Reducing Communication Overhead for Average Consensus

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Consensus

Mahmoud El Chamie, Giovanni Neglia, Konstantin Avrachenkov

To cite this version:

Mahmoud El Chamie, Giovanni Neglia, Konstantin Avrachenkov. Reducing Communication Overhead

for Average Consensus. [Research Report] RR-8025, INRIA. 2012, pp.22. �hal-00720687v2�

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0 2 4 9 -6 3 9 9 IS R N IN R IA /R R -- 8 0 2 5 -- F R + E N G

RESEARCH REPORT N° 8025

July 2012

Reducing

Communication

Overhead for Average Consensus

Mahmoud El Chamie, Giovanni Neglia, Konstantin Avrachenkov

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RESEARCH CENTRE

SOPHIA ANTIPOLIS – MÉDITERRANÉE

MahmoudEl Chamie

∗

, Giovanni Neglia

†

, Konstantin

Avrahenkov

‡

Projet-Teams Maestro

Researh Report n°8025July201222pages

Abstrat: Averageonsensus isaniterativeprotoolwhere nodesina network, eah havingan

initialsalar valuealled estimate,performa distributedalgorithm toalulatetheaverageofall

estimatespresentedinthenetworkbyusingonlyloalommuniation. Witheveryiteration,nodes

reeive theestimates from theirneighbors, and theyupdate their own estimateby theweighted

average of the reeived ones. Whilethe average onsensus protool onverges asymptotially to

onsensus,implementingaterminationalgorithmishallengingwhennodesarenotawareofsome

globalinformation (e.g. thediameter ofthe network or the number ofnodes presented). Inthis

report,weareinterestedindereasingtherateofthemessagessentinthenetworkastheestimates

areloser to onsensus. Wepropose a totally distributed algorithm foraverageonsensus where

nodessendmoremessageswhenthenodeshavelargedierenesintheirestimates,andreduetheir

rateof sending messages when theonsensus is almost reahed. The onvergene of the system

is guaranteed to be within a predened margin

ǫ

^from ^the ^true^average ^and ^theommuniation overheadislargelyredued.

Key-words: averageonsensus, energyredution,termination protool,distributed algorithms

∗

InriaSophiaAntipolis,Frane,Mahmoud.El_Chamieinria.fr

†

InriaSophiaAntipolis,Frane,Giovanni.Negliainria.fr

‡

InriaSophiaAntipolis,Frane,K.Avrahenkovsophia.inria.fr

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Résumé : Le onsensus de moyenne est un protoole itératif où les n÷uds d'un réseau,

ayanthaununeestimationinitiale,exéutentunalgorithmedistribuépouralulerlamoyenne

de es estimations en utilisant uniquement les ommuniation loales. A haque itération, les

n÷udséhangent leursestimationsaveleursvoisins. Ces estimationsserontremplaéesparla

moyennepondéréede ellesreçues. Laonvergeneduonsensusdemoyenneest asymptotique

etlamiseen÷uvre d'unprotooledeterminaison estdiile lorsquelesn÷udsneonnaissent

pas l'estimation global (par exemple, le diamètre du réseau ou le nombre de n÷uds). Dans

e rapport, nous intéressons à larédution du taux de messages envoyés dansle réseau quand

lesestimations deviennent prohe du onsensus. Nous présentons un algorithmede onsensus

demoyenne totalementdistribué, oùlesn÷uds envoientplusde messages lorsque ladiérene

entre leurs estimationsest grande et moins demessages lorsque lesystème est à peuprèson-

vergeant. La onvergene dusystèmeest garantied'être prohedela vraiemoyenneet le oût

desommuniations estfortementréduit.

Mots-lés: onsensusdemoyenne,rédutiondel'énergie,protooledeterminaison,algorithme

distribué

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1 Introdution

Average onsensusprotools areusedto ndtheaverageaross initialmeasurementspresented

at nodesin a network ina distributed manner havingno entralentity toontrol and monitor

thenetwork. Thisproblem isgaining interestnowadaysdueto itswidedomain ofappliations

as in ooperative robot motions [1 ℄, resoure alloation [2 ℄, and environmental monitoring [3℄,

in addition to the existeneof large networks suh as wireless sensor networks for whih suh

onsensusalgorithmsareneessary[4 ℄. Underthisdeentralizedapproah,nodesseletaweight

fortheestimate ofevery neighbor andperforma weightedaverageofthevaluesin theirlosed

neighborhood. Byonsideringspeialonditionsontheseletedweights,theprotoolisguaran-

teedtoonvergeasymptotiallytotheaverageonsensus. Aftereverylineariteration,eahnode

must send its newvalueto its neighbors so that theyan use it in the next iteration. For an

extensiveliteratureonaverageonsensus protool andits appliations,hekthesurveys[5 , 6 ℄

andthereferenestherein.

To speed up the onvergene, some approahes have foused on seleting the weights so

that theonvergene beomesfaster. Xiao and Boyd in [7 ℄ have formulated theproblem as a

SemiDenite Program that an beeientlyand globally solved. However,speeding upthe

onvergene does not redue the number of messages that are sent in the network even when

usingtheoptimalweights. Thereasonisthattheonvergeneisreahedonlyasymptotially,and

even ifnodes' estimatesare very lose tothe average, nodes keep onperformingtheaveraging

andsendingmessages totheirneighbors.

Thereportisorganizedasfollows: Setion2givesthenotationusedandaformulationofthe

problem. Setion3givesthepreviousworkontheterminationoftheaverageonsensusprotool.

Setion4motivatestheworkbyanimpossibilityresultfornitetimetermination. Setion5gives

theproposedalgorithm, itsanalysis, andthesimulationsofthe algorithm. Setion 6 onludes

thereport.

2 Problem Formulation

Consider a network of

n

^nodes ^that ân êxhange ^messages ^betweenêah ôther ^through ôm-

muniation links. Every node in this network has a ertain measurement (e.g. pressure or

temperature),andweneedeahnodetoknowtheaverageoftheinitialmeasurementsbyfollow-

ing a distributed linear iterationapproah. The networkof nodes an bemodelledas a graph

G = (V, E)

^where

V

^is^the^set^of^verties⁽

|V | = n

⁾^and

E

îs^the^set ôfêdges^suh^the

{i, j} ∈ E

if nodes

i

^and

j

âre ônnetedând ân ômmuniate ^(they âre neighbors). Letalso

N i

^be^the

neighborhood set of node

i

^. ^Let

x i (0) ∈ R

be the initial value at node

i

^. ^We ^are ^interested

in omputing theaverage

x ave = (1/n) P n

i=1 x i (0)

^, ⁱⁿ ^a deentralized manner with nodesonly ommuniatingwiththeirneighbors. Ournetworkmodelwillbetheaveragingdoneona xed

networkwithasynhronizationlok. Whentheloktiks,allnodesinthesystemperformthe

iterationoftheaveragingprotoolatthesametime(thisisthesynhronousupdateassumption).

