Average Consensus and Gossip Algoriths in Networks with Stochastic Asyetric Counications Duarte Antunes, Daniel Silvestre, Carlos Silvestre Abstract We consider that a set of distributed agents desire to reach consensus on the average of their initial state values, while counicating with neighboring agents through a shared ediu. This counication ediu allows only one agent to transit unidirectionally at a given tie, which is true, e.g., in wireless networks. We address scenarios where the choice of agents that transit and receive essages at each transission tie follows a stochastic characterization, and we odel the topology of allowable transissions with asyetric graphs. In particular, we consider: (i) randoized gossip algoriths in wireless networks, where each agent becoes active at randoly chosen ties, transitting its data to a single neighbor; (ii) broadcast wireless networks, where each agent transits to all the other agents, and access to the network occurs with the sae probability for every node. We propose a solution in ters of a linear distributed algorith based on a state augentation technique, and prove that this solution achieves average consensus in a stochastic sense, for the special cases (i) and (ii). Expressions for absolute tie convergence rates at which average consensus is achieved are also given. I. INTRODUCTION The average consensus proble is a distributed control proble in which a set of agents ais to agree on the average of their initial state values by exchanging essages dictated by a given counication topology. Several ultidisciplinary applications of average consensus algoriths have been reported in the literature. These include distributed optiization [], []; otion coordination tasks, such as flocking, leader following [3], and rendezvous probles [4]; and resource allocation in coputer networks [5]. An elegant theory is now available in the literature to solve consensus probles using linear distributed algoriths, in which each agent coputes a weighted average between its state value and the state values of the agents to which it can counicate (see, e.g, [6], [7]). Many variations of this proble have been addressed in the literature considering, e.g., stochastic packet drops and link failures [8], [9], quantized data transissions [0], and tie-varying counication connectivity [6]. On the other hand, randoized gossip algoriths have been proposed, e.g., in [], as a decentralized solution to the average consensus proble that can deal with several features such as the absence of a centralized entity, and the possibly varying topology of the network caused by agents that join and leave the network. The preise of gossip D. Antunes is with the Departent of Mechanical Engineering, Eindhoven University of Technology, the Netherlands, D.Antunes@tue.nl D. Silvestre, C. Silvestre are with the Dep. of Electrical Eng. and Coputer Science, Instituto Superior Técnico, ISR, 046-00 Lisboa, Portugal. {dsilvestre,cjs}@isr.ist.utl.pt algoriths is that each agent counicates with no ore than one neighbor at each transission tie. In [], it is assued that at each counication tie, the sender and the receiver set their state to the average of their current values, which subsues that the counication is bidirectional. In the present work, we consider the average consensus proble in scenarios where counication is unidirectional at each tie slot, i.e., at each transission tie a single agent transits data to one or several agents, but does not receive data. Note that at a different tie slot receiver and sender agents ay invert their roles, i.e., the word unidirectional refers only to counication at a given transission tie. For concreteness, we consider the two following scenarios: (i) randoized gossip algoriths in wireless networks, where each agent becoes active at randoly chosen ties, transitting its data to a single neighbor; (ii) broadcast wireless networks, where each agent transits to all the other agents, access to the network occurs with the sae probability for every agent, and the intervals between transissions are independent and identically distributed. As we shall see, the unidirectionality counication constraint precludes in general the existence of a linear distributed algorith where associated to each agent there is a single scalar state, updated based on the state of the other agents, as in related probles where the counication topology of the network is also tie-varying, but satisfies different assuptions (see [6], [7]). We assue a syetric counication topology, eaning that if an agent a can counicate with an agent b then the agent b can counicate with the agent a, although this does not take place at the sae transission tie, i.e., at each transission tie the graphs odeling counications are in general asyetric. Note that this is typically the case in wireless networks, and therefore this assuption is reasonable to assue in both scenarios (i) and (ii). The ain contribution of the present paper, is to propose a linear distributed algorith using a state augentation technique to achieve average consensus when counication constraints ipose tie-varying unidirectional transissions, and to prove convergence in expectation to average consensus for scenarios (i) and (ii). Moreover, the stochasticity of the ties between transissions, in both gossip and broadcast scenarios considered herein, render non-trivial the coputation of the rates of convergence of the linear distributed algoriths to consensus in absolute tie, as opposed to discrete-tie, i.e., in ters of the nuber of transissions. We provide expressions for the absolute tie rates in both scenarios. A recent work [] addresses a siilar proble, consid-
