i

#

(32)

E





k−1

X

i=1

Φ(k, i)Z

i

2



 = E





k−1

X

i=1

Φ(k, i) ¯

Z

2



 + E





k−1

X

i=1

Φ(k, i)B

i

2





+ 2E





k−1

X

i=1

Φ(k, i) ¯

Z

!

T

_k−1

X

i=1

Φ(k, i)B

i

!



(33)

As noise ve tors

B

i

are entered, ross terms of equations (32) and (33) vanishesthroughlinear ombinations:

E

"

Z

k

T

k−1

X

i=1

Φ(k, i)Z

i

#

= E

"

¯

Z

T

k−1

X

i=1

Φ(k, i) ¯

Z

#

+ E

"

B

k

T

k−1

X

i=1

Φ(k, i)B

i

#

(34)

E





k−1

X

i=1

Φ(k, i)Z

i

2



 = E





k−1

X

i=1

Φ(k, i) ¯

Z

2



 + E





k−1

X

i=1

Φ(k, i)B

i

2





(35)

Thelimit of the four termsremaining in (34) and (35)are obtained sepa-rately: Proposition6.

lim

k→∞

E

"

¯

Z

T

k−1

X

i=1

Φ(k, i) ¯

Z

#

= n¯

θ

2

−

¯

Z

2

(36) Proposition7.

lim

k→∞

E





k−1

X

i=1

Φ(k, i) ¯

Z

2



 =

¯

Z

2

− n¯

θ

2

(37)

Proposition 8.

E

"

B

T

k

k−1

X

i=1

Φ(k, i)B

i

#

−−−−−→

k→+∞

0

Proposition 9.

lim

i→+∞

E





k−1

X

i=1

Φ(k, i)B

i

2



 = 0

Theproofofthese fourpropositionsisthefastidioustaskweannoun edfor provingproposition(4),andisprovidedinappendix. Propositions6and7area reformulationof lassi alresultsongossipaveragingalgorithms([16℄)adaptated totheterminologyofthisarti le,anddonot ontainsmajordi ulties. Onthe ontrary,theproofof propositions 8and 9ismorete hni al andrelies onthe fa t thattheperturbationsdue tothemeasurement/estimationnoisevanish if theassumptionmadeon ovarian esin (12)isvalid.

Now,thanks to propositions 5to 9the limitof

E

h

kX

k

k

2

i

when

k

goesto

+∞

an bederived:

E



_X

_k

T

_X

_k



_{−−−−→}

k→∞

¯

Z

+ 2(n¯θ

2

−

¯

Z

) + (

¯

Z

_{− n¯θ}

2

)

(38)

= n¯

θ

2

(39)

Equation(24)andrelation(38)letusnishtheproofofproposition 4:

lim

k→∞

E

h

X

_k

_{− ¯}

_θ1

2

i

_{= 0}

(40)

3.5 Convergen e in probability

Proposition4ensuresthe onvergen ein probabilityof

X

k

towardtheuniform ve tor

θ1

¯

. Thisistheprobabilisti ounterpartofthe onvergen eofsequen es onanormedve torspa e. Proposition10statesthattheprobabilityofhaving anerrorgreaterthan agiventhreshold anbemadearbitrarysmall, thetime indexbeinggreaterthanase ondthresholddependingontheerror.

Proposition10. Foranypositiverealnumbers

α

and

δ

,thereisaninteger

N

, su hthat:

∀k ≥ N, Pr



X

k

− ¯

θ1

≥ δ



≤ α

(41)

Proof. If

(Ω, B, P)

isaprobabiltyspa e,and

f

isameasurablereal-valued fun -tionon

Ω

,Markov'sinequalitystatesthat forany

t ∈ R

+

:

P

_{({ω ∈ Ω : |f(ω)| ≥ t}) ≤}

1

t

E

P

[|f|]

(42)

Inparti ular, for anyreal randomvariable

X

and

δ > 0

, this isequivalent to:

Applyinginequality(43)to

X

k

− ¯θ1

,oneobtains:

Pr



X

k

− ¯

θ1

≥ δ



= Pr

h

X

k

− ¯

θ1

2

_{≥ δ}

2

i

(44)

≤ δ

−2

E

h

X

_k

_{− ¯}

_θ1

2

i

(45)

Byproposition(4),on anndaninteger

N

su hthat

∀k ≥ N, E

h

X

k

− ¯θ1

2

i

_{≤ αǫ}

2

(46)

Together,equations (45)and (46) ensurethat

∀k ≥ N

, one anguarantee that:

Pr



X

k

− ¯

θ1

≥ δ



≤ α

(47)

Corollary 1.