Atiteration

k + 1

^,^node

i

^updates^its^state^value

x i

^:

x i (k + 1) = w ii x i (k) + X

j ∈ N i

w ij x j (k)

⁽¹⁾

where

w ij

^is ^the ^weight ^seleted ^by ^node

i

^for ^the ^value ^sent ^by ^its ^neighbor

j

^and

w ii

^is ^the

weightseletedbynode

i

^forîtôwn^value. ^The^matrix^formêquationîs:

x (k + 1) = W x (k)

⁽²⁾

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where

x (k)

îs^the^state^vetorôf^the^systemând

W

^is^the^weight^matrix.

Under some onditions on the weight matrix

W

^given ⁱⁿ ^{[8 ℄,} ^the ^system ^is ^guaranteed ^to

onverge to theaverage. Inthis report, we onsider

W

^to ^be^a ^doubly ^stohasti ^matrix^that

satises these onditions and is onstruted loally, some methods foronstruting theweight

W

^using ^only^loal information an befound is[9 ℄. Let

1

bethe vetor havingthe value

1

ⁱⁿ

everyentry,theonvergene oftheaverageonsensusgivenaboveis ingeneralasymptoti:

k lim →∞

x (k) = x ave 1 .

⁽³⁾

Theaveragingworksasfollows: afterseletingtheweightsfortheirneighbors, alineariteration

phasestarts. Eah nodealulates its newvalue depending on theweightsand theneighbors'

values,andthenitbroadaststhenewvaluetoallitsneighbors beforethegloballoktiksfor

thenextiteration. Sinethelimitin(3 )istypiallyreahedatinnity,thenodeswillbealways

busysendingmessages. Let

M (k)

^be^the^number^of^nodestransmittingatiteration

k

^,^so^without

a termination protool all nodesare transmitting at iteration

k

^,

M (k) = M = n

^, independent from theonvergene of estimates. So how an nodes knowthat theyare lose enough to the

averageandthusstop sendingmessagestotheirneighborsandsavebandwidthandenergy? We

propose in this report an algorithm that redues ommuniation overhead aused by sending

messagesatevery iteration.

3 Related Work

Somework onsiders protools foraverageonsensus protoolto terminate in nite time. One

approah is that estimates of nodes performing suh a protool reahes the exat average, as

in [10 ℄, their solution is based on the onept of the minimal polynomial of the matrix

W

^,

wheretheyshowedthatanodeusingoeientsofthispolynomial anuseitsestimateover

K

suessiveiterationstoalulatetheaverage,butnodesmustalsohavehighmemoryapabilities

to storethe

n × n

^matrix, ^and^high ^proessing apabilities to solvethe

n

^linearlyindependent equationsto nd the oeientsof theminimal polynomial. Anotherapproah for nite time

termination,isgivenin [11 ℄,but insteadof exataverage,guaranteed tobewithin apredened

thresholdfromtheaverage. Thisapproahrunsthreeonsensusprotoolsatthesametime: the

maximumandtheminimumonsensusrestartedevery

U

^iterations^where

U

îsânûpper ^bound

of thediameter of the network, and average onsensus. The dierene betweenthe maximum

andminimumonsensusprovidesastoppingriteriafornodes.

Inadierenttimesettingforaverageonsensus,usingtheasynhronousrandomizedgossip-

ing,theauthorsin[12 ℄proposedanalgorithmthatleadstotheterminationofaverageonsensus

in nite timewith high probability. In theirapproah, eah nodehas a ounter

c i

^that ^ounts

howmanytimesthedierenebetweenthenewestimateandtheoldonewaslessthanaertain

threshold

τ

^, ^when ^the ôunter ^reahes â êrtain^value, ^say

C

^, ^the ^node^will ^stop ^initiating^the

algorithm. They proved that by a orret hoie of

C

^and

τ

^(depending ^on ^some ^parameters

as the maximum degree in the network, the number of nodes, and the number of edges) the

terminationiswithhighprobability.

However,amajordrawbakofthesealgorithms(withouttakingintoonsiderationthemem-

oryapabilitiestheyassumednodestohaveor therobustness ofthesystem)theassumptionof

knowledge of some global variables at eah nodes whih violates the deentralized senario of

averageonsensus. Designingadeentralizedalgorithmforaverageonsensusthatterminatesin

nitetimewithoutusing anyglobalestimate(as thediameterofthenetwork orthenumber of

nodes)inthealgorithmisstillanopenproblem.

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Figure1: Line graph

G

^with

3

^nodes.

4 Motivation

Weaddresstheproblemoftermination ofaverageonsensus inthisreport. Wewill startbyan

impossibilityresultforterminationoftheaverageonsensusprotoolinnitetimewithoutusing

of a global estimate (mainly an upper bound on the diameter) and by only using the history

estimatesofthenode.

Theorem 1. In the average onsensus protool on axedgraph(no link failure), wherenodes

perform synhronous iterations as in (1 ) to onverge to the average of initial estimates, it is

impossible to reate a deterministi distributed algorithm that terminates in nite time to give

theaverage ofinitialestimatesoranyboundedapproximateoftheaverage usingonlythe history

of the node'sestimate.

Proof. Consider aline graph

G

^of^three ^nodes,

a

^,

b

^,^and

c

^as ⁱⁿ^Fig.^1,^where^the^weight^matrix

is real doubly stohasti matrix that satises the onvergene onditions (as a result we an

assume

w aa , w cc > 0

^). ^Suppose ^there^exists ^atermination algorithmforaverageonsensus that terminates in nite time, then applying this algorithm on this graph implies that their exists

an iteration

K > 0

^and

ǫ > 0

^where ^node

a

^deides ^to ^terminate ând ^this âlgorithm ûses

only the history of estimates of node

a

^:

x a (0), x a (1), x a (2), ..., x a (K)

^, ^to ^deide ^to ^stop ^and

|x a (K) − x ave | < ǫ

^, ^where

x ave = α = ^x ^a ^(0)+x ^b ₃ ^(0)+x ^c ⁽⁰⁾

^. ^We ^will ^dene ^the ^extended ^mirror

graphof

G

^to^be^a^line ^graph^of

6

^nodes

a 1 , a 2 , b 1 , b 2 , c 1 ,

^and

c 2

^,^formed^by^two^idential^graphs

as

G

^but^adding ^a

{c 1 , c 2 }

êdge. ^Theînitial êstimatesâre^the^same ^forâll^nodes^(e.g. ^for^node

a

^we^have

x a 1 (0) = x a 2 (0) = x a (0)

^),^weight^matrixîsâlso ^the^same êxept^for

c 1

^and

c 2

^,^where

w c 1 c 1 = w c 2 c 2 = w c 1 c 2 = ^w ₂ ^cc

^(see ^Fig. ^2). ^Notie ^that ^on ^the ^new ^generated ^graph ^and ^the

old one,

x a 1 (k) = x a (k) ∀k ≤ K

^, ^so ^node

a 1

^on ^the^new ^graph ^will ^deide^to ^terminate ^after

exatly

K

iterations. Notie now that we an ontinue doing this proedure of graph mirror extension till wehave a line graph where

n > K

^, ^all ^this ^graph

H

^.

a 1

ⁱⁿ ^graph

H

^still ^have

thesame estimatesduring theiterations

k ≤ K

^, ^so ît^will âlways^terminateâfter

K

iterations.

