Average Consensus with Asynchronous Updates and Unreliable Communication (with proofs)

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1 Average Consensus wt Asyncronous Updates and Unrelable Communcaton (wt proofs) Ncoletta Bof Ruggero Carl Luca Scenato All autors are wt te Department of Informaton Engneerng, Unversty of Padova, Va Gradengo 6/a,35131 Padova, Italy. Abstract: In ts work we ntroduce an algortm for dstrbuted average consensus wc s able to deal wt asyncronous and unrelable communcaton systems. It s nspred by two algortms for average consensus already present n te lterature, one wc deals wt asyncronous but relable communcaton and te oter wc deals wt unrelable but syncronous communcaton. We sow tat te proposed algortm s exponentally convergent under mld assumptons regardng te nodes update frequency and te lnk falures. Te teoretcal results are complemented wt numercal smulatons. Keywords: Mult-agent systems, Dstrbuted control and estmaton, Control and estmaton wt data loss, Asyncronous algortms, Average consensus 1. INTRODUCTION In recent years, substantal effort as been dedcated towards te desgn of dstrbuted algortms for large-scale systems. Te man drver for ts s tat, nowadays, te avalablty of small and ceap computatonal unts s becomng wdespread. As a consequence, t s affordable to develop extended systems to montor and control many dfferent envronments. However, due to te sze of ts systems, t s not always possble to collect all te nformaton n a sngle computatonal unt and sometmes t s neter advsable, snce some of te nformaton collected by te system could be sensble. Moreover, eac unt s endowed wt some computatonal power wc wll not be fully exploted oterwse. Anoter addtonal advantage of usng a dstrbuted approac s tat te wole system s n a way safer, snce t does not rely on a sngle unt but s assgned to many dfferent ones. Dstrbuted algortms present two maor drawbacks: memory and computatonal constrants and need of a relable communcaton system. Te frst s due to te fact tat te unts wc compose te system can be qute lmted n ter computatonal and storage capabltes, te second s ntrnsc to te dstrbuted settng, snce eac unt can excange nformaton only wt ts negbours n order to perform a global task. It s terefore fundamental to always consder te propertes of te communcaton system adopted. One very mportant caracterstc s te communcaton protocol, wc can be syncronous or asyncronous. Syncronous protocols requre substantal coordnaton between te nodes, and wen te number of agents n te system ncreases, ts coordnaton can become dffcult to aceve. An asyncronous communcaton protocol, on te oter and, as no coordnaton requrements, but an algortm wc uses suc a protocol could n general requre more teratons because n eac teraton of te algortm only a subset of te nodes n te networks are actvated. Anoter mportant feature of te communcaton system s ts relablty due to te possblty tat packets are lost durng te transmsson. Obvously, te system sould not lose packets but perfect relablty could be dffcult or too expensve to enforce; to deal wt possble packet losses, eter an acknowledgement sceme s developed or te algortm s mplemented n suc a way tat te loss of a packet s not detrmental for te convergence. In ts paper we descrbe and study a dstrbuted algortm for average consensus. Bascally, eac unt as a gven scalar quantty and te am for eac node s to compute te average of all tese quanttes. Sensor networks represent a remarkable doman were te evaluaton of te average of te measured quanttes s requred n several applcatons Xao et al. (2005), Bolognan et al. (2010), Garn and Scenato (2010). However, dfferently from te rc lterature on ts topc, we adopt an asyncronous and unrelable communcaton system, and we allow te communcaton not to be b-drectonal, tat s f a unt communcates wt anoter one, te converse s not assured. In a syncronous and relable communcaton scenaro, mportant works are Boyd et al. (2004), Olsevsky and Tstskls (2009), Oreskn et al. (2010), Domínguez-García and Hadcosts (2011). Wen unrelablty n te communcaton s ntroduced, some works ave adopted te acknowledgement sceme Cen et al. (2010) or assumed tat eac unt can determne weter te communcaton works Patterson et al. (2007), Xao et al. (2005). However, an acknowledgement sceme requres addtonal secondary transmssons, wc slow down te entre algortm and consume extra energy. Terefore, n context were te energy consumpton s constraned, te latter sceme s not adoptable and te transmsson as to be reduced only to essental nformaton. In an asyncronous settng, Bénézt et al. (2010) ntroduce an algortm tat reaces

