Degree Fluctuations and the Convergence Time of Consensus Algorithms

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1 Degree Fluctuatons and the Convergence Tme of Consensus Algorthms Alex Olshevsy John N. Tstsls Abstract We consder a consensus algorthm n whch every node n a tme-varyng undrected connected graph assgns equal weght to each of ts neghbors. Under the assumpton that the degree of any gven node s constant n tme, we show that the algorthm acheves consensus wthn a gven accuracy ɛ on n nodes n tme O(n 3 ln(n/ɛ)). Because there s a drect relaton between consensus algorthms n tme-varyng envronments and nhomogeneous random wals, our result also translates nto a general statement on such random wals. Moreover, we gve smple proofs that the convergence tme becomes exponentally large n the number of nodes n under slght relaxatons of the above assumptons. We prove that exponental convergence tme s possble for consensus algorthms on fxed drected graphs, and we use an example of Cao, Spelman, and Morse to gve a smple argument that the same s possble f the constant degrees assumpton s even slghtly relaxed. I. INTRODUCTION Consensus algorthms are a class of teratve update schemes that are commonly used as buldng blocs for the desgn of dstrbuted control laws. Ther man advantage s robustness n the presence of tme varyng envronments and and unexpected communcaton ln falures. Consensus algorthms have attracted sgnfcant nterest n a varety of contexts such as dstrbuted optmzaton [8],[7] or coverage control [2], and many other contexts nvolvng networs n whch central control s absent and communcaton capabltes are tmevaryng. Whle the convergence propertes of consensus algorthms n tme-varyng envronments are well understood, much less s nown about the correspondng convergence tmes. An nspecton of the classcal convergence proofs ([3], [3]) leads to convergence tme upper bounds that grow exponentally wth the number of nodes. It s then natural to loo for condtons under whch the convergence tme only grows polynomally, and ths s the subject of ths paper. In our man result, we show that a consensus algorthm n whch every node assgns equal weght to each of ts neghbors n an undrected, connected graph (where the graph can be tme-varyng) has polynomal convergence tme f the degree of any gven node s constant n tme. Because there s a drect relaton between consensus algorthms n tme-varyng envronments and nonhomogeneous random wals, our result also translates nto a general statement on such random wals. Research partally supported by the NSF under grant CMMI Alex Olshevsy s wth the Department of Mechancal and Aerospace Engneerng, Prnceton Unversty, Emal: aolshevs@prnceton.edu John N. Tstsls s wth the Laboratory for Informaton and Decson Systems, Department of Electrcal Engneerng and Computer Scence, Massachusetts Insttute of Technology. Emal: jnt@mt.edu. A. Model, notaton, and bacground In ths subsecton, we defne our notaton, the model of nterest, and some bacground on consensus algorthms. Gven a drected graph G, we wll use N (G) to denote the set {j (, j) s an edge} of drect successors of node n G, and d (G) to denote the cardnalty of N (G). Gven a sequence of drected graphs G(0), G(),..., G( ), we wll use the smpler notaton N (t), d (t) n place of N (G(t)), d (G(t)), and we wll mae a smlar smplfcaton for other varables of nterest. We are nterested n analyzng a consensus algorthm n whch a node assgns equal weght to each one of ts neghbors. We consder n nodes and assume that at each dscrete tme t, node stores a real number x (t). We let x(t) = (x (t),..., x n (t)). For any gven sequence of drected graphs G(0), G(), G(2),..., and any ntal vector x(0), the algorthm s descrbed by the update equaton x (t + ) = d (t) j N (t) whch can also be wrtten n the form x(t + ) = A(t)x(t), x j (t), =,..., n, () for a sutably defned sequence of matrces A(0), A(),..., A(t ). The graphs G(t), whch appear n the above update rule through d (t) and N (t), correspond to nformaton flow among the agents; the edge (, j) s present n G(t) f and only f agent uses the value x j (t) of agent j n ts update at tme t. To reflect the fact that every agent always has access to ts own nformaton, we assume that every graph G(t) contans all the self-loops (, ). Note that we have [A(t)] j > 0 f and only f (, j) s an edge n G(t). It s well nown ([8], [3]) that, subject to some natural condtons on the graph sequence, every component of x(t) converges to a common value. In ths paper, we focus on the convergence rate of ths process n some natural settngs. To quantfy the progress of the algorthm towards consensus, we wll use the functon S(x) = max x mn x. For any ɛ > 0, we wll say that a sequence of graphs G(0), G(),..., G( ) (alternatvely, a sequence of matrces A(0), A(),..., A( )) results n ɛ-consensus f S(x()) ɛs(x(0)) for all ntal vectors x(0). We wll focus on graph sequences n whch every graph G(t) s bdrectonal, meanng that f (, j) s an edge n