However,if weadd a newnodeto

H

ât ^theênd ôf ^the^line, ^havingân êstimate

β >> nα

^,^the

eet would only reah

a 1

âfter â ^number ôf îterations

k > K

^. ^Therefore, ^on ^this ^graph ^(the

graphwhereweaddedonenodeattheendofthelinegraph

H

^),^the^node

a 1

^will^terminate^with

|x a 1 (K) − α| < ǫ

^,^but îtîs ^not^possible ^to^guaranteeît îs^lose ^to^theâverage ^beause^the^new

averagenowis

x ave = α + ^β ⁻ _n ^α >> x a 1 (K)

^.

Theaboveproofanbeextendedtoinlude anygraph

G

^, ^not^just^line^graphs, ^by^using^the

same tehniqueof generatingtheextendedmirrors graphs of

G

^to ^show^that ^we^annot^simply

applytheterminationprotoolonanykindof networksnotjustthelinegraph.

Thetermtermination in this deentralized senario must referto themessages sentbythe

nodes in the network but not to the exeution of the algorithm at the level of nodes. The

algorithms touse mustallownodes torefrain fromsending theirestimateas messages in some

iterations and send it in another. The termination ours when the number of messages sent

in thenetwork disappears, even if thenodesare still running the algorithm internally, but no

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Figure2: Extendedmirrorgraphof

G

^with

6

^nodes.

more messages are sent in the network. In a less restritive objetive, we an also dene the

asymptotiterminationwhentherateofsending messagesinthenetworkisvanishing,i.e.:

t lim →∞

P t

k=1 M (k)

t = 0.

⁽⁴⁾

Inotherwords,therateofmessagesinthenetworkmustgiveusanideaabouttheonvergeneof

nodes,soifwemonitorthenetworkforanintervalofiterationsandsawthatnotmanymessages

are generated, we should beableto onlude that the nodesare almost onvergedto thetrue

averageor toanapproximatewhih isbounded byathresholdfromthetrueaverage.

5 Our Approah

5.1 Centralized Termination

5.1.1 Introdution

Even if thenodeswill not terminate in nite time, we areinterested in dereasing the rate of

themessages sentinthenetworkin suha waytoberelatedtotheamountofimprovementwe

aregaining in terms ofthe onsensus. For example, ifthe nodeshavewide rangeof dierene

in their estimates, sending messages an beeient to derease the error in the network and

reah onsensus. However, when the nodes are almost onverged, sending a lot of messages

will not be eient, as the improvement is just a user perspetive. So from an engineering

perspetive, our network must send more messages when the nodes have large dierenes in

their estimates, and lessmessages when the system is almost onverging. This subsetion on

entralized termination just provides basis and intuition for the more pratial deentralized

terminationalgorithmproposedinthisreport.

5.1.2 Overview

Inthis setion, wedisuss a simpleentralizedalgorithm for termination of averageonsensus

protools. Weallitaentralizedprotoolsineinthisprotool,eahnodeansendabroadast

signaltothenetworktoperformaniteration. Atanyiteration,ifanyofthenodesinthenetwork

sent this signal, all the nodes will respond by sending the new estimates to their neighbors

aordingtotheaveraging equation:

x (t + 1) = W x (t),

⁽⁵⁾

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where

t

îs^the^time^whenâ^broadast^signalîs^sentⁱⁿ^the^network. Ôn^theôntrary^,^we^see^that

ifnosignalissent,thenodeswillpreservethesame estimate:

x (t + 1) = x (t),

⁽⁶⁾

where

t

îs ^the îteration^where ^no^broadast ^message ^was ^sent. În ^the^following,^we^will ^give â

simplealgorithmwherethefrequenyofsendingabroadastsignalwillbedereasing(onverging

tozeroi.e. nonodeina networkwillbesendingsuhamessage), sothemessages foraveraging

will alsobesentlessoftenas nodes onverges tothe average. This algorithmisan intuitionto

howthedeentralizedterminationprotoolworks.

5.1.3 Analysisand Convergene

Let us deneformally the method. Let

e(t)

^beân înreasing^salar ^visible ^by âll ^the^nodes ⁱⁿ

the network suh that

e(0) = 0

^and

0 ≤ e(t) < ǫ(t)

^, ^where

ǫ(t)

^is ^a ^boundary^threshold^whih

is alsoan inreasingsalar suhthat

ǫ(0) = ǫ = constant

^. ^Let

W

^be^the ^weight ^matrix^of^the

network satisfyingonvergene onditions of averageonsensus and

x (t)

^be^the^state ^vetor ^of

thesystematiteration

t

^. ^We^dene^a ^logial^expression

L t

âsâ^Boolean^variable^(either^trueôr

false)denedatevery iteration

t

^of^the^followingexpression:

L _t : e(t − 1) + ||W x (t − 1) − x (t − 1)|| ∞ < ǫ(t − 1),

⁽⁷⁾

with

L 0 = F alse

^. Âording^to ^thisôndition,âtionsâre^takenⁱⁿîteration

t

^. ^If^this ^ondition

wasfalse, a broadast signal issentin thenetwork andall nodeswill performan iteration;we

have

x (t −1)

^hanges,

ǫ(t −1)

âlsoîsînreased^byâ^valueôf

ǫ/m ²

^,^where

m

îsâôunterîndiating

howmanytimesthevalueof

ǫ

^has^inrementedⁱⁿ ^the^past,^but^the^other^salar

e(t − 1)

^is^kept

thesame. On theontrary, if

L t

îs ^true,^then ^there îs^no^signal ⁱⁿ ^the ^network, ând ^the^nodes

keepthesameestimateas theprevious iteration,theboundarythresholdisalsokeptthesame,

but

e(t − 1)

^is ^inreased^by

y(t − 1) = ||W x (t − 1) − x (t − 1)|| ∞

^. ^In^partiular,^the^hanges^to

thenetworkvariablesdueto

L t

^are^givenⁱⁿ^the^following^equation:

e(t) =

( e(t − 1) + y(t − 1)

^if

L _t = T rue,

e(t − 1)

^if

L t = F alse,

⁽⁸⁾

x (t) =

( x (t − 1)

^if

L t = T rue,

W x (t − 1)

^if

L t = F alse,

⁽⁹⁾

ǫ(t) =

( ǫ(t − 1)

^if

L t = T rue,

ǫ(t − 1) + ǫ/k ²

^if

L t = F alse,

⁽¹⁰⁾

where

k

îsâôunter îndiating^how^many^times^the^valueôf

ǫ(t)

^has^inrementedⁱⁿ ^the^past.

When

L _t

^is^true,^we^all

t

â^silentîteration^beause^the^nodes^have^the^same êstimateâs^the

previous iteration (i.e.

x i (t) = x i (t − 1)

⁾ ând ^there îs ^no ^need ^to êxhange ^messages ôf ^these

estimates in the network. On the other hand, when

L t

^is ^false, ^we ^all

t

âs â ^busy îteration

beause nodes will perform an averaging (i.e.

x (t) = W x (t − 1)

⁾ ^and ^the ^estimates ^must ^be

exhangedin thenetwork. Letusdene

T

^as^the^set ^of^busy^period:

T = {t | L t = F alse}.