2 average consensus usng te so-called rato consensus. A very nterestng dea s ntroduced n Domnguez-Garca et al. (2011) and Vadya et al. (2011) were te adopted communcaton s syncronous and unrelable. In tese works a robust and syncronous algortm nspred by Bénézt et al. (2010) s ntroduced. Adoptng te dea of mass transfer gven n Vadya et al. (2011), but developng an asyncronous algortm as done n Bénézt et al. (2010), we descrbe an algortm for average consensus wc s provably convergent to te average n an asyncronous and unrelable communcaton scenaro. Te convergence proof reles on te ntroducton of two assumptons concernng te communcaton sceme, one regardng te frequency of wakng up of eac node and te oter regardng ow many consecutve tmes a gven lnk can fal. Our nterest n ts algortm s ustfed by ts possble executon n more complex algortms. In fact, even toug t s an nterestng stand-alone algortm, tere are some algortms wc need average consensus algortm as a buldng block, e.g. te Newton-Rapson algortm for convex optmzaton (see Varagnolo et al. (2016)), some dstrbuted versons of te Kalman flter (see Cattvell and Sayed (2010)) or some algortms for energy resources dstrbuton n power grds (see Domnguez-Garca and Hadcosts (2010)). However, to be used n suc algortms, te average consensus as to be exponentally convergent. Te aforementoned works by Bénézt et al. (2010) and Vadya et al. (2011), for example, do not sow weter exponental convergence s guaranteed n ter algortms By provng ts knd of convergence for our algortm, we make possble to use t n more advanced algortms wc could ten be appled n a realstc communcaton scenaros (.e. asyncronous and unrelable cannels). 2. NOTATION & COMMUNICATION PROTOCOLS Gven a scalar x R, x denotes ts absolute value. Gven a matrx A R N N, [A] denotes ts (, ) t element, and A ndcate ts transpose. A vector x s strctly postve f x > 0, {1,..., N}. Gven two vectors x, z R N, wt x or [x] we denote ts t element and wt x/z te Hadamard dvson of te two vectors. I N ndcates te N N dentty matrx. A grap G s represented by te couple (V, E), wt V te set of nodes {1,..., N} and E V V te set of edges. Te number of edges n te grap s E. We consder drected, strongly connected and statc graps, wt all te nodes avng a self loop. Gven a node V, te set N n contans all te negbours wc communcate to, tat s N n = { V,, (, ) E}, wle te set N out contans all te negbours to wc communcates, tat s N out = { V,, (, ) E}. For a set A V, A denotes te cardnalty of te set. A matrx P R N N s row stocastc f P 1 N = 1 N, were 1 N s te all-ones vector of dmenson N. Fnally, f a, b R, a < b, wt [a, b] we ndcate te nterval between a and b, extremes ncluded. In te followng we brefly descrbe some of te communcaton protocols wc are usually adopted n wreless sensors networks gven a pre-assgned communcaton grap, namely, te syncronous protocols, as opposed to te broadcast asymmetrc coordnated broadcast asymmetrc gossp gossp symmetrc Fg. 1. Asyncronous communcaton protocols: te opaque red node s te one tat wakes up (for te frst 3 protocols), te oter glgted nodes are tose wc excange nformaton wt t. In gossp symmetrc tere s no erarcy n te nodes selected. asyncronous ones lke broadcast asymmetrc, coordnated broadcast, gossp asymmetrc and gossp symmetrc. In a syncronous protocol all te nodes actvate at te same tme nstants and perform te updatng and communcaton operatons (almost) syncronously. Ts protocol requres a common noton of tme among te nodes; ndeed all te agents ave to wake up smultaneously and so a perfect coordnaton s requred. If te grap s moderately small, requrng syncronzaton may not be a great deal, but as te dmenson of te network ncreases, syncronzaton can become an ssue. Instead, n asyncronous protocols, durng eac teraton, only a small subsets of all te nodes n te network perform te communcaton and updatng steps. Specfcally, n te broadcast asymmetrc protocol, at eac teraton, tere s only one node transmttng nformaton to ts out-negbours, wc, based on te receved messages, update ter nternal varables. At a gven teraton, te (unque) transmtter node s sad to be te one tat wakes up (or turns on). Te same termnology s appled to te node tat performs te frst step of te communcaton n te two protocols we descrbe next. Te coordnate broadcast can be consdered as te dual protocol of te broadcast asymmetrc. Indeed, at eac teraton, tere s only one node wc wakes up, but, nstead of sendng nformaton, t polls all ts n-negbours n order to receve from tem some desred messages. In asymmetrc gossp agan only one node wakes up but t sends nformaton to only one of ts out-negbours, typcally randomly cosen. Fnally, te symmetrc gossp s a protocol tat requres bdrectonal communcaton, tat s te communcaton grap G as to be undrected (mplyng Nn = N out for all V); durng eac teraton an edge of te grap s selected and only te two nodes wc are ponted by ts edge excange nformaton wt eac oter. Fgure 1 gves a pctoral descrpton of te asyncronous protocols ust descrbed. 3. PROBLEM FORMULATION Consder N agents (also called nodes) wc can communcate wt eac oter accordng to a grap G and trougout some asyncronous communcaton protocol. We assume te communcatons to be unrelable, tat s, some packet losses mgt occur durng te transmssons of te messages. Eac node {1,..., N} as a prvate scalar quantty 1 v R, wc can be collected n vector v R N, and te problem to solve s te evaluaton of te mean of tese v, tat s of v = v /N. Te evaluaton as to be carred out by eac node n a dstrbuted way, 1 Te algortm can be modfed to manage multdmensonal quanttes.