2 G(t), then so s (j, ). In practce, graphs that capture nformaton flows are often bdrectonal. For example, G(t) s bdrectonal f: () G(t) contans all the edges between agents that are physcaly wthn some dstance of each other; () G(t) contans all the edges between agents that have lneof-sght vews of each other; () G(t) contans the edges correspondng to pars of agents that can send messages to each other usng a protocol that reles on acnowledgements. It s an mmedate consequence of exstng convergence proofs ([3], [3]) that any sequence of Cn n ln(/ɛ) connected bdrectonal graphs, wth self-loops at every node, results n ɛ-consensus. Here, C s a constant that does not depend on the problem parameters n and ɛ. We are nterested n smple condtons under whch the undesrable O(n n ) scalng becomes polynomal n n. B. Our results Our contrbutons are twofold. Frst, n Secton II, we prove our man result. Theorem. Consder a sequence G(0), G(),..., G( ) of connected bdrectonal graphs, wth self-loops at each node. Suppose, furthermore, that the degree of each node stays constant n tme,.e., d (t) = d (t ),, t, t. If the length of the graph sequence s at least n 3 ln 2n2 ɛ, then ɛ-consensus s acheved. To put Theorem n perspectve, we note that polynomal convergence tmes were only nown for the cases where: (a) the graphs G(t) are the same at each tme t, bdrectonal, connected, wth all self-loops present [4]; or (b) n addton to some natural connectvty assumptons, the underlyng teraton matrces are doubly stochastc, whch, for the consensus algorthms consdered here amounts to an assumpton that each graph G(t) s regular [6]. Theorem can be vewed as a generalzaton of the above two results. In Secton III, we gve an nterpretaton of our results n terms of Marov chans. Theorem can be nterpreted as provdng a suffcent condton for a random wal on a tmevaryng graph to forget ts ntal dstrbuton n polynomal tme. In Secton IV, we show through examples that relaxng the assumptons of Theorem even slghtly can lead to a convergence tme whch s exponental n n. Specfcally, we show the followng. () If we do not requre the graphs G(t) to be bdrectonal, exponental convergence tme s possble, even f the graphs G(t) do not change wth tme; () If we replace the assumpton that each d (t) s ndependent of t wth the weaer assumpton that the sorted degree sequence (say, n non-ncreasng order) s ndependent of t (thus allowng nodes to swap degrees), exponental convergence tme s possble. Whle ths fact was nown (although unpublshed) [5], our contrbuton s to provde a smple proof. The presentaton of our counterexamples captalzes on the correspondence wth Marov chans dscussed n Secton III. In summary: for connected bdrectonal graphs wth selfloops, unchangng degrees s a suffcent condton for polynomal tme convergence, but relaxng t even slghtly by ether allowng the nodes to swap degrees or by losng ln symmetry leads to the possblty of exponental convergence tme. C. Prevous wor There s consderable and growng lterature on the convergence tme of consensus algorthms. We only menton papers that are closest to our own wor, omttng references to the lterature on varous aspects of consensus convergence tmes that we do not address here, such as topology desgn, performance n geometrc random graphs, etc. Worst-case upper bounds on the convergence tmes of consensus algorthms have been establshed n [8], [6], [7], [], [2], [], [0], [9]. The papers [8], [6], [7] consdered a settng slghtly more general than ours, and establshed exponental upper bounds. The papers [], [2] addressed the convergence tmes of consensus algorthms n terms of spannng trees that capture the nformaton flow between the nodes. It was observed that n several cases ths approach produces tght estmates of the convergence tmes. The wors [], [0], [9] tae a geometrc approach, and consder the convergence tme n a somewhat dfferent model, nvolvng nteractons between geographc nearest neghbors. These papers fnd that the convergence tme s qute hgh (ether sngly exponental or terated exponental, dependng on the model). The orgnal papers [8], [3] also consdered the effect of delays on convergence; some more recent wor on ths subject may be found n [7] and [4]. Our wor dffers from these papers n that our convergence tme bounds are polynomal n n. To the best of our nowledge, polynomal bounds on the partcular consensus algorthm consdered n ths paper had been derved earler only n [4] and [6]. Our wor encompasses a much wder class of stuatons than [4], whch requred the graphs G(t) to be constant n tme. Moreover, our results may be vewed as complementary to those n [6], whch proved a polynomal convergence tme bound for averagng algorthms, nvolvng doubly stochastc matrces. II. PROOF OF THEOREM As n the statement of Theorem, we assume that we are gven a sequence of bdrectonal connected graphs G(0), G(),..., wth self-loops at each node, and such that d (t) s the same n each G(t). We wll thus drop the parameter t and refer to the degree of node smply as d. Observe that d > 0 due to the connectvty of the graphs G(t).