ering gossip algoriths also with asynchronous counication between the agents. In [], a ethod is proposed to achieve average consensus, also using a state augentation technique, and this ethod is proved to converge alost surely to consensus. The counication topologies considered in [] encopass general directed graphs. Hence, [] does not need to assue a syetric counication topology, which is crucial to obtain our convergence results. Note, however, that in the ethod proposed in [], the state updates depend in a nonlinear way of the current state, while the algorith that we propose is a linear distributed algorith, proving a solution that parallels existing algoriths for the standard gossip [] and linear distributed algoriths [7]. The reainder of the paper is organized as follows. We provide soe preliinaries and state the average consensus proble in Section II. The proposed solution is given in Section III and our ain results are stated in Section IV. Concluding rearks and directions for future work are given in Section VI. Notation : The transpose and the spectral radius of a atrix A are denoted by A and r σ (A), respectively. For vectors a i, (a,...,a n ) := [a...a n ]. We let n := [...] and 0 n := [0...0] indicate n-diension vector of ones and zeros, and I n denotes the identity atrix of diension n. Diensions are oitted when no confusion arises. The vector e i denotes the canonical vector whose coponents equal zero, except coponent i that equals one. The notation diag([a...a n ]) indicates a block diagonal atrix with blocks A i. The Kronecker product is denoted by. II. PRELIMINARIES AND PROBLEM STATEMENT We consider that a set of agents with scalar state x i (t), i, desire to obtain the average of their initial states, i.e., li t x i(t) =x av := x i (0). () i= We refer to this proble as the average consensus proble. The counication topology is odeled by a directed graph G =(V,E), where V is the set of agents, also denoted by nodes, and E V V is the set of counication links. The node i can send a essage to the node j, if (j, i) E. If there exists at least one i V such that (i, i) E we say that the graph has self-loops. A standing assuption will be that G is strongly connected and aperiodic [7]. We associate to the graph G an adjacency atrix N with entries {, if (i, j) E N ij :=. () 0, otherwise More generally, we can consider al weighted adjacency atrix W associated with G with entries [W ] ij = w ij R, if (i, j) E and [W ] ij =0, otherwise. A. Previous Work A linear distributed algorith is defined by the iteration x(t k+ )=W k x(t k ), (3) where {W k,k 0} is a set of weighted adjacency atrices associated with the graph G, and t k are the ties at which a transission occurs, with t 0 := 0. It is subsued throughout the paper that between transission ties, the state variables do not change their values, i.e., e.g., ẋ(t) = 0,t [t k,t k+ ). If W k = W c is constant for every k 0, where W c is a stochastic atrix (W c = ), and the graph is strongly connected and aperiodic, the linear distributed algorith (3) converges to consensus (cf. [7]), eaning that li k x k = c, for a given constant c. See [7] for conditions under which consensus is achieved in the case where W k are tie-varying. Moreover, if all the atrices {W k,k 0} are doubly stochastic (W k = and W k = ) and (3) converges to consensus then (3) converges to average consensus, i.e., () is achieved. A randoized gossip algorith is a special case of a linear distributed algoriths. In gossip algoriths, each of the nodes has a clock which ticks at a rate Poisson process, i.e., inter-tick ties are rate exponential distributed. It is clear that at a given tick tie, each node has the sae probability of being the node whose clock ticked. Such node, denoted by i, chooses a rando out-neighbor j (corresponding to the non-zero entries of colun i of the adjacency atrix N) according to a given distribution p ij, which denotes the probability that node j N out (i) is chosen, where N out (i) denotes the set of out-neighbors of node i and j N p out(i) ij =. Both nodes update their current state values according to the averages of their values, while the reaining nodes keep their state intact, i.e., if node i ticks at tie t k and chooses node l to counicate with, then x(t k+ )=W k x(t k ), where W k = M il, M il := I (e i e l )(e i e l ) B. Stability and convergence rate As in gossip algoriths, there are several works in the literature (see, e.g., [8], [9], and references therein), where the atrices W k in (3) are assued to be stochastic atrices, i.e., Prob[W k = M j ]=p j, for soe atrices M j, j n M, and probabilities p j, n M i= p j =. This ay be due to several sources of randoness, such as packet drops or link failures. In such cases the definition of achieving average consensus (cf. ()) ust be adapted to a stochastic setting. Definition : We say that a linear distributed algorith taking the for (3), where {W k,k 0} are stochastic atrices: (i) converges alost surely to average consensus if li t x i(t) =x av := alost surely. x i (0), i {,...,} i= We use this forulation, instead of siply considering discrete-tie variables z k = x(t k ) indexed by the transission nuber k, because we are interested in deterining convergence rates to consensus in absolute tie t.