3 and te nodes can excange nformaton among temselves only f tey are negbours n te grap G. Formally, denoted by x (k) te estmate of te mean v stored n memory by node at tme k, and ntroducng te vector x(k) = [x 1 (k),..., x N (k)] T s to develop an algortm suc tat lm x (k) = v, {1,..., N} k R N, te problem lm x(k) = v1 N k and suc tat te update of x (k) depends only on quanttes tat belong to te negbours of node n Nn. Dstrbuted algortms to solve te gven problem already exsts f te communcaton s syncronous, and te real callenge we want to face s represented by te fact tat te communcaton s asyncronous and not relable. In ts paper we assume te agents communcate wt eac oter troug a broadcast asymmetrc communcaton protocol; owever te analyss we propose n next sectons can be adapted to te oter communcaton protocols we ave descrbed n te prevous Secton. Due to te communcaton scenaro, some assumptons are needed concernng ow many tmes a gven node wakes up and ow unrelable te communcaton s. We make te followng (determnstc) assumptons Assumpton 1. (Communcatons are persstent). For any tme nstant k N tere exsts a postve nteger τ suc tat eac node performs at least one broadcast transmsson wtn te nterval [k, k + τ]. Assumpton 2. (Packet losses are bounded). Tere exsts a postve nteger L suc tat te number of consecutve communcaton falures over every drected edge n te communcaton grap s smaller tan L. A drect consequence of tese assumptons s tat, consderng any nstant k 0, n te nterval k, k + 1,..., k + L( τ + 1) 1 eac lnk of te grap G s successfully used at least once. 4. ASYNCHRONOUS AND ROBUST CONSENSUS Te asyncronous and robust Average Consensus algortm (ra-ac ) we present takes nspraton from te algortm presented n Vadya et al. (2011) but under asyncronous communcaton. In partcular we adopt a broadcast asymmetrc communcaton protocol, tat s we only allow one node and, n a second moment, all ts out-negbours tat receve nformaton, to update part of ter varables at eac teraton. In Vadya et al. (2011), nstead, all te nodes at eac teraton perform some computatons. Tanks to our assumptons on te communcaton, we are able to prove tat te convergence of te algortm s exponental. As te algortm n Vadya et al. (2011), also te ra- AC algortm s based on te average rato consensus ntroduced n Bénézt et al. (2010). Accordng to te rato consensus, varable x R of node tat reaces consensus on te mean of te vector v s obtaned as te rato of two approprate scalar quanttes y and s ; te update of y and s are made by node as a lnear combnaton of ts own varable and of te companon varables of ts negbours. However, dfferently from Bénézt et al. (2010)), were te communcatons were assumed to be relable, n our communcaton scenaro te packets excanged between two nodes can be lost. In ts case, we Algortm 1 ra-ac (tme k, node wakes up) 1: % Node updates ts varables 2: y (k + 1) = y (k) N out +1; 3: s (k + 1) = s (k) N out +1; 4: x (k + 1) = y (k+1) 5: σ (y) 6: σ (s) s (k+1) ; (k + 1) = σ(y) (k) + y (k + 1); (k + 1) = σ(s) (k) + s (k + 1); 7: % Node broadcasts varable σ (y) (k + 1) and σ(s) (k + 1) to all Nout 8: f node receves σ (y) (k + 1) and σ(s) (k + 1) ten 9: y (k + 1) = σ (y) (k + 1) ρ(,y) (k) + y (k); 10: s (k + 1) = σ (s) (k + 1) ρ(,s) (k) + s (k); 11: x (k + 1) = y(k+1) s ; (k+1) 12: ρ (,y) 13: ρ (,s) (k + 1) = σ (y) (k + 1); (k + 1) = σ (s) (k + 1); 14: end f 15: % Te varables of te oter nodes are not canged need to ensure tat all te nformaton sent by node to ts negbour s receved by at least every once n a wle. Te remarkable dea tat allows to meet ts requrement s tat of ntroducng te use of counters: n partcular node as a counter σ (y) (k) (σ (s) (k) respectvely) to keep track of te total y-mass (total s-mass) 2 sent by tself to ts negbours from tme 0 to tme k, wle node as a counter ρ (,y) (k) (ρ (,s) (k) resp.) to take nto account te total y-mass (total s-mass) receved from ts negbour from tme 0 to tme k (one suc varable for all N n ). In ts way, f at tme k node receves nformaton from node, te nformaton comng from node used n te update of te varable y (k) (s (k) resp.) wll be σ (y) (k) ρ (,y) (k) (σ (s) (k) ρ (,s) (k) resp.); n ts way te nformaton sent by an agent but not receved due to packet losses s only delayed and not lost. We ave nerted te dea of usng counters from te algortm n Vadya et al. (2011). Our ra-ac algortm, takng nspraton from te latter deas, carres out a rato consensus accordng to an asyncronous communcaton protocol, and te generc k-t teraton s descrbed n Algortm 1. To be mplemented, eac node {1,..., N} n te network as to keep n memory te followng scalar quanttes: y (k), s (k), σ (y) (k), σ (s) (k) and ρ (,y) (k), ρ (,s) (k), (, ) E, wle te quantty of nterest x (k) s evaluated as y (k)/s (k). We collect varables y (k) and s (k) resp. n te N-dmensonal vectors y(k) and s(k). Suppose tat at a gven teraton node wakes up. Ten, te man steps of ra-ac are te followng: frst node updates ts varables y and s dvdng ter prevous value by te cardnalty of ts out-negbours set augmented by 1 (steps 2-3). Note tat ts operaton leaves n fact uncanged te value of varable x. Ten t updates te 2 As n Vadya et al. (2011), we ntercangeably use te word mass for nformaton, snce te pyscal dea of te transferrng of mass quanttes can be elpful n understandng ow te algortm works.