3 We wll use G to refer to the class of bdrectonal connected graphs wth self-loops at every node such that the degree of node s d. We let D be the n n dagonal matrx whose th dagonal entry s d. We wll use E (G) to denote the edge set of a graph G. We wll sometmes fnd t convenent to use the notaton E(G) to refer to the set of unordered pars (, j) such that the ordered pars (, j) and (j, ) belong to E (G). Defnton. We defne the nner product, d by n x, y d = d x y. = Note that because d > 0 for all, x, y d s a vald nner product. Defnton 2. Gven a drected graph G, we defne the update matrx A(G) by { /d (G), f j N (G), [A(G)] j = 0, otherwse. We use A(t) as a shorthand for A(G(t)), so that Eq. () can be wrtten as x(t + ) = A(t)x(t). (2) Conversely, gven an update matrx A of the above form, we wll use G(A) to denote the graph G whose update matrx s A. We use N (A) as shorthand for N (G(A)); the quanttes d (A), E(A), and E (A) are defned smlarly. Fnally, we use A to denote the set of update matrces A(G) assocated wth graphs G G. Note that DA s the adjacency matrx assocated wth the graph correspondng to A. Gven that we restrct to bdrectonal graphs, DA s symmetrc for every A A. Lemma. For any A A, we have x, Ay d = (,j) E (A) x y j. Proof: We have x, Ay d = x T DAy = n = j N (A) x y j = (,j) E (A) x y j Lemma 2. Each A A s self-adjont wth respect to the nner product, d. Proof: Usng the fact that DA s symmetrc and D s dagonal, we have x, Ay d = x T DAy = x T (DA) T y = x T A T Dy = Ax, y d. Lemma 2 s the reason for ntroducng the nner product, d. The fact that matrces n the set A are self-adjont plays a central role n the analyss of the algorthm (2). One of ts consequences s that matrces n A have real egenvalues. We use the notaton λ (A) to denote the th largest egenvalue of a matrx A A. Note that every A A s a stochastc matrx, and therefore λ (A) =. Next, we dentfy a weghted average that s preserved by the teraton x(t + ) = A(t)x(t). For any x, we let x = x, d, d. We observe that for any A A, x, d = x T D = x T DA = x T A T D = Ax, d, where the second equalty used the fact that A s a stochastc matrx. Therefore, Ax = x, A A. Wth these prelmnares n place, we now proceed to the man part of our analyss, whch s based on the Lyapunov functon n V (x) = x x, x x d = d (x x) 2. = The next lemma quantfes the decrease of V ( ) when a vector x s multpled by some matrx A A. Lemma 3. For any A A and any vector x, we have V (Ax) λ 2 (A 2 )V (x). Proof: Fx some A A and some vector x. Snce Ax = x, the lemma asserts that Ax x, Ax x d λ 2 (A 2 ) x x, x x d. Let y = x x, and note that y, d = 0. It therefore suffces to show that Ay, Ay d λ 2 (A 2 ) y, y d, for all y wth y, d = 0; or, equvalently, that y, y d Ay, Ay d y, y d λ 2 2(A), f y, d = 0 and y 0. To establsh Eq. (3), we note that y, y d Ay, Ay d y, y d y, A 2 y d mn = mn y, d =0 y, y d y, d =0 y, y d y, (I A 2 )y d = mn, y, d =0 y, y d where n the mnma above we only consder nonzero vectors y; note that these mnma are attaned t suffces to consder vectors y on the unt ball, a compact set. Now, observe that I A 2 s also self-adjont under the nner product, d. Moreover, snce A (and, therefore, A 2 as well) s stochastc, the smallest egenvalue of I A 2 s 0, wth an assocated egenvector of. Consequently, by the Courant-Fscher varatonal characterzaton of egenvalues, the expresson on the rght s the second smallest egenvalue of I A 2 : y, y d Ay, Ay d mn = λ n (I A 2 ) = λ 2 (A 2 ), y, d =0 y, y d (3)