∀α > 0

,

δ > 0

,

∃N ∈ N

, su h that

∀k ≥ N

, the two following inequalities hold: i)

Pr



max

1≤i≤n

|x

(k)

i

− ¯

θ| ≥ δ



≤ α

ii)

∀i ∈ [1, n], Pr

h

|x

(k)

i

− ¯

θ| ≥ δ

i

≤ α

Proof. Thisisa onsequen eofthe lassi alnorminequalitystatingthat

∀X ∈ R

n

, kXk

∞

≤ kXk

2

(48)

Convergen einprobabilityisthentriviallydedu ed fromthisinequality.

4 Simulation and analysis

No formal result is still available on the rate of onvergen e of JEGA. Some resultsmaybeexpli itlydedu edfromtheproof,butitseemsmoreinteresting to observe dire tly the behaviour of quadrati error through time under the inuen eofparameters(noise,meanparameters,...). Ourgoalistogetintuitive and empiri alknowledge ofglobal tenden ies. Forthispurpose, we onsidera wireless network modeled asa random unit dis graph

5

G

where

n

nodes are distributed unifomily on a square simulation plane of width

d

. For the sake of simpli ity, ea h node updates its lo al estimateat ea h iteration

k

(it gets onesample). However,only onepairof nodes will intera t. For thispurpose, an initiator

i

is hosen randomly among the set of verti es of

G

a ordingto uniform distribution,whilethedestinator

j

is hosenuniformilyrandomly too inthesetofneighborsof

i

. Wethen omputeanestimateoftheaveragevalueof aparameter. Measurementnoisearei.i.d. zero-meangaussianrandomvariables withstandarddeviation

σ

i

at node

i

.

5

4.1 Impa t of noise varian e

Inthis set of simulations, theentries of the steady stateve tor

Z

¯

were taken on eat random uniformilyin the interval

[0, 3]

6

and kept onstant duringthe simulations. Weanalysetheevolutionofthemeansquaredeviationunder sev-eralvaluesof

σ

intheset

{10, 1, 0.1, 0.01}

,takenidenti allyforallnodes,and alsofornoiselessmeasurements(

σ = 0

). Thislast ase orrespondtothe stan-dard gossip averagingpro ess. Werun the algorithm for

N

iter

iterationsand pro ess

N

avg

simulations. ThetermMSEdenotesthemeansquarederrornorm ofthestateve tor

X

k

,i.e.

E

h

X

k

− ¯

θ1

2

i

.

n

40

d

100

r

2

σ

{0, 0.01, 0.1, 1, 10}

N

iter

1000

N

avg

3000

Table1: Simulationparameters

Inthenoiseless ase,onere ognizethe lassi alsum-exponential onvergen e rateof gossipaveragingalgorithms, dominatedforlarge

k

asfollows:

E

h

X

_k

_{− ¯}

_θ1

2

i

≤ [λ

2

(W )]

k

E

h

X

0

− ¯

θ1

2

i

(49)

Proof of this inequality an be found in [14℄. In the general ase, i.e. when

σ 6= 0

,theerrorduetonoise an ellationshouldbesuperposedtothenoiseless error. We used standard sample mean estimators on ea h node, whi h are known to onverge proportionnaly to the inverse of the number of samples. This trend is onserved in the global mean squared error (MSE) as seen on gure2,whereweplottedthe inversenormalizedMSE. Thus,for small

σ

,the onvergen eshouldbeseenasexperien ingtwodierentsstates. Therststate orrespondsto the oarsehomogeneizationof the omponentsof

X

k

just asif all estimates to be averaged were onstant ( lassi al gossip averaging, see [6℄ and [16℄), while these ond orresponds to the slow an ellation of estimation noise,asifspatialaveragingwasinstantaneous. Howeveroneshouldthinkthat there are asymptotes towards any non-zero values. This false impression is givenby thelogarithmi s aleand the fa t that estimatorvarian esde reases proportionnalyto

k

−1

).