4 counters σ (y) and σ (s) (steps 5-6) and sends tese updated values to ts out-negbours. Now, f node N out receves te packet from node, t updates te varables y and s as descrbed n steps 10-11, ten t adourns x and t fnally stores n memory te new values for ρ (,y) and ρ (,s) (steps ). Te followng Teorem sows tat, wt a proper ntalzaton of te varables, te ra-ac algortm works as an average consensus algortm, tat s, te varables x, {1,..., N}, wc are updated dstrbutvely and teratvely, converge to te average of te N components of te vector v. Teorem 3. Under Assumptons 1 and 2 and under te followng ntalzaton for te varables y(0) = v, s(0) = 1 N, σ (y) (0) = σ (s) (0) = 0, {1,..., N}, ρ (,y) (0) = ρ (,s) (0) = 0, (, ) E, te evoluton, obtaned usng ra-ac algortm, of te varable x(k) exponentally converges to v1 N, v = v 1 N /N, tat s, tere exst sutable constants C > 0, 0 < d < 1 suc tat ( ) k x(k) v1 N 2 C d 1 τ x(0) v1n 2, (1) were τ = NL( τ + 1). Snce te proof of te Teorem s qute nvolved and conssts of several steps we postpone t n te next Secton. Te bound n (1) depends on our communcaton scenaro troug τ. For a fxed number of nodes N, τ mgt ncreases, eter because eac node wakes up less often, or because eac communcaton lnk may fal for a longer perod of tme (or bot), wc mples tat te dssemnaton of nformaton may become more dffcult. In Equaton (1), f τ ncreases te upper bound becomes larger and larger, wc s coerent wt te fact tat te nformaton s spread troug te network n a slower way. Remark 4. If no packet losses occur, te varables σ (y) (k), σ (s) (k), ρ (,y) (k) and ρ (,s) (k) can be dscarded (and varables y and s of te nodes tat receve te nformaton are updated drectly usng te packets tey receve). Te algortm we obtan n ts case s subsumed by tose presented n (Bénézt et al., 2010). 5. PROOF OF CONVERGENCE Te proof of Teorem 3 s based on te teory of ergodc coeffcents for postve matrces Seneta (2006), appled to te partcular case of stocastc matrces. In te frst part, we follow wat s done n Vadya et al. (2011). Te Assumptons 1 and 2 allow us to state te results n Vadya et al. (2011) wtout resortng to probablty teory. However, we explot te ergodcty teory n a way suc tat t allow us to conclude te exponental convergence of te algortm. To proceed wt te proof, we frst rewrte te algortm teraton n a matrx form, ten we study te property of te matrces nvolved and we fnally explot ergodcty teory to prove te convergence of te algortm. Matrx form for ra-ac We start by ntroducng te ndcator varables χ (k) and χ (,) (k),, {1,..., N}. Te varable χ (k) s equal to 1 f node wakes up at tme k, oterwse s 0; at ts regard, recall tat snce we adopt a broadcast asymmetrc protocol only one node turns on at eac teraton. Concernng χ (,) (k), te varable s 1 f node wakes up at tme k, f Nout and f te edge (, ) E s relable at tme k, wle t s 0 oterwse. Formally { 1 f node wakes up at tme k χ (k) = (2) 0 oterwse and χ (,) (k)= { 1 f χ (k)=1, (,) E actve at tme k 0 oterwse 3 (3) Observe tat N =1 χ (k) = 1. Now, we only analyze te matrces wc descrbe te evoluton of varable y(k), snce te same matrces drve te evoluton of varable s(k). Usng te ndcator varables we can rewrte te update for te total sent-mass counter σ (y) total receved-mass counter ρ (,y) (k) as ρ (,y) σ (y) (k + 1) = σ (y) (k + 1) = ρ (,y) (k) ( χ (k)χ (,) (k) Let us ntroduce varables (,) y (k) (k) + χ (k) Nout + 1 ρ (,y) (k) and for te ) (k) σ (y) (k + 1) (k) = σ(y) (k) ρ (,y) (k), (, ) E. Tese varables ndcate ow muc of te mass sent by node s stll to be receved by node. If at tme k 1 node turns on and te communcaton between node and (were s a negbour of ) s successful, ten (,)(k) s 0, oterwse t contans te nformaton mssng n node. Usng equatons (4) and (5) te update of tese varables can be wrtten as (,) (k + 1)=[ 1 χ (k)χ (,) (k) ][ ] σ (y) (k + 1) ρ (,y) (k) = [ 1 χ (k)χ (,) (k) ] [ ] y (k) χ (k) Nout ν(y) (,) (k). (6) We wll explot tese varables to rewrte te update of vector y(k) n a matrx form. Note tat tese quanttes are not actually computed by te nodes, and are ust auxlary varables used to enable te matrx verson of te update. (4) (5) To rewrte te update for te y (k) varable, we ave to consder tree dfferent cases: f χ (k) = 1, t olds y (k + 1) = y(k) N out +1; f χ (k) = 1 and N n and χ (,)(k) = 1, ten y (k + 1) = σ (y) (k + 1) ρ (,y) (k) + y (k) = (,) (k) + y (k) N out y (k); f χ (k) = 1 and N n and χ (,)(k) = 0 or f / N ten t olds y (k + 1) = y (k). 3 χ (,) (k) s consdered dentcally 0 for all k, f (, ) / E n,