4 whch concludes the proof. Thus, to bound how much V (x) decreases at each step, t suffces to obtan an upper bound on λ 2 (A 2 ), for matrces A A. Ths can be done usng the next lemma, whch s the the man result of [4]. We nclude a short proof for completeness. Lemma 4. Let A A, and let l be the dameter of the graph G(A). Then, λ2 (A 2 )= max{ λ n (A), λ 2 (A)} where d max s the largest of the degrees d. nd max l, Proof: The frst nequalty follows because the egenvalues of A 2 are the squares of the egenvalues of A. For the second nequalty, usng agan the Courant-Fsher characterzaton, and some easy algebra, we have λ 2 (A) = max x, d =0 x,x d = = mn x, d =0 x,x d = x, Ax d = (,j) E(A) max x, d =0 x,x d = (,j) E (A) (x x j ) 2. Thus, t suffces to show that (x x j ) 2 mn x, d =0 x,x d = (,j) E(A) nd max l x x j Towards ths purpose, we carry out a varaton of an argument frst used n [4]. Fx some x that satsfes x, d = 0 and x, x d =. Wthout loss of generalty, we assume that () node has the largest value of d x 2, () node has the smallest value of x, and () the shortest path from node to s (, 2), (2, 3),..., (, ). The condton x, x d = mples that d x 2 /n, and consequently that x / nd max ; the requrement x, d = 0 mples that x < 0. Thus, x x / nd max, whch we wrte as (x x 2 ) + (x 2 x 3 ) + + (x x ) Applyng the Cauchy-Schwarz nequalty, we get = (x x + ) 2 nd max. ndmax. We then use the fact that l, to obtan the clamed bound on λ 2 (A). As for λ n, we observe that the dagonal entres of A are at least /n and the row sums are. The Gershgorn crcle theorem mmedately gves λ n + /n, whch s stronger than the bound we have clamed. We can now complete the proof of Theorem. Lemma 3 descrbes the decrease n the varance V (x(t)) n terms of λ 2 (A 2 (t)), and Lemma 4 gves us a way to upper bound the latter quantty. Proof of Theorem : Usng Lemmas 3 and 4, and the bounds d max n, l n, we see that for every A A, and every x, we have V (Ax) λ 2 (A 2 )V (x) = max{ λ n (A) 2, λ 2 2(A)}V (x) ( ) 2V (x). n 3 Because the defnton of ɛ-consensus s n terms of S(x) rather than V (x), we need to relate these two quanttes. On the one hand, for every x, we have n n V (x) = d (x x) 2 n (x x) 2 n 2 S 2 (x). = = On the other hand, for every x, we have V (x) max (x x) 2 4 (max x mn x ) 2 = 4 S2 (x). Suppose that t n 3 ln(2n 2 /ɛ). Then, S(x(t)) ( 4V (x(t)) 2 ) n 3 ln(2n 2 /ɛ) V (x(0)) n 3 2n 2( n 3 ) n 3 ln(2n 2 /ɛ) S(x(0)) 2n 2 e ln(2n2 /ɛ) S(x(0)) = ɛs(x(0)). (We have used here the nequalty ( c/n) n e c, for c > 0.) III. MARKOV CHAIN INTERPRETATION In ths secton, we gve an alternatve nterpretaton of the convergence tme of a consensus algorthm n terms of nhomogeneous Marov chans. In the next secton, we wll use ths nterpretaton to gve some examples of graph sequences that do not satsfy Theorem and whch have exponentally large convergence tmes. We consder an nhomogeneous Marov chan whose transton probablty matrx at tme s A(). We fx some tme t and defne P = A(0)A() A(t ). Ths s the assocated t-step transton probablty matrx: the j-th entry of P, denoted by p j, s the probablty that the state at tme t s j, gven that the ntal state s. Let p be the vector whose th component s p ; thus p T s the th row of P. We address a queston whch s generc n the study of Marov chans, namely, whether the chan eventually forgets ts ntal state,.e., whether for all, j, p p j converges to zero as t ncreases, and f so, at what rate. We wll say that the sequence of matrces A(0), A(),..., A(t ) s ɛ-forgetful f for all, j, we have 2 p p j ɛ. The above quantty, 2 max,j p p j s nown as the coeffcent of ergodcty of the matrx P, and appears often n the study of consensus algorthms (see, for example, [8]). The matrx P can also be nterpreted n terms of consensus updates: an ntal vector x s multpled by the matrces A(t