4.2 Impa t of message ex hange rate

The rst set of simulations shows an in rease of onvergen e speed when

σ

de reases. Thus, one should onsider two way of redu ing virtually

σ

. On onehand,themessageex hangerate ould beredu edbysomefa tor,say

K

. Ontheotherhand,theaveragingpro essowsnormallybut datasamplesare

6

1e-10

1e-08

1e-06

0.0001

0.01

1

100

10000

0

1000

2000

3000

4000

5000

6000

MSE

iteration number

sigma=10

sigma=1

sigma=0.1

sigma=0.01

sigma=0 (gossip)

Figure 1: MSE

E

h

X

k

− ¯

θ1

2

i

(log-s ale)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0

1000

2000

3000

4000

5000

6000

normalized inverse MSE

iteration number

sigma=10

sigma=1

sigma=0.1

sigma=0.01

Figure2: InversenormalizedMSE

E

h

1

σ

2

X

k

− ¯

θ1

2

i

−1

(log-s ale)

buered and sent in bursts to estimators: this should avoid the propagation of information of poor quality. In fa t, the rst proposal indu es a laten y proportionalto

K

but leadsto energysavings fora hievingagivenerror(see gures3&4: lessex hangesareneededforaa hievingagivenerror),whilethe se ond s hemes seems not to improve onvergen e. In fa t, simulationsshow thatbuering reatesjigsawos illationsaroundtheerrora hievedbystandard s heme

7

(see gures5&6).

5 Con lusion

Inthis paper, we introdu ed anew distributed algorithm for joint estimation and averagingwhi h generalizes the spa e-time diusion s heme presented in [9℄, and named itJEGA. We provedthe onvergen eof our solutionin terms of rst and se ond order moments of deviation to thetrue average,and then dedu ed the onvergen e in probability of ea h lo al averaged estimate. As itis here basedon peer-to-peer intera tions, this algorithm is learly adapted forsensornetworksappli ations. However,weproposetogeneralizetheproofs given hereto the ase of syn hronousintera tions hara terizedby a onstant transition matrix: su h an approa hrelies onnding ne essary onditionsfor ergodi ityand verifying their onsequen esonnetworking models. Theability oeredbyJEGAofperformingestimationandaveraginginparallelgivesriseto appli ations in artography, lo alization, orsyn hronization in wireless sensor networks. Despiteitssimpli ity,themain defaultof thisalgorithmisthe di- ultytonda losedexpressionforits onvergen erate. Nevertheless,bymean of simulation, our analysis provides a good heurisiti for qualitatively predi t themeanbehaviourofdeviation throughtime. Inafuture work,wewillfo us ourattention ona deepest analysis of onvergen e rate and on nding better solutionswiththehelpofpredi tive/polynomiallters([21℄).

7

1e-05

0.0001

0.001

0.01

0.1

1

10

100

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

MSE

delay (iteration number x subsampling factor)

K=1

K=2

K=4

K=5

K=10

K=25

K=50

K=100

Figure 3: MSE

E

h

X

k

− ¯

θ1

2

i

(log-s ale)

1e-05

0.0001

0.001

0.01

0.1

1

10

100

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

MSE

iteration number

K=1

K=2

K=4

K=5

K=10

Figure 4: MSE

E

h

X

k

− ¯

θ1

2

i

(log-s ale)

0.0001

0.001

0.01

0.1

1

10

100

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

MSE

iteration number

buffersize=1

buffersize=10

buffersize=100

Figure5: MSE

E

h

X

k

− ¯θ1

2

i

(log-s ale)

0.001

2000

2100

2200

2300

2400

2500

MSE

iteration number

buffersize=1

buffersize=10

buffersize=100

Figure6: MSE

E

h

X

k

− ¯

θ1

2

i

(log-s ale): zoomof g5

Appendix A: Proofs for propositions in part 3.4 Proof of proposition 6

E

"

¯

Z

T

k−1

X

i=1

Φ(k, i) ¯

Z

#

= ¯

Z

T

_E

"

_k−1

X

i=1

Φ(k, i)

#

¯

Z

(50)

wherethemiddlesum anbeeasily simplied:

E

"

_k−1

X

i=1

Φ(k, i)

#

= E

"

_k−1

X

i=1

(Ψ(k, i) − Ψ(k, i + 1))

#

(51)

= E [Ψ(k, 1) − Ψ(k, k)]

(52)

= E [W

k−1

W

k−2

. . . W

1

] − I

(53) Asthematri es

W

k

arei.i.d.,one anaveragethemseparatelyandobtains:

E

"

_k−1

X

i=1

Φ(k, i)

#

= W

k−1

− I

(54)

Thislast expression anbereinje tedineq. (50):

E

"

¯

Z

T

k−1

X

i=1

Φ(k, i) ¯

Z

#

= ¯

Z

T

W

k−1

− I

¯Z

(55)

Bymeanofergodi ityof

W

, onehave:

¯

Z

T

W

k−1

− I
¯Z −−−−→

k→∞

¯

Z

T



11

T

n

− I



¯

Z = n¯

θ

2

−

¯

Z

2

(56) Proof of proposition 7

E





k−1

X

i=1

Φ(k, i) ¯

Z

2



 = E





k−1

X

i=1

Φ(k, i)

!