5 Te above tree cases are all captured by te followng update y (k) y (k + 1) = χ (k) Nout [1 χ (k)] [ ( )] χ (k)χ (,) (k) (,) (k)+ y (k) N out +1 +y (k). (7) Now let us ntroduce te column vector (k) = [ (,) (k)] RE, wc collects all dfferent (,)(k). Moreover let us defne te row vector φ (y) (k) = [y(k) (k) ] R N+E. Our am s to fnd matrx M(k) R (N+E) (N+E) accordng to wc t olds φ (y) (k + 1) = φ (y) (k)m(k). (8) Let us frst consder te t row of matrx M(k), wt {1,..., N}. Te element [M(k)] ndcates ow y (k) nfluences y (k + 1), so [M(k)] = χ (k) N out [1 χ (k)]. Te element [M(k)], {1,..., N} \ {} ndcates ow y (k) nfluences y (k + 1). It olds [ ] χ (k)χ (,) (k) [M(k)] = [1 χ (k)] Nout. + 1 Fnally, f l {N + 1,..., N + E} s suc tat [φ (y) (k)] l = (r,) (k), te element [M(k)] l ndcates ow y (k) nfluences (r,)(k). It olds { [1 χ (k)χ [M(k)] l = (,) (k) ] [ ] χ (k) Nout +1 f r = 0 f r We now analyse te t row of M(k), wt {N + 1,..., N + E}. Let us suppose tat [φ (y) (k)] = (r,l) (k). Reasonng as before, we ave [M(k)] = 1 χ r (k)χ (r,l) (k), [M(k)] l = [1 χ l (k)] [ χ r (k)χ (r,l) (k) ]. and all te oter elements n te t row are 0. Usng te matrces M(k) ust defned and ntroducng varables ν (s) (,)(k) = σ(s) (k) ρ (,s) (k), (, ) E and φ (s) (k) = [s(k) ν (s) (k) ], te evoluton of φ (y) (k) and φ (s) (k) s gven by { φ (y) (k + 1) = φ (y) (k)m(k) φ (s) (k + 1) = φ (s) (9) (k)m(k) We recall tat te frst N elements of vectors φ (y) (k) and φ (s) (k) corresponds respectvely to y(k) and s(k). Propertes of matrces M(k). Introducng te set M, wc collects all possble matrces M(k), te followng lemma olds true. Lemma 5. Te set of matrces M satsfes (1) M s a fnte set; (2) eac M M s a row-stocastc matrx; (3) eac postve element n any matrx M M s lower bounded by a postve constant c; (4) gven τ = NL( τ + 1), for all k 0, te stocastc matrx V (τ) (k) = M(k)M(k+1) M(k+τ 1), M(t) M, s suc tat ts frst N columns ave all te elements wc are strctly postve. Proof. (1) Eac matrx M M depends on wc node wakes up and on wc communcaton lnks from ts node to ts negbours work. Snce te number of all possble combnatons s fnte (and n partcular equal to N =1 2 N out ) te property s verfed. (2) Consder frst te t row of M, wt {1,..., N}. Ten, eter χ (k) = 0, from wc t follows { [M(k)] = 1 [M(k)] = 0 f {1,..., N} \ {}, [M(k)] = 0 f {N + 1,..., N + E} or χ (k) = 1 (and all oter χ (k) = 0, ), mplyng 1 [M(k)] = Nout +1 and [M(k)] = χ (,)(k) N out +1 f {1,..., N} \ {} [M(k)] l = 1 χ (,)(k) Nout + 1 for tose l {N + 1,..., N + E} for wc tere exsts a {1,..., N} \ {} suc tat ψ l (k) = (,)(k) and [M(k)] l = 0 oterwse. Note tat n bot cases te sum of te row s 1. Consder now te t row of matrx M(k), wt suc tat [φ (y) (k)] = (r,l) (k). If χ r(k) = 0 t olds { [M(k)] = 1 [M(k)] l = 0 f l {1,..., N + E} \ {}. On te oter and, f χ r (k) = 1 [M(k)] = 1 χ (r,l) (k) [M(k)] l = χ (r,l) (k) [M(k)] = 0 f {1,..., N + E} \ {, l} In bot cases te row sums up to 1. (3) Ts drectly follows from te constructon of M(k). (4) Let us defne V () (k) = M(k)M(k + 1)... M(k + 1), k 0, 1, V (0) (k) = I N, k 0, wc can be dvded as V () (k) = [ A () 11 (k) A() A () 21 (k) A() ] 12 (k) 22 (k), wt A () 11 (k) RN N, A () 22 (k) RE E, A () 12 (k) RN E and A () 21 (k) RE N. Snce every matrx M M s suc tat [M] > 0 f {1,..., N}, t olds, for 1, tat f n te product wc yelds V () (k) tere exsts a matrx wt te element n poston (, ) strctly greater tan 0, ten also [V () (k)] > 0. Due to Assumptons 1 and 2, after L( τ + 1) teratons, all te lnks n grap G ave successfully transmtted at least once. Moreover, f at tme k +, 0 L τ, te communcaton lnk of te edge (, ) E s relable, ten consderng ndex s suc tat [φ (y) ( )] s = (,) ( ), t olds [M(k + )] s > 0. As a consequence, eac row of A (L τ) 21 (k) as at least one non zero element. Usng a smlar reasonng, for all (, ) E,