⁽¹¹⁾

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Let

t k

^where

k = 1, 2, ...

^be^theînreasing^sequeneôf^theêlementsôf^the^set

T

^. ^Let

α k

^be^the

numberofsilentiterationsbetween

t k

^and

t k+1

^. ^We^an^see^that

t k+1 = α k +t k + 1

^. ^We^say^that

thealgorithm isterminatingasymptotiallyifthesilentperiodisdiverging,i.e. wehavethat:

t→∞ lim α k = ∞.

⁽¹²⁾

First,it is not diultto see that if

t → ∞

^, ^then

k → ∞

^too,^beause ânûpper ^bound ôn

k

impliesanupperboundon

t

^. ^Therefore,^it^is^suient^to^prove^that

lim k →∞ α k = ∞

^.

Let

z (k) = W x (t _k ) − x (t _k )

^,^weân^see^that âording^to^thisâlgorithm,

α k = ⌊ ǫ(t k ) − e(t k )

|| z (k)|| ∞

⌋

≥ ǫ(t k − 1) + ǫ/k ² − e(t k − 1)

|| z (k)|| ∞

− 1

≥ ǫ

k ² || z (k)|| 2

− 1.

⁽¹³⁾

Moreover,

z (k)

êvolveâording ^to^the^followingêquation:

z (k) = (W − G) z (k − 1) = (W − G) ^k z (0),

⁽¹⁴⁾

where

G = 1/n 11 ^T

. Finally,

|| z (k)|| 2 ≤ Cρ ^k (W − G),

⁽¹⁵⁾

where

C = || z (0)||

^and

ρ(W − G)

^is^the^spetral^radius^of^the^matrix

W − G

^,^so

0 < ρ < 1

^sine

W

^satises ^theôndition ôfâ ônverging^matrix^(see ^{[8 ℄).} ^Puttingêverything ^together, ^we^get

nallythat:

α k ≥ ǫ

Ck ² ρ ^k − 1,

⁽¹⁶⁾

and hene

α k → ∞

^as

k → ∞

^. Consequently, the rate of messages sent in the network is vanishing,namely

t lim →∞

P t i=1 M (i)

t = lim

t →∞

M k t = lim

t →∞

M k k + P k −1

i=1 α i

= 0.

⁽¹⁷⁾

5.2 Deentralized Environment

5.2.1 Introdution

Inadeentralizedalgorithm,eahnodeworksindependently,thenodesonlyagreeonsynhronous

timestepstodotheiteration. Thesilentperiodona nodeisjustloal, soanodean besilent,

whileits neighbor is not. In thissenario, we seethat within aniteration, some nodes willbe

transmitting and others will be silent. This an ause instability in the network beause the

average with every iterationis not now onserved, and the salar

e(k)

^dened ⁱⁿ ^the^previous

subsetionisnowavetor

e (k)

^where

e i (k)

îs^loal^toêvery^node. ^Toônserve^theâverageⁱⁿ^the

deentralized setting, this salar must take part in the stateequation as wewill showin what

follows.

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5.2.2 System Equation

In ourapproah, weonsider a more generalframework foraverageonsensus where westudy

theonvergene ofthefollowingequation:

x (k + 1) + e (k + 1) = W x (k) + e (k).

⁽¹⁸⁾

Someworkhavestudied,thefollowingequationasaperturbedaverageonsensusandonsidered

e (k)

^to ^be^zero ^mean ^noise ^with ^vanishing ^variane ^(see ^{[13 ,} ^14℄). ^However, ⁱⁿ ^our^model, ^we

onsider

e

as a deterministi part of the state of the system and not a random variable. We onsidersuientonditions forthesystemtoonverge,andweusetheseonditionsto givean

algorithmthat anreduethenumber ofmessagessentinthenetwork.

Inthestandardonsensus algorithms, thestateofthesystemisdened bythestatevetor

x

,but inthemodiedsystem,thestateequationisdenedbytheouple

{ x , e }

^.

Theorem 2. Let

e (k + 1) = e (k) − F(k) x (k)

^, ^and

A(k) = W + F (k)

^. ^Assume ^the ^following

onditions onthe matrix

A(k)

^:

(a)

a _ij (k) ≥ 0

^for^al^l

i

^,

j

^,^and

k

^,^and

P n

j=1 a _ij (k) = 1

^for^all

i

^and

k

^.

(b) Lowerboundon positiveoeients: thereexistssome

α > 0

^suh^that ^if

a ij (k) > 0

^,^then

a ij (k) ≥ α

^,^for^all

i

^,

j

^,^and

k

^.

() Positivediagonaloeients:

a ii (k) ≥ α

^,^for^all

i

^,

k

^.

(d) Cut-balane: forany

i

^with

a ij (k) > 0

^,^we^have

j

^with

a ji (k) > 0

^.

(e)

lim k →∞ x (k) = x ^⋆ ⇒ lim k →∞ F (k) x (k) = 0

.

Then

lim k →∞ x (k) = x ^′ _ave 1

,wherewehavethat

x ^′ _ave ∈ [min j x j (0), max j x j (0)]

^iffurthermore wehave

e (0) = 0

and

e i (k) < ǫ

^,^then

|x ave − x ^′ _ave | < ǫ

^.

Proof. Letus rstprovethat

x (k)

^onverges. ^Bysubstituting theequationof

e (k + 1)

ⁱⁿ^{(18 ) ,}

weobtain:

x (k + 1) = A(k) x (k),

⁽¹⁹⁾

where

A(k) = W + F (k)

^. ^F^rom ^the^onditions (a),(b),(), and(d) on

A(k)

^, ^we^have^from ^{[ 15 ℄}

that

x

onverges, i.e.

lim k →∞ x (k) = x ^⋆

. Sine the systemis onverging, then from equation (18 ) ,weanseethat:

x ^⋆ + lim

k→∞ ( e (k + 1) − e (k)) = W x ^⋆ ,

so,

x ^⋆ = W x ^⋆ .

⁽²⁰⁾

Therefore,

x ^⋆

isan eigenvetor orresponding to the highesteigenvalue(

λ 1 = 1

⁾ ^of

W

^. ^So^we

an onludethat

x ^⋆ = x ^′ _ave 1

where

x ^′ _ave

^is^a^salar(Perron-Frobeniustheorem).

Theondition

1 ^T W = 1 ^T

onthe matrix

W

ⁱⁿ ^equation^{(2 )} ^leads ^to^thepreservation ofthe averagein the network,

1 ^T x (k) = nx ave ∀k

^. ^This ^ondition^is ^not^neessary ^satised ^by

A(k)

^,

soletusprovenowthatthesystempreservestheaverage

x ave

^:

1 ^⊤ ( x (k + 1) + e (k + 1)) = 1 ^⊤ (W x (k) + e (k))

⁽²¹⁾

= 1 ^⊤ ( x (k) + e (k)).