¯

Z

2





(57)

= E

h

(Ψ(k, 1) − I) ¯

Z

2

i

(58)

= E

h

Z

¯

T

Ψ(k, 1)

T

Ψ(k, 1) ¯

Z

i

(59)

− 2E

 ¯Z

T

_{Ψ(k, 1) ¯}

_Z



₊

_¯

_Z

2

(60)

Equation(59)isequivalenttothesquarednormofthestateve torin stan-dardgossipaveragingalgorithms[16℄,andthustendsnaturallytowardthenorm oftheaverage onsensusve tor:

E

h

_Z

¯

T

_{Ψ(k, 1)}

T

_{Ψ(k, 1) ¯}

_Z

i

_{−−−−→}

k→∞

¯

_θ1

2

= n¯

θ

2

(61)

Aseq. (60)isequivalentto eq. (56),we an omputethelimitof eq. (57) foraninnite

k

:

E





k−1

X

i=1

Φ(k, i) ¯

Z

2



 −−−−→

k→∞

¯

Z

2

− n¯

θ

2

(62)

Proof of proposition 8

Thisproofrelies ontheuseof thefollowingstatement, whi h is aspe ial ase ofthetheorems provedin[22℄:

Proposition11([22℄). Let

E

beeither

R

or

C

,

(u

n

)

n∈N

asequen eofelements of

E

su hthat

lim

n→∞

ku

n

k = 0

,and

z ∈ E

with

|z| < 1

. Then,

lim

n→+∞

n

X

k=1

u

k

z

n−k

= 0

(63)

W

is learly symmetri and positive-semidenite. It implies that

W

an be diagonalized by an orthogonal matrix. In the following,

P

j

denotes the eigenve torof

W

asso iatedwiththe

j

-th eigenvalue

λ

j

.

E

"

B

T

k

k−1

X

i=1

Φ(k, i)B

i

#

=

k−1

X

i=1

E



_B

T

k

E

[Φ(k, i)] B

i



(64)

=

k−1

X

i=1

E



_B

_k

T

_W

k−i

_{− W}

k−i−1

_B

_i



(65)

=

n

X

j=1

k−1

X

i=1

λ

k−i

j

− λ

k−i−1

j

_E



B

T

k

P

j

P

T

j

B

i



(66)

where equation (65) results from the independan e of estimation noises and ex hangematri es. For

λ

j

= 1

,

λ

k−i

j

− λ

k−i−1

j

vanishes. Ontheotherside,the

ovarian eterm anbeboundedbyanon-negativesequen ehavingazerolimit atinnity:



E



B

T

k

P

j

P

T

j

B

i



 =











X

u,v



P

j

P

j

T



uv

C

ki

uv











(67)

≤ n

2

max

u,v

max

k∈N

k>i

|C

ki

uv

|

∆

= n

2

C

i

(68)

The prerequisite ondition on

C

ki

uv

ensures that

C

i

onverges to

0

. Thus,

E



_B

T

k

P

j

P

T

j

B

i



does similarly for all normalized ve tor

P

j

. On another side,

|λ

j

| < 1

whenever

λ

j

6= 1

. Bysplitting the summation overeigenvalues and developping the multipli ation by

λ

k−i

j

− λ

k−i−1

j

, equation (66) an be

easilyexpressedasalinear ombinationof

2(n − 1)

sums,ea hofwhi hsaties proposition 11. This impliesthat (66) goes to0 as

k

grows,independantlyof thede reaserateof ovarian es

8 .