6 t olds tat [V (L τ) (k)] > 0. Snce grap G s connected, t olds tat all te elements of A ((N 1)L( τ+1)) 11 (k) are strctly postve. Due to te last two propertes, coosng τ = NL( τ + 1), matrx V (τ) (k) as te frst N columns wt all te elements strctly postve. Remark 6. Te constant τ as been evaluated n te worst possble scenaro. As a matter of fact we assumed tat n grap G tere are at least two nodes tat communcate wt eac oter n no less tan N 1 steps and we also assumed tat te communcaton along one lnk fals L 1 tmes consecutvely. Ts mples tat n a random network G, were te dameter of te grap s usually muc smaller tan te number of nodes, te actual constant τ accordng to wc te frst N columns of V (τ) (k) are strctly postve wll be, n general, muc smaller. Ergodcty teory and convergence of ra-ac We wll brefly recall some useful concepts of ergodcty teory to be later appled to prove te convergence of te algortm. An exaustve explanaton for ergodcty teory can be found n Seneta (2006). Gven a stocastc matrx P R N N, a coeffcent of ergodcty for P quantfes ow muc ts rows are dfferent from eac oter. Two well-known coeffcents of ergodcty for a stocastc matrx P are δ(p ) := max max [P ] 1 [P ] 2, 1, 2 λ(p ) := 1 mn 1, 2 mn {[P ] 1, [P ] 2}. Tese coeffcents are proper (tat s δ(p ) = 0 and λ(p ) = 0 f and only f P = 1 n w, wt w suc tat w 1 n = 1), and, as all te coeffcents of ergodcty, 0 δ(p ) 1 and 0 λ(p ) 1. Consder now a stocastc matrx P suc tat δ(p ) < ψ. Selectng two elements n any column of P, te dfference between tese two elements s necessarly smaller tan ψ. Consder now a vector y R N wc sums to 0, tat s 1 N y = 0. Let us defne te related quanttes y pos = y 1, y neg = y 0, y pos +y neg = 0, y >0 y <0 and also, for any {1,..., N}, te quanttes P = max [P ], P = mn[p ], P ψ P P. Suppose now tat 4 y pos > 0. We am at fndng an upper and lower bound for [y P ], for any 1 N. Te maxmum value for ts product s aceved n case te postve elements of vector y are multpled by P and te negatve elements of te same vector are multpled by P, tat s [y P ] y pos P + y neg P y pos P + y neg P y neg ψ wc reduces to [y P ] y neg ψ. Te mnmum value of [y P ] s nstead produced f te negatve elements are multpled by P and te postve ones by P,.e. [y P ] y neg P + y pos P y neg P + y pos P y pos ψ wc mples [y P ] y pos ψ. From tese two bounds we obtan 4 It s possble to verfy tat te property we are nterested n, tat s te bound n Equaton 10, s stll verfed f y = 0. [y P ] ψ N y (10) Ts bound wll be used to prove te convergence of te algortm. In partcular, te stocastc matrx nvolved wll be te forward product of te matrces tat defne te evoluton of te algortm as seen n (9), tat s matrx T (k) = M(0)M(1) M(k). Ts matrx allows to evaluate φ (y) (k + 1) gven φ (y) (0) (supposng te ntal tme s 0). In order to evaluate te coeffcent δ(t (k)), we wll explot an mportant property tat olds for δ( ) and λ( ) wen we are dealng wt te product of row stocastc matrces: gven stocastc matrces P 1,..., P, ten δ(p 1 P 2 P ) λ(p ). (11) =1 =1 As a consequence f some of te matrces P are suc tat λ(p ) < 1, ten also δ(p 1 P 2 P ) wll be strctly less tan 1. A stocastc matrx P suc tat λ(p ) < 1 s called scramblng, and a suffcent condton for P to be scramblng s tat at least one column s strctly postve,as can be verfed by te defnton of λ( ). Let us apply ts teory to our forward product of matrces, startng by defnng te followng W () = τ 1 k=( 1)τ M(k), 1, M(k) M wc, by Lemma 5, ave strctly postve columns. As a consequence, λ(w ()) < 1 for all 1. Moreover, te number of dfferent W () s fnte snce matrces M(k) are fnte and te Assumptons 1 and 2 ave to be satsfed. Collectng all W () n set W, t s possble to defne value d = max λ(w ), W W wc s strctly smaller tan 1. Te followng lemma olds Lemma 7. Te constant β = d 1/(2τ), 0 < β < 1, s suc tat δ(t (k)) β k for k τ. Proof. If k τ, T (k) can be rewrtten as T (k) = W (1) W ()M(τ)M(τ + 1) M(τ + ) wt = k/τ and = k τ, 0 τ 1. As a consequence, usng Formula (11) δ(t (k)) λ(w (1)) λ(w ())λ M(τ + ) d. =0 Snce k/(2τ), d d k/(2τ), so coosng β = d 1/(2τ), δ(t (k)) β k. Lemma 7 mples tat te coeffcent of ergodcty for T (k) converges to 0 as k goes to nfnty. Before sowng te convergence of ra-ac, we need to sow tat eac component of s(k) s lower bounded by a constant µ > 0, snce te varable x(k) s obtaned troug te Hadamard dvson by s(k). Note tat f k τ, te elements of te frst N columns of T (k) are strctly bgger tan c τ. As a consequence, s (k+1) c τ N =1 s (0) c τ. On te oter and, snce te frst N elements of te dagonals of matrces M(k) are strctly postve, f 1 k τ 1, s (k + 1) c k s (0) c τ, snce 0 < c < 1 and