⁽²²⁾

Thelastequalityomesfromthefat that

W

^is^sum^preserving^sine

1 ^⊤ W = 1 ^⊤

.

(13)

Finally by a simple reursion we have that

1 ^⊤ ( x (k) + e (k)) = 1 ^⊤ x (0) = nx ave

^, ^and ^the

averageisonserved. Moreover,sine

|e i (k)| ≤ ǫ ∀k

^,^so ^we^have:

|(1/n) 1 ^⊤ x (k) − x ave | ≤ ǫ ∀k.

⁽²³⁾

Butwe justproved that

lim k →∞ x (k) = x ^′ _ave 1

, so this onsensus is within

ǫ

^from ^the ^desired

x ave

^:

|x ^′ _ave − x ave | ≤ ǫ.

⁽²⁴⁾

Thisendstheproof.

5.3 Message Reduing Algorithm

Wetry to solve the termination problem through a fully deentralized approah. We onsider

large-sale networks where nodes have limited resoures (in terms of power, proessing, and

memory), do notuse any global estimate(e.g. diameter of the network or number of nodes),

keeponlyoneiterationhistory,andanonlyommuniatewiththeirneighbors. Ourmaingoalis

thatthenumberofmessagesin thenetworkisaetedbyhowlosethenodesaretoonsensus.

Namely, the loser they are to onsensus, the less nodes transmit messages at eah iteration.

Thealgorithmmust guaranteethat thesystemreahesa onsensus within aertainpredened

marginfrom theaverage.

Themain ideais thata node,say

i

^forêxample,^willômpareîts^newâlulated^value^with

theoldone. Aordingto the hange in theestimate,

i

^will ^deide^either ^to ^broadast ^its^new

valueor not to doso. Wedivide an iterationinto two parts, in the rstpart ofthe iteration,

onlynodeswithsignianthangein theirestimatesareallowedto sendmessages. However,in

theseondpart of theiteration,onlynodespolledbytheirneighbors from phase1are allowed

to send an update. As a node approahes the onsensus, the hange in its estimate beomes

smaller, and more nodes will stop sending messages during an iterationof the algorithm, and

themessagesinthesystemwilldisappearasymptotially. Duringthesilentperiodofeahnode,

itaumulates anerrorausedbytheabseneof itsmessages. Whenthesilentnodetransmits

again,it an alwaysredue theaumulatederror by redistributingit in thenetworkand thus

savingmessages whilepreservingtheaverage.

Before starting the linear iterative equation, nodes will selet weights as in the standard

onsensus algorithm. The weight matrixonsidered here must be doubly stohasti with

0 <

α < w _ii < 1 − α < 1

^for^some ^onstant

α

^. ^Eah ^node

i

ⁱⁿ^the^network^keeps^two^state^values^at

iteration

k

^:

x i (k)

^: ^the^estimate^of^node

i

ûsedⁱⁿ ^theîterativeêquations^by^theôther^nodes.

e i (k)

^: â ^real ^value ^that ^monitors ^the ^shift ^from ^the âverage ^due ^to ^the îterations ^where

node

i

^did^not^sendâ^message ^toîts^neighbors. Îtîsînitially^set ^to^zero,

e i (0) = 0

^.

Eahnodealsokeeps itsownboundarythreshold

ǫ _i (k)

^where

ǫ _i (1) = ^ǫ ₂ = constant ∀i

^. ^Note

thatthisepsilonisinreasedaftereverytransmissionasintheentralizedase,butthedierene

hereisthat itisloaltoevery node.

Eahiterationisdividedintotwophases:

Intherstphase,anode

i

ân ^beⁱⁿ ôneôf^the^two ^following^states:

Transmit: Theset ofnodesorrespondingtothisstateis

T _k

^,^where^the^subindex

k

^orre-

spondsto the fat that theset an hangewith every iteration

k

^. ^The^nodes ⁱⁿ

T k

^send

theirnewalulatedestimatetotheirneighbors. Theyalsopollthenodeshavingmaximum

andminimumestimatesintheirneighborhood totransmitinphase2.

(14)

Algorithm 1TerminationAlgorithm-node

i

^-^Phase¹

1:

{x i (k), e i (k)}

^are ^the ^state ^values ^of ^node

i

^at ^iteration

k

^,

0 < α < w ii < 1 − α < 1

^,

counter i = 1

îs^theôunter^for^the^numberôftransmissionssofar.

ǫ i (1) = ǫ/2 ∈ R

,

T k

^is^set

of

T ransmit

^state.

W k

^set orrespondingto

W ait

^state. ^Initially ^we ^have

T k = W k = ∅

. Every node

i

^follows^the^followingâlgorithmâtîteration

k

^.

2:

y i (k + 1) ← w ii x i (k) + P

j∈N i w ij x j (k)

3:

d i ← y i (k + 1) − x i (k) + e i (k)

4: if

|d i | < ǫ i (counter i )

^then

5:

i

^hanges^to^a ^W^ait ^state.

\ \ i ∈ W k

6: else

7:

counter i = counter i + 1

8:

ǫ i (counter i ) = ǫ i (counter i − 1) + ǫ i (1)/counter _i ²

9:

c i ← ₍₁ ₋ ^α _w

ii )

³ y i (k + 1) − x i (k) ´

10: if

|c i | ≤ |e i (k)|

^then

11:

x i (k + 1) ← y i (k + 1) +

^sign

(c i .e i (k))c i

12:

e i (k + 1) ← e i (k) −

^sign

(c i .e i (k))c i

13: else

14:

x _i (k + 1) ← y _i (k + 1) + e _i (k)

15:

e i (k + 1) ← 0

16: endif

17:

i

^hanges^to^a ^T^ransmit ^state.

\ \ i ∈ T k

18: Notify theneighbors havingmaximumand minimumvalues.

19: endif

20: GotoPhase2

Wait: Thesetofnodesorrespondingtothisstateis

W k

^. ^The^node's^deision^will^be^taken

in the seond phaseof theiteration based on the ation of nodes in the Transmit state

(dependingiftheywerepolledbyanyoftheirneighbors).

Intheseondpartoftheiteration,nodesthatarein

W k

^will^be^lassied ^as^follows:

Silent: Theset of nodesorresponding to this stateis

S k

^. ^These^are ^the^nodes^that ^will

remainsilentwith nomessagesentfromtheirpart in thenetwork. Thenodesin

S k

^have

thatnonoftheirneighborssendingthemanypollmessage.

Cut-Balane: Theset of nodesorrespondingto this stateis

B k

^. ^They ^are^alled

Cut − Balance

^beause ^they însure ^the ût-balane ôndition ^(d) ôf ^theorem

2

^. ^They ^are ^the

nodesin

W k

^that ^have^been^polled^by^at^least^one^neighbor ⁱⁿ

T k

^.