Proof of proposition 9

Letus startbyrewrittingequation(9):

E





k−1

X

i=1

Φ(k, i)B

i

2



 =

k−1

X

i=1

k−1

X

j=1

E

h

B

T

j

Φ(k, j)

T

_{Φ(k, i)B}

i

i

(69) 8

thepartialsum(65)mustbe arefullymanipulatedinordertoavoiddivergen eatlarge

Aswedeveloptheterm

Φ(k, j)

T

_{Φ(k, i)}

,weobserveon eagainthat the ompo-nents of ea h

B

i

and

B

j

that are olinearto

1

(asso iatedwith eigenvalue1) arenottransmittedthrough

Φ(k, i)

:

Φ(k, j)

T

Φ(k, i) = Ψ(k, j)

T

Ψ(k, i) − Ψ(k, j + 1)

T

Ψ(k, i)

− Ψ(k, j)

T

Ψ(k, i + 1) + Ψ(k, j + 1)

T

Ψ(k, i + 1)

(70) Forthe sakeof simpli ity,

Ξ

k

(i, j)

denotes

E

h

Ψ(k, j)

T

Ψ(k, i)

i

. It is usefulto noti ethat

Ξ

k

(i, j)

an befa torizedbyexternalizingtermsin

i

:

Ξ

k

(i, j) = E

h

Ψ(k, j)

T

Ψ(k, i)

i

(71)

= E

h

Ψ(k, j)

T

Ψ(k, j)Ψ(j, i)

i

(72)

= E

h

Ψ(k, j)

T

Ψ(k, j)W

j−i

i

(73)

= Ξ

k

(j, j)W

j−i

(74)

Followingtheapproa hof[14℄helpsboundingthespe tralradiusof

Ξ

k

(j, j)

. Forany

n

-dimensionalve tor

X ⊥ 1

, thefollowinginequalitieshold:

X

T

_Ξ

k

(j, j)X = X

T

E

h

Ψ(k, j)

T

Ψ(k, j)

i

X

(75)

= X

T

_E

h

_{Ψ(k − 1, j)}

T

_{W Ψ(k − 1, j)}

i

_X

(76)

≤ λ

2

X

T

E

h

Ψ(k − 1, j)

T

Ψ(k − 1, j)

i

X

(77)

≤ λ

k−j

2

kXk

2

(78)

Usingthisinequalityin onjun tionwiththeRayleigh-Ritz hara terization theorem,oneobtains:

ρ

Ψ

(k, j)

= ρ

∆



Ξ

k

(j, j) −

11

T

n



(79)

= max

X⊥1

kXk=1

n

X

T

_E

h

_{Ψ(k, j)}

T

_{Ψ(k, j)}

i

_X

o

(80)

≤ λ

k−j

2

(81)

Re alling that any omponent of

B

i

olinear to

1

an be an elled, the modulusofits oordinatesonabasisofeigenve torsof

W

arethenres aledby afa torlessthanor equalto

λ

j−i

2

whenweapply

W

j−i

. Thisgivesrisetothe followinginequality:



E



B

T

j

V

k

V

k

T

W

j−i

_B

i



 =







X

λ

j−i

p

E

h

B

T

j

V

k

V

k

T

V

p

V

T

p

B

i

i





(82)

≤ (n − 1)n

2

λ

j−i

2

max

_u,v

|C

uv

ij

|

(83) where

V

k

isanyunitaryve torof

R

n

,and

V

p

isaneigenve torof

W

(andthen of

W

j−i

) asso iated with eigenvalue

λ

p

. Let

S

ij

onstitutedofeigenvalues

λ

ψ

ofasso iatedeigenve tor

V

λ

Ψ

. Usingthisnotation,

E



_B

T

j

Ξ

k

(i, j)B

i



anbebroughtintoausefulform forbounding purposes:



E



B

T

j

Ξ

k

(i, j)B

i



 =



E



B

T

j

Ξ

k

(j, j)W

j−i

B

i





(84)

≤

X

λΨ∈S

ij

_k

λψ 6=1



λ

Ψ

E



B

T

j

V

λ

Ψ

V

T

λ

Ψ

W

j−i

_B

i





(85)

≤ (n − 1)

2

ρ

Ψ

(k, j)n

2

λ

j−i

2

max

_u,v

|C

ij

uv

|

(86)

Couplingequations(81)and(83),thefollowingmajorizationstates:



E



B

T

j

Ξ

k

(i, j)B

i



 ≤ (n − 1)

2

_n

2

_λ

k−i

2

max

_u,v

|C

ij

uv

|

(87) Letus dene

ξ

∆

=

√

λ

2

, and rememberthat

i < j

: immediately

0 ≤ λ

k−i

2

=

ξ

2k−2i

_{≤ ξ}

2k−i−j

. Thisinequalityhen eimplies:













X

i<j

E



_B

T

j

Ξ

k

(i, j)B

i















≤ (n − 1)

2

_n

2

X

i<j

λ

k−i

2

max

_u,v

|C

ij

uv

|

(88)

≤ (n − 1)

2

_n

2

X

i<j

ξ

2k−i−j

_max

u,v

|C

ij

uv

|

|

{z

}

−−−−→

k→∞

0

(89)

The onvergen e toward

0

of theright-handside is ensuredby proposition 12:

Proposition 12. ([22℄)Let

E

be either

R

or

C

,

(u

ij

)

(i,j)∈N

2

∈ E

N

2

su hthat

lim

(i+j)→∞

ku

ij

k = 0

,and

z ∈ E

with

|z| < 1

. Then

lim

n→+∞

X

1≤i,j≤n

u

ij

z

2n−i−j

= 0

(90)

When

i = j

,a lassi alresultfrom [14℄statesthat:

E



_B

T

_i

_Ξ

_k

_{(i, i)B}

_i



_{≤ λ}

k−i

2

E

h

kB

i

k

2

i

(91)

Then,summingover

i

givesthediagonallimit:

0 ≤

k−1

X

i=1

E



_B

_i

T

_Ξ

_k

_{(i, i)B}

_i



_≤

k−1

X

i=1

λ

k−i

2

E

h

kB

i

k

2

i

|

{z

}

−−−−→

k→∞

0

(92)

In the same way, it is easy to bound terms with

Ξ

k

(i + 1, j)

,

Ξ

k

(i, j + 1)

or

Ξ

k

(i + 1, j + 1)

in equation (69). We obtain 4 sums in the spirit of the ombinationofequations(88)and(92)thattendto0as

k

goesto innity.

Appendix B: Optimality of

1/2

weights

Given a weight

α ∈ [0, 1]

for ex hanges, the transition matrix

W

ij

takes the form:

W

ij

= I − α(e

i

− e

j

)(e

i

− e

j

)

T

(93) Under theassumption that

W

k

i.i.d. randommatri es drawn from theset of

W

ij

,performan eboundsarederivedgiventhese ondsmallesteigenvalueof

E



_W

T

k

W

k



(see[15℄),i.e.:

λ

(α)

2

∆

= ρ



E



_W

_k

T

_W

_k



₋

11

T

n



(94)

= max

X⊥1

kXk=1

X

T

E



_W

T

k

W

k



X

(95)

One antrytooptimizethis riteriona ordingtoparameter

α

. Proposition 13answerssimplyto thisproblem.

Proposition13. Forpeer-to-peerex hangeswithxedlinkprobabilitiesandx ex hange weight

α ∈ [0, 1]

,

λ

(α)

2

isoptimalfor

α = 1/2

.

Proof. Let

p

ij

betheprobabiltythat,ifnode

i

is hosenasinitiator,it onta ts node

j

. Then

E



W

T

k

W

k



anbeeasily rewrittenwiththesamenotationsasin equation(93):

E



_W

_k

T

_W

_k



₌

1

n

X

i,j

p

ij

W

ij

T

W

ij

(96)

= I − 2α(1 − α) (e

i

− e

j

) (e

i

− e

j

)

T

(97) Then,writting

λ

i

(M )

the

i

-theigenvalueof

M

inde reasingorder,onehas:

λ

(α)

₂

= ρ





1

n

X

i∼j

p

ij

W

ij

T

W

ij





(98)

= λ

2



I − 2α(1 − α) (e

i

− e

j

) (e

i

− e

j

)

T



(99)

= 1 − 2α(1 − α)λ

n−1





X

i∼j

p

ij

E

ij





(100)

= 1 − 2α(1 − α)µ

(101) Now,derivating

λ

(α)

2

w.r.t. weight

α

gives:

∂λ

(α)

2

∂α

= (−2 + 4α)µ

(102)

Sin e

E

[W

k

]

is ergodi ,

µ

annot be null. Thus the only way for

λ

(α)

2

to

haveanullderivativeisthat

α = 1/2

. Thematrix

E



W

T

k

W

k



istriviallydoubly

sto hasti and then

λ

(α)

2

≤ 1 = λ

(0)

2

. As a onsequen e,

λ

(α)

2

is minimum for

α = 1/2

.

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