7 s(0) = 1 N. Terefore we take µ = c τ. Let us fnally prove convergence: we wll frst prove te exponental convergence n case vector v s zero mean, v = 0, and ten we wll generalze to te case, v 0. For k τ, by Lemma 7 we ave δ(t (k)) β k, were T (k) s suc tat φ (y) (k +1) = φ (y) (0)T (k). Startng from Formula (10), and rememberng tat te frst N elements of φ (y) (0) are y(0) and te oter elements are 0, for all 1 N t olds [φ (y) = [φ (k + 1)] (y) (0)T (k + 1)] β k y Now, snce for 1 N, y (k + 1) = [φ (y) (k + 1)] and te elements of s(k) are strctly greater tan µ, we ave y (k + 1) = s (k + 1) s (k + 1) y (k + 1) βk y wc mples x (k + 1) = y (k + 1) s (k + 1) 1 s (k) βk y 1 µ βk y and ten ( ) 2 x(k+1) 2 N µ 2 β 2 (β2 ) k+1 y N 2 µ 2 β 2 (β2 ) k+1 x(0) 2 were te last nequalty s a consequence of te Caucy- Scwarz nequalty and te fact tat x(0) = y(0). Defnng C := N 2 µ 2 β 2 te latter becomes x(k) 2 C (d 1/τ ) k x(0) 2, k τ + 1. If 0 k τ, T (k) s stll stocastc and t surely olds tat δ(t (k)) 1, so applyng a smlar reasonng we ave x(k) 2 C k (d 1/τ ) k x(0) 2, C k = N 2 µ 2 (d 1/τ ) k. Introducng C = max{c 1,..., C τ, C } = C τ we ave x(k) 2 C(d 1/τ ) k x(0) 2, k 0, (12) tat s we ave te exponental convergence of te algortm wen vector v s 0, snce te mean v s 0, and te vector x(k) s convergng to 01 N. Let us fnally generalze to te case n wc v s suc tat v 0. Introducng vector v 0 = v v1 N, we consder two evolutons of te algortm, one ntalzed usng v 0 and te oter ntalzed to v. At eac tme step k te same matrx M(k) s appled for bot ntalzatons. We use te subscrpt 0 to ndcate te varables of te evoluton startng from te zero-mean vector v 0 and te subscrpt v to ndcate tose startng from vector v (vectors s(k) and φ (s) (k) do not ave a subscrpt snce tey are te same n bot evolutons). Rememberng tat s(0) = 1 N, we ave x 0 (0)= y 0(0) s(0), x v = y v(0) s(0) = y 0(0)+ v1 N = y 0(0) s(0) s(0) + v1 N so x v (0) = x 0 (0) + v1 N. Moreover, t s possble to verfy tat φ (y) v (0) = φ (y) 0 (0) + vφ(s) (0), accordng to wc we ave for all 1 N [ ] [ ] φ (y) 0 [x 0 (k)] = (k) φ (y) 0 (0)T (k 1) = φ (s) (k) φ (s) (0)T (k 1) [ ] [ ] φ (y) v (k) φ (y) v (0)T (k 1) [x v (k)] = = φ (s) (k) φ (s) (0)T (k 1) [ ] [ ] φ (y) 0 (0)T (k 1) = + v φ(s) (0)T (k 1) φ (s) (0)T (k 1) φ (s) (0)T (k 1) = [x 0 (k)] + v. We ave ust proved tat x v (k) can be always obtaned as x 0 (k) + v1 N for all k 0, and terefore, snce for x 0 (k) Equaton (12) olds, we also ave x v (k) v1 N 2 C(d 1/τ ) k x v (0) v1 N, k 0, tat s te exponental convergence of x to v1 N olds for any vector v R N. Remark 8. It s possble to adequately modfy te algortm n order to work also n te oter asyncronous communcaton scenaros ntroduced n Secton 2. If some determnstc assumptons equvalent to te ones n ts paper old, te proof of convergence of te modfed algortm can be obtaned by te one ust descrbed, ntroducng approprate modfcatons n te constructons of matrces M(k) and sowng tat te propertes n Lemma 5 are stll verfed. Remark 9. Te dea of usng a consensus algortm wt an augmented state n order to prove te convergence of ts partcular rato consensus s taken from Vadya et al. (2011). However, n te latter te communcaton s syncronous, tat s at eac teraton all te nodes perform some updates, and moreover te results concernng te convergence are gven n probablty. In te set-up llustrated n ts paper, te algortm s asyncronous and te convergence result s stated consderng a worst-case scenaro. Ts s a consequence of te two assumptons we made, wc, remarkably, also allow us to prove tat te convergence s exponental. 6. SIMULATIONS In ts secton we sow te results of some smulatons done for ra-ac. Te set-up of te smulatons s te followng: te number of agents consdered s N = 50, te underlyng communcaton grap s random geometrc, wt te agents arranged n a squared envronment of edge equal to 1 and wt maxmum dstance between negbourng nodes equal to r. In addton, n order to work on drected graps, some of te lnks ave been forced to be undrectonal. Te value of τ and L for Assumptons 1 and 2 are respectvely 75 and 10. In partcular, we consder a probablty 0 < p < 1 of losng a gven packet, but f te lnk tat s selected as faled to transmt for L 1 prevous consecutve tmes, ten te lnk s forced to be relable wtout consderng te packet loss probablty. In Table 1 we gve te averaged root mean squared error (ARMSE) of te results. In partcular, for eac value of d and p selected, we run M = 500 Monte Carlo runs (MCR) for dfferent grap realzatons. Denotng wt x {} (k) te value x(k) obtaned n te t MCR, ten ARMSE(k) = 1 M [ ] 1 x {} (k) v1 N 2 M N =1