The two phases of the termination protool implemented at eah node are desribed by

pseudoodeinAlgorithm1and2. Nodesinthe

T k

^set^(the^setôf^nodes^thatâreⁱⁿâ

T ransmit

state)willbroadasttheirestimatetotheirneighborsattheendoftherstphase,whilenodesin

W k

^set^(or

W ait

^state)^will^postpone^their^deision^to^send^or^not^till^the^next^phase. ^Nodes^that

donotreeiveamessage fromtheirneighbors ata ertainiteration,usesthelast seenestimate

fromthespeiedneighbors(note: abseneofmessagesfromaneighborduringaniterationdoes

not meanthe failureof link,it meansthat theneighbor is broadasting thesame oldestimate

as before, so we may dierentiate the link failure by a keep alive message sentfrequently to

maintainonnetivityand set ofneighbors). Theinput forthealgorithm are theestimatesof

the neighbor of

i

^, ^the ^weights ^seleted ^for^these ^neighbors, ^and ^the^state ^values

{x i (k), e i (k)}

^.

(15)

Algorithm 2TerminationAlgorithm-Phase 2

1:

{x i (k), e i (k)}

^are^the^state^values ^of^node

i

^at^iteration

k

^.

2: for allnodes

i

^having^Wait ^state^do

3:

y i (k + 1) ← w ii x i (k) + P

j ∈ N i w ij x j (k)

4: if

i

^reeived^a ^poll^message^from ^any^neighbor^then

5:

z i (k + 1) ← (w ii + P

j∈N i ∩W k w ij )x i (k) + P

j∈N i ∩T k w ij x j (k)

6:

x i (k + 1) ← z i (k + 1)

7:

e i (k + 1) ← y i (k + 1) − z i (k + 1) + e i (k)

8:

i

^hanges^to ^a

Cut − Balance

^state.

\ \ i ∈ B k

9: else

10:

x i (k + 1) ← x i (k)

11:

e i (k + 1) ← y i (k + 1) − x i (k) + e i (k)

12:

i

^hanges^to ^a^Silent ^state.

\ \ i ∈ S k

13: endif

14: endfor

15:

k + 1 ← k

Theoutputoftherstphaseisthenewstatevalues

{x i (k + 1), e i (k + 1)}

^for^nodesⁱⁿ

T k

^and

theoutput of theseond phase is thenew statevalues

{x _i (k + 1), e _i (k + 1)}

^for ^nodes ⁱⁿ

W _k

^.

Let us go through the lines of the algorithm. In phase 1,

y i (k + 1)

^of ^line

2

^is ^the ^weighted

average oftheestimates reeivedby node

i

^; ^without^the terminationprotoolthis valuewould be sent to all itsneighbors. The protool evaluateshowmuh

y i (k + 1)

^diers^from ^the^state

value

x i (k)

^. ^This^dierene^aumulatesⁱⁿ

d i

ⁱⁿ^line

3

^. Îf^this^shiftîs^less^thanâ^given^threshold

ǫ i

^,^the ^node^will ^wait ^for^next^phase^to ^take ^deision. ^If^the ^onditionⁱⁿ ^line

4

^is^not^satised,

thatmeansthenodewillsenda newvalueto itsneighbors. Lines

7 − 8

^onerns^the^extending

ofthe boundarythreshold

ǫ i (k)

^after ^every transmission. Note that bythis extensionmethod, wehave

ǫ i (k) < ǫ ∀i, k

^sine

k lim →∞ ǫ i (k) = ǫ i (1)(

∞

X

i=1

1/k ² )

< ǫ i (1) × 2

= ǫ.

⁽²⁵⁾

Weintrodueinline

9

^a^new^salar

c i

^used^for^deiding^whih^portion^of

e i (k)

^the^node^will^send

inthenetwork. Inlines

11 − 12

^and

14 − 15

^,^the^algorithm^satises^the^equation^{(18 ).} ^Then^the

newstatevalue

x i (k + 1)

^is^sent^to^the^neighbors^and

e i (k + 1)

îsûpdatedâordingly. În^Phase

2

ôf ^the âlgorithm ^(Algorithm^2), ^nodes înitially ⁱⁿ ^the ^wait ^state^will ^deide êither ^to ^send â

ut-balanemassageortoremainsilent,theutbalanemessagesaresentwhenanodereeives

apollmessagefromanyofitsneighbors.

5.4 Convergene study

Theonvergeneofthepreviousalgorithmismainlyduetothefatthattheproposedalgorithm

satisestheonditionsofonvergenegivenin5.2.2. Infat,thealgorithmisdesignedtosatisfy

alltheseonditions thatguaranteeonvergene. Startingwiththestateequation,weannotie

fromthealgorithm1giventhatwhatevertheonditionthenodesfae,itisalwaystruethatthe

sumofthenewgeneratedstatevalues

{x i (k + 1), e i (k + 1)}

^is^as ^follows:

x i (k + 1) + e i (k + 1) = y i (k + 1) + e i (k),

(16)

where

y i (k + 1) = w ii x i (k) + P

j ∈ N i w ij x j (k)

^. Âsâ^result^the^systemêquationîs^theône^studied

in setion5.2.2(equation(18)).

Before going through the dierent onditions in the theorem

2

^, ^we ^should ^also ^show ^that

e (k + 1) = e (k) − F(k) x (k)

^for ^some ^matrix

F (k)

^suh ^that

F(k) 1 = 0

. Aording to the algorithmwean write,

e i (k + 1) = e i (k) − v i (k),

⁽²⁶⁾

where

v i (k)

^diers^aording ^to ^the^state^of^the ^node

i

^, ^but îtônly^depends ôn^theêstimateôf

node

i

^and^its^neighbors:

v i (k) =



 



 



± ₍₁ ₋ ^α _w

ii )

³ y i (k + 1) − x i (k) ´

if

i ∈ T k − (1),

± ₍₁ ₋ ^αγ _w

ii )

³ y i (k + 1) − x i (k) ´

if

i ∈ T k − (2), x i (k) − y i (k + 1)

^if

i ∈ S k , z i (k + 1) − y i (k + 1)

^if

i ∈ B k ,

(27)

where

T k − (1)

^is ^the^set ^of ^nodes ^subset ⁱⁿ

T k

^where

|c i | ≤ |e i (k)|

^, ^and

T k − (2)

^set ^of ^nodes

where

|c i | > |e i (k)|

^. ^In ^the^latter^ase,

e i (k + 1) = 0

^, ^but ^we^an ^always ^nd

γ < 1

^suh ^that

e i (k + 1) = e i (k) − γ(

^sign

(c i .e i (k)c i ) = 0

^where

c i = ₍₁ ₋ ^α _w

ii ) (y i (k + 1) − x i (k))

^.

y i (k + 1)

^and

z i (k + 1)

âreâs îndiatedⁱⁿ^theâlgorithmândâreâ ^linearômbinationôf^theêlementsôf

x (k)

^.

From theequation of

v _i (k)

^, ^weân âlso ^see ^that ît îs â ^linear ômbination ôf ^the êlements ôf

x (k)

^,^suh^that^the^oeients^sum^to

0

^. ^A^row

i

ⁱⁿ

F (k)

^will^be^theôeientsôf^theêstimates

x (k)

ⁱⁿ

v i (k)

^,^so

F (k) 1 = 0

.

Nowweanstudytheonditionsmentionedinthetheorem

2

^on^the^matrix

A(k) = W +F (k)

^.