5 ), A(t 2),..., A(0), to produce the vector P x; note that the dfferent matrces are now appled n the reverse order. The result that follows relates the tmes to acheve ɛ-consensus or ɛ-forgetfulness, and s essentally the same as Proposton 4.5 of [5]. IV. SOME COUNTEREXAMPLES A. The bdrectonalty requrement n Theorem cannot be relaxed Proposton 2. The sequence of matrces A(0), A(),..., A(t ) s ɛ-forgetful f and only f the sequence of matrces A(t ), A(t 2),..., A(0) results n ɛ-consensus (.e., S(P x) ɛs(x), for every vector x.) Proof: Suppose that the matrx sequence A(0), A(),..., A(t ) s ɛ-forgetful,.e., that 2 p p j ɛ, for all and j. Gven a vector x, let c = (max x + mn x )/2. Note that x c = (max x mn x )/2 = S(x)/2. We then have [P x] [P x] j = (p p j )x = (p p j )(x c) p p j x c 2ɛS(x)/2 = ɛs(x). Snce ths s true for every and j, we obtan S(P x) ɛs(x), and the sequence A(t ), A(t 2),..., A(0) results n ɛ- consensus. Conversely, suppose that the sequence of matrces A(t ), A(t 2),..., A(0) results n ɛ-consensus. Fx some and j. Let x be a vector whose th component s /2 f p p j and /2 otherwse. Note that S(x) =. We have 2 p p j = (p T p T j )x = [P x] [P x] j ɛs(x) = ɛ, where the last nequalty made use of the ɛ-consensus assumpton. Thus, the sequence of matrces A(0), A(),..., A(t ) s ɛ-forgetful. We wll use Proposton 2 for the specal case of Marov chans that are random wals. Gven a drected graph G(t), we let, as before, N (t) be the set of nodes j for whch the edge (, j) s present. In the random wal assocated wth ths graph, f the state at tme t s, the state at tme t+ s chosen to be one of the elements of N (t), wth equal probablty. We let A(t) be the assocated transton probablty matrx. We wll say that a sequence of graphs s ɛ-forgetful whenever the correspondng sequence of transton probablty matrces s ɛ-forgetful. Proposton 2 allows us to renterpret Theorem as follows: random wals on tme-varyng bdrectonal connected graphs wth self-loops forget ther ntal dstrbuton n a polynomal number of steps. Proposton 2 also has a corollary whch we wll use later. It s based on the followng observaton: concatenatng two sequence of graphs, each of whch acheves ɛ-consensus, results n a sequence whch acheves ɛ 2 -consensus. Corollary 3. Suppose that a sequence of graphs s ɛ-forgetful. Then, concatenatng ths sequence wth tself tmes results n a sequence whch s ɛ -forgetful. Fg.. The graph used n Proposton 4. In ths subsecton, we show that t s mpossble to drop the assumpton that the graph s bdrectonal, even when the graph does not change wth tme. Proposton 4. Let G be the graph shown n Fgure. Consder the graph sequence consstng of G, repeated t tmes. For ths graph sequence to result n (/8)-consensus, we must have t 2 n/2 /6. Proof: Suppose that ths graph sequence of length t results n ɛ-consensus. By Proposton 2, t s (/8)-forgetful. In partcular, p p n /4 (recall that p j stands for the t-step transton probablty from to j). For n/2, let T be the frst tme that a random wal that starts at state vsts the bottom part of the graph, and let δ be the probablty that T s less than or equal to t. Note that condtonal on startng at and never transtonng to the bottom half, the probablty of beng at state at tme t s /2; thus, p 2 ( δ ). Furthermore, p n δ by symmetry. Therefore, 4 p p n 2 ( δ ) δ = δ, whch yelds δ /6. Usng a straghtforward couplng argument, we have δ δ /6, for = 2,..., n/2. By vewng perods of length t as a sngle attempt to get to the bottom half of the graph, wth each attempt havng probablty at least /6 to succeed, we conclude that E[T ] 6t. On the other hand, T s just the frst tme untl the wal moves to the rght n/2 consecutve tmes. Each tme that the random wal returns to state, we have a new tral wth probablty of success equal to 2 n/2. Thus, 2 n/2 E[T ] 6t, whch yelds the desred result. Remar: It s not hard to see that f we add a self-loop to each node n Fgure, the result holds wth mnor modfcatons.