8 log 10 (ARMSE(k)) ARMSE(2000) p = 20% p = 50% p = 80% r = r = r = Table 1. Values of ARMSE at tme k = 2000, computed over M = 500 Monte Carlo runs for dfferent values of r and p p = 20% p = 50% p = 80% ,000 1,500 2,000 k Fg. 2. ARMSE as a functon of tme for 3 dfferent values for te packet loss probablty, evaluated over M = 500 MCR. Te value for te maxmum dstance r between nodes s 1/ ,000 k v x 14 (k) y 14 (k) s 14 (k) Fg. 3. Tme evoluton for te scalar varables x 14 (k), y 14 (k) and s 14 (k) for one run of te algortm. In ts smulaton r = 1/3 and p = 20%. Te results of te smulatons sow tat te more connected te grap s, te faster te convergence s. On te oter and, te packet loss probablty, as expected, makes te convergence slower. Note tat for r = 0.25, even at teraton 2000 te convergence s stll not good. However, even n te best case, at teraton 2000 all te nodes ave woken up at most 40 tmes, and so, due to te presence of packet losses and te fact tat eac node ave only few negbours, ts s not surprsng. Fgure 2 sows te tme evoluton for te ARMSE(k), n case r = 1/3. For all te dfferent values of te packet loss probablty t s possble to apprecate te exponental convergence of te algortm. Fnally, Fgure 3 sows te tme evoluton of te varables of a sngle node n te network. It s nterestng to see tat, wle te rato between y 14 (k) and s 14 (k) exponentally converges to te mean v, te sngle varables do not converge but keep oscllatng. Ts beavour s typcal n te rato consensus algortm n presence of undrectonal lnks and packet losses. 7. CONCLUSIONS In ts paper we presented an algortm wc allows eac node n te network to reac consensus on te mean of some prvate constants wc belong to eac agent. We gave te proof for te exponental convergence of te algortm under mld condtons and we carred out some smulatons to furter verfy ts performance. As future researc, we want to apply ra-ac as te fundamental block for consensus n te Newton-Rapson algortm presented n Varagnolo et al. (2016). Newton- Rapson s an algortm tat allows to dstrbutvely mnmze te sum of convex cost functons and one of te steps of ts teratons s a consensus. Introducng ra- AC n ts mplementaton would make Newton-Rapson an asyncronous and robust algortm, were te latter features are mportant n real applcatons. REFERENCES Bénézt, F., Blondel, V., Tran, P., Tstskls, J., and Vetterl, M. (2010). Wegted gossp: Dstrbuted averagng usng non-doubly stocastc matrces. In Proceedngs ISIT 10, EPFL-CONF IEEE. Bolognan, S., Favero, S.D., Scenato, L., and Varagnolo, D. (2010). Consensus-based dstrbuted sensor calbraton and least-square parameter dentfcaton n wsns. Internatonal Journal of Robust and Nonlnear Control, 20(2), Boyd, S., Dacons, P., and Xao, L. (2004). Fastest mxng markov can on a grap. SIAM revew, 46(4), Cattvell, F.S. and Sayed, A.H. (2010). Dffuson strateges for dstrbuted Kalman flterng and smootng. IEEE Transactons on automatc control, 55(9), Cen, Y., Tron, R., Terzs, A., and Vdal, R. (2010). Correctve consensus: Convergng to te exact average. In 49t IEEE Conference on Decson and Control (CDC), IEEE. Domnguez-Garca, A.D. and Hadcosts, C.N. (2010). Coordnaton and control of dstrbuted energy resources for provson of ancllary servces. In Smart Grd Communcatons (SmartGrdComm), 2010 Frst IEEE Internatonal Conference on, IEEE. Domínguez-García, A.D. and Hadcosts, C.N. (2011). Dstrbuted strateges for average consensus n drected graps. In t IEEE Conference on Decson and Control and European Control Conference, IEEE. Domnguez-Garca, A.D., Hadcosts, C.N., and Vadya, N.H. (2011). Dstrbuted algortms for consensus and coordnaton n te presence of packet-droppng communcaton lnks-part I: Statstcal moments analyss approac. arxv preprnt arxv: Garn, F. and Scenato, L. (2010). A survey on dstrbuted estmaton and control applcatons usng lnear consensus algortms. In Networked Control Systems, Sprnger. Olsevsky, A. and Tstskls, J.N. (2009). Convergence speed n dstrbuted consensus and averagng. SIAM Journal on Control and Optmzaton, 48(1), Oreskn, B.N., Coates, M.J., and Rabbat, M.G. (2010). Optmzaton and analyss of dstrbuted averagng wt sort node memory. Sgnal Processng, IEEE Transactons on, 58(5), Patterson, S., Bame, B., and El Abbad, A. (2007). Dstrbuted average consensus wt stocastc communcaton falures. In Decson and Control, t IEEE Conference on, IEEE. Seneta, E. (2006). Non-negatve matrces and Markov cans. Sprnger Scence & Busness Meda.

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