Lemma 1.

A(k)

îsâ^stohasti^matrix^that ^satisesônditions(a),(b),and() ofTheorem

2

^.

Proof. First,weanseethat

A(k) 1 = 1

sine

W 1 = 1

and

F(k) 1 = 0

. Itremainstoprovethat allentries inthematrix

A(k)

^are^non^negative.

Wewillprovethisby onsideringeah row

i

^of

A(k)

^aording ^to ^the^ation^taken^by^node

i

^. ^We^andistinguishfourases:

1. Node

i ∈ T _k

^-^ondition^1:

|c _i | ≤ |e _i (k)|

a ii = w ii − ₁ ₋ ^α _w

ii × (1 − w ii ) = w ii − α > 0

^sine

w ii > α

and

a ij = w ij + ₁ ₋ ^α _w

ii × w ij ≥ w ij > 0 ∀j ∈ N i

^.

or

a ii = w ii + ₁ ₋ ^α _w

ii × (1 − w ii ) > α

^sine

w ii > α

and

a ij = w ij − ₁ ₋ ^α _w

ii × w ij > 0 ∀j ∈ N i

^sine

α < w ii < 1 − α

^.

2. Node

i ∈ T k

^-^ondition^2:

|c i | > |e i (k)|

sine

|c i | > |e i (k)|

^,^we^an^always^nd^positive

γ < 1

^,^suh^that

a ii = w ii − ₁ ₋ ^γα _w

ii × (1 − w ii ) = w ii − γα > 0

^sine

w ii > α

and

a ij = w ij + ₁₋ ^γα _w

ii × w ij ≥ w ij > 0 ∀j ∈ N i

^.

or

a ii = w ii + ₍₁₋ ^γα _w _ii ₎ × (1 − w ii ) > α

^sine

w ii > α

and

a ij = w ij − ₁ ₋ ^γα _w

ii × w ij > (1 − ^α _α )w ij > 0 ∀j ∈ N i

^.

3. Node

i ∈ S k

^:

then

a ii = w ii + (1 − w ii ) = 1

and

a ij = 0 ∀j 6= i

^.

(17)

4. Node

i ∈ B k

^:

then

a ii ≥ w ii > 0

and

a ij ∈ {w ij , 0} ∀j 6= i

^.

Therefore,

A(k)

îs^stohastiâtêveryîteration

k

^.

Denition1. Twomatries,AandB,aresaidtobeequivalentwithrespettoavetor

v

ifand onlyif

A v = B v

.

Notiethat

A(k)

^satisesônditions^(a),(b), ând⁽⁾ôf ^Theorem

2

^,^but ^possibly^not^the^ut

balaneondition(d)beausefor anode

i ∈ T _k

^that ^transmits,

a _ij (k) > 0 ∀ j ∈ N _i

^, ^but^it^an

bethat

∃j ∈ N _i

^suh ^that

a _ji = 0

^if

j

^was ^silent^at^that ^iteration⁽

j ∈ S _k

^). ^However,^the^next

theoremshowsthatthereisamatrix

B(k)

^equivalent^to

A(k)

^with^respet^to

x (k)

^that^satises

alltheonditions.

Lemma 2. For all

k

^, ^there ^exists ^a^matrix

B(k)

^equivalent ^to

A(k)

^with ^respet ^to

x (k)

^suh

that

B(k)

^satises^the ^onditions(a),(b),(),and(d)of Theorem

2

^.

Proof. Let

m(k) = argmin _j ∈ N i x j (k)

^and

M (k) = argmax _j ∈ N i x j (k)

^,^we^will^proof^the^existene

of

B(k)

^by^modifying

A(k)

ⁱⁿ^suhâ^way^to^preserve^the^properties^(a)^to⁽⁾ând^toâdd^the^new

property(d). For simpliity ofnotationwewill drop

k

^from ^the^variables^sine ^this^is ^true^for

every

k

^. ^Note^rst^that ^the^ondition^(d)^is^not^satisedⁱⁿ

A

^only^for^the^rows^where

i

^belongs

to

T k

^, ^let

a _i

denotethe row

i

^of

A

^and

b _i

denotetherow

i

^of

B

^. ^Sine ^any^node

i

ⁱⁿ

T k

^must

pollthenodes

m

^and

M

^to^transmitⁱⁿ^Phase ^2,^then^we^are^sure^that^the^olumn

i

^has^at^least

threenonzeroelements(

a ii , a mi ,

^and

a M i

^). ^Let

C i = {j | a ji > 0 , i 6= j}

^, ^so^we^are ^sure^that

C i

ôntainsât^least^two êlements

m

^and

M

^. ^Theût^balaneôndition^requires^that ^the^rowôf

i

^mustônly^have^positive^values ât^theîndex^where^theôlumnîs^positive. ^Weân ^write^that

a _i x (k) = a ii x i (k) + X

j ∈ N i

a ij x j (k)

= a ii x i (k) + X

j ∈ C i

a ij x j (k) + X

j ∈ N i − C i

a ij x j (k)

= a ii x i (k) + X

j∈C i

a ij x j (k) + hx M (k) + f x m (k),

⁽²⁸⁾

where

h =

P

j∈ Ni − Ci a ij (x j − x m ) x M − x m

,

f =

P

j∈ Ni − Ci a ij (x M − x j ) x M − x m

. Sine

h ≥ 0

^,

f ≥ 0

^and

f + h = P

j ∈ N i − C i a ij

^,^we^an^dene^the^row^vetor

b _i

tobe:

b _i :=



 



 



b ij = a ij

^if

j = i, b ij = a ij + f

^if

j = m, b ij = a ij + h

^if

j = M,

b ij = a ij

^if

j ∈ C i − m − M, b ij = 0

^if

j ∈ N i − C i .

(29)

Finally,

b _i x (k) = a _i x (k)

ând ît^satises ^theônditions ^(a) ^to ^(), ^so

∀i ∈ T k

^, ^we^replae

a _i

by

b _i

andwegetthenewmatrix

B

^whih^is^equivalent^to

A

^with^respet^to

x (k)

^.

Lemma3. Themessageredutionalgorithm(Phases1,2)satisesondition(e)ofTheorem

2

^.

(18)

Proof. Wewill proveit byontradition. Suppose

lim k →∞ x (k) = x ^⋆

, but

lim k →∞ F (k) x (k) 6=

0

, then there exists a node

i

^suh ^that

lim k →∞ x i (k) = x ^⋆ _i

^and

lim k →∞ y i (k + 1) = w ii x ^⋆ _i + P

j ∈ N i w ij x ^⋆ _j = y ^⋆ _i

^, ^but

y ^⋆ _i − x ^⋆ _i = δ ^⋆ > 0

^. ^From ^Algorithm ^{1 ,} ^we ^an ^see ^that ^the ^node ^will

enteratransmitstateinnitelyoften(beause

d i

^inreases ^linearly^with

δ ^∗

^and^it^will^reah^the

threshold

ǫ i

^). ^Then, ^the^node

i

^willûpdateîtsêstimateâording^to^theêquation

x i (k + 1) = y i (k + 1) + α

1 − w ii

(y i (k + 1) − x i (k)).