6 E[T ] 4t, whch yelds the desred result. Fg. 2. The top-left fgure shows graph G(0); top-rght shows G(); bottom-left shows G((n/2) 2); bottom-rght shows G((n/2) ). As these fgures llustrate, G(t + ) s obtaned by applyng a crcular shft to each half of G(t). All edges are bdrectonal, and every node has a self-loop whch s not shown. For aesthetc reasons, nstead of labelng the nodes as,..., n, we label them wth,..., n/2 and,..., (n/2). B. The unchangng degrees condton n Theorem cannot be relaxed In ths subsecton, we show that t s mpossble to relax the condton of unchangng degrees n Theorem. In partcular, f we only mpose the slghtly weaer condton that the sorted degree sequence (the non-ncreasng lst of node degrees) does not change wth tme, the tme to acheve ɛ-consensus can grow exponentally wth n. Ths s an unpublshed result of Cao, Spelman, and Morse [5]; we provde here a smple proof. Proposton 5. Let n be even and let t be an nteger multple of n/2. Consder the graph sequence of length t = n/2, consstng of perodc repettons of the reversal of the lengthn/2 sequence descrbed n Fgure 2. For ths graph sequence to result n (/4)-consensus, we must have t 2 (n/2) /8. Proof: Suppose that ths graph sequence of length t results n (/4)-consensus. Then Proposton 2 mples that the sequence of length n/2 consstng of perodc repettons of the length n/2 sequence descrbed n Fgure 2 s (/4)- forgetful. Let p j be the assocated t-step transton probabltes. Let T be the tme that t taes for a random wal that starts at state n/2 to cross nto the rght-hand sde part of the graph. Let δ be the probablty that T s less than or equal to t. Let R = {... (n/2) }. We have j R p (n/2),j δ and, usng symmetry, j R p (n/2),j δ. Usng the fact that the graph sequence s (/4)-forgetful n the frst nequalty below, we have 2 j R p (n/2),j p ((n/2),j p (n/2),j j R j R ( δ) δ = 2δ, p (n/2),j whch yelds δ /4. Argung as n the proof of Proposton 4 (node n/2 s the least favorable non-central startng state n the left-hade sde of the graph), we obtan E[T ] 4t. Note that for a wal that starts at state n/2 to cross nto the rght-hand sde part of the graph, t must frst tae a selfloop (n/2) consecutve tmes. Consequently, 2 (n/2) V. CONCLUSIONS The man contrbuton of ths paper s Theorem, whch shows that consensus algorthms converge n polynomal tme for a large class of graph sequences. We also gave smple proofs showng that ths fndng s fragle, and even slght relaxatons of the hypotheses cause the concluson to fal. A smlar result s avalable for the case of doubly stochastc update matrces A(t), and our result can be vewed as complementary [6]. Interestngly, both results rely on a sutable quadratc Lyapunov functon as well as on the fact that all update matrces share a common left egenvector. REFERENCES [] D. Angel, P.-A. 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