Letting

k → ∞

^yields

(1 + α

1 − w _ii )δ ^⋆ = 0.

Thus,

δ ^⋆ = 0

^whih^is^aontradition,andthealgorithmsatisesondition(e)ofTheorem2.

Thealgorithmalsoprovidesthat

|e i (k)| ≤ ǫ i (k) ∀k, i

^and

ǫ i (k) < ǫ ∀i, k

^,^as ⁱⁿ^the^rst^phase

thisonditionissatisedbyonstrution,andfortheseond phaseoftheiteration,nodesfrom

Phase

1

ân ^hek ^for^worst âseânalysis ând ^theyônly ênter înto ^Wait ^stateîf^they âre^sure

that theonditionanbesatisedinthenextphaseiteration.

Nowwearereadytostatethemain Theoreminthissetion:

Theorem 3. The nodes applying the message reduing algorithm given in pseudoode by Al-

gorithm 1 and 2, have estimates onverging to a onsensus within a margin

ǫ

^from

x ave

^, ^i.e.

lim k →∞ x (k) = x ^′ _ave 1

and

|x ^′ _ave − x ave | ≤ ǫ.

Proof. Thetheorem isdue to thefat that theLemmasgiven in thissubsetion show thatthe

algorithmsatisesalltheonvergeneonditions ofTheorem

2

^.

5.5 Asymptoti Termination

Wehaveprovedintheprevioussetionthatthealgorithmproposedisonvergingtoaonsensus.

We will usethis to show that any node

i

^an ^inrease^the^boundary^threshold

ǫ i (k)

^at ^ertain

times

k s , s = 0, 1, 2, ...

^(i.e.

ǫ i (k s ) = ǫ i (k s −1 ) + ǫ(0)/s ²

⁾^suh^that ^the^silent^period ^of^this^node

(

α _i (s)

^,^theônseutive^numberôfîterations^thatâ^node

i

^did^not^send^any^message)^is^growing

toinnity. Thesystemequationwehadfollowingourmessage savingalgorithmisasfollows:

x (k + 1) = A(k) x (k),

⁽³⁰⁾

whereA(k)was astohastimatrix.

LetusdeneforamatrixAtheproperoeientofergodiity

τ(A)

^to^study^the^onvergene

rateofthesystem(formoreinformationonproper oeientsofergodiitysee[16 ℄),

τ(A) = 1 2 max

i,j n

X

s=1

|a is − a js |.

⁽³¹⁾

Letalso

U p,r

^be^the

r

^-step^bakward^produt^of^the^matrix^A(k):

U p,r = A(p + r)...A(p + 2)A(p + 1).

⁽³²⁾

Theorem 4. Ergodiity of bakwards produts

U p,r

^formed^from ^a^given ^sequene

A(k), k ≥ 1

^,

obtainsifandonlyifthereisastritlyinreasingsequeneofpositiveintegers

{k s }, s = 0, 1, 2, ...

suhthat

∞

X

s=0

{1 − τ (U k s ,k s+1 − k s )} = ∞.

⁽³³⁾

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Proof. [16 ,p.155℄.

Sineoursystemisergodi(onverges)asweprovedinprevioussetion,thistheoremimplies

that foranypositiveonstant

δ < 1

^we^have â ^stritly înreasing^sequene ôf^positiveîntegers

{k s }, s = 0, 1, 2, ...

^suh^that

τ(U k s ,k s+1 − k s ) ≤ δ.

⁽³⁴⁾

By theinrementofthethreshold

ǫ i (k)

^at^instanes

k s

^,^we^have^that ^the^silent^period

α i (s)

ofa node

i

îs^the ^number ôfîterations^node

i

îs ^silent âfter îts^last înrement ât ^time

k s

^. ^The

silentperiod satisesthefollowingequation:

α i (s) ≥ ǫ

s ² ¡

W x (k s ) − x (k s ) ¢

i

,

⁽³⁵⁾

let

α(s)

^be^the^minimum^aross ^all^loal

α i (s)

^,^we^have:

α(s) ≥ ǫ

s ² ||W x (k s ) − x (k s )|| ∞

.

⁽³⁶⁾

Sine

W

îsâ^stohasti^matrix,^thenâll^the^valuesôf^the^vetor

W x (k _s )

^belong^to^the^interval

[min i x i (k s ), max i x i (k s )]

^,^so^we^have,

||W x (k s ) − x (k s )|| ∞ ≤ max

i,j |x i (k s ) − x j (k s )|,

⁽³⁷⁾

butfromthedenition oftheoeientofergodiitywehavethat

max i,j |x _i (k _s ) − x _j (k _s )| ≤ τ(U _k _s−1 _,k _s − k s−1 )×

{max i,j |x i (k s − 1 ) − x j (k s − 1 )|},

⁽³⁸⁾

andweanonlude thatthesilentperiod isgrowingexponentiallyin

s

^:

α(s) ≥ ǫ

Cs ² δ ^s ,

⁽³⁹⁾

where

C

îs^justâ ônstant^dependsôn^theînitial^state^vetor

x (0)

^.

Andtheasymptotiterminationisobtained:

s lim →∞ α(s) = ∞.

⁽⁴⁰⁾

Eventhoughnodesarenotawareofthesetime

k s

^where^they^have^to^inrement^the^threshold,

withaninrementation

ǫ i (m)

^at^times

m

^,^the ^system^is^very ^robust^is ^the^sense ^that ^nodes^do

nothavetobesynhronizedanderrorsin wrongtimeinrementation analsobetolerated and

donotaet theonvergeneas longas

m ² = o(δ ^s )

^,^where

o(.)

^is^the^small-oh^notation⁽ⁱⁿ^the

studyofonvergene

m

^was^equal^to

s

^). ^In^thesimulationsandasshowninthealgorithm1,we inremented

m

⁽

counter i

ⁱⁿ ^theâlgorithm)^whenever â^nodeîs ^going^to ^transmitât ^phaseône,

andtheresultsaresatisfatory.

5.6 Simulations

To simulate the asymptoti termination of the algorithm desribed above, we onsidered two

typesof graphs with

n = 50

^, ^the ^Random ^Geometri ^Graphs^(RGG) ^with ^onnetivity ^radius

r = 0.234

^, ând ^the Êrdos ^Renyi ^(ER) ^with âverage ^degree

4

^. Âll ^the ^graphs ônsidered âre

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0 1000 2000 3000 4000 5000

−6

−5

−4

−3

−2

−1 0

iteration number

log(normalized error)

RGG n=50 r=0.234

0 1000 2000 3000 4000 5000

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5 0

iteration number

log(normalized error)

ER n=50 average degree=4

Figure3: Normalized erroronRGG andERgraphs.

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0 1000 2000 3000 4000 5000 0

5 10 15 20 25 30 35 40 45 50

iteration number

number of nodes transmiting

RGG n=50 r=0.234

0 1000 2000 3000 4000 5000

0 5 10 15 20 25 30 35 40 45 50

iteration number

number of nodes transmitting

ER n=50 average degree=4

Figure4: Messagessentwithevery iteration.