(ii) converges in expectation to average consensus if li t E[x i(t)] = x av, i {,...,}. (iii) converges in second oent to average consensus if li E[(x i(t) x av ) ] 0, i {,...,}. t In applications it is typically easier to prove convergence in expectation and in second oent, which are the two stability definitions we shall focus in the present paper. It is clear that convergence in second oent to consensus is stronger than convergence in expectation and also stronger than alost sure convergence. As argued in [] it is soeties possible to prove that convergence in expectation iplies alost sure convergence. However, we ake no such assertion here. Provided that consensus is achieved one ay ask at which rate is it achieved. In the present paper we shall focus on deterining the following rate corresponding to the convergence rate to zero of the second oent of the error between state and average consensus. Definition : Suppose that the following condition holds for soe γ > 0 and c>0 and for every x(0) R, E[(x i (t) x av ) ] ce γt (x i (0) x av ), i {,...,}. (4) Then, the second oent convergence rate is defined as α:=sup{γ : (4) holds for soe c>0 and every x(0) R }. C. Proble Stateent We consider the following counication constraint, inspired by wireless counication constraints: Counication Constraint 3: At each transission tie t k, a single agent i sends data to one or various of its outneighbors, and cannot receive data fro its in-neighbors. For concreteness, we consider the two following scenarios. Gossip algoriths: We adapt the set-up proposed in [] and described in Section II-A to accoodate the counication constraint 3. In other words, we consider that each of the agents has a clock which ticks at a rate Poisson process, randoly choosing one of its out-neighbors according to the distribution p ij associated with node i. Node j can change its state (not necessary a scalar) at this transission tie based on the data received fro node i. However, node i does not receive data at this transission tie. We allow p ii to be different fro zero, eaning that at soe clock ticks, node i can update its state without counicating to any other node. Broadcast wireless networks: At each transission tie an agent sends data to every other agent in the network. We consider that nodes have the sae probability of accessing the networks at each transission tie and assue that the ties between transissions are independent and identically distributed as it is typical to consider in such networks (cf. [3]). The probles we are interested in are the following. (i) In the setup of scenarios ) and ), design a linear distributed algorith that achieves average consensus according to one of the stability notions of Definition. (ii) Copute the second oent convergence rate to average consensus of this linear distributed algorith. Although we shall provide a ethod to obtain average consensus that can be applied to any counication topology described by (), our stability results given for scenarios ) and ) assue that N is syetric, which is reasonable to assue in wireless networks, in which if a node a can counicate to a node b then typically b can counicate with a. The next exaple illustrates why, in general, it is not possible to achieve average consensus by using the standard solution described in Section II-A, where each node updates a scalar state according to the recursion (3). Exaple 4: Suppose [ that ] =and the adjacency atrix is given by N =. When node i {, } transits, node j {, }, j i, updates its inforation based on the received state of x i and its own state x j, and node i can only update its state based on its own state inforation. Thus, a linear distributed algorith using a single scalar state per node, [ is described ] by [(3), where ] W k {Z,Z }, and α α Z = β 0, Z 0 α =, where we assue that 3 β β 3 α 0, and β 0, since otherwise there would be no counication between the nodes. It is clear that W k cannot therefore be doubly stochastic atrices. In the next section, we propose a linear distributed algorith that solves this proble by using a state augentation technique. III. PROPOSED SOLUTION We augent the original state x(t) with an auxiliary vector y(t) R ny, and define z =(x, y). (5) We consider a linear distributed algorith taking the for z(t k+ )=U k z(t k ), (6) where z(0) = (x(0),y(0)),y(0) = 0. Intuitively, the purpose of y it to assure that at each iteration the total state average is kept constant, i.e., that x i(t k+ )+ n y y i(t k+ ) + n y = x i(t k )+ n y y i(t k ) + n y. (7) If we initialize y to zero and guarantee that y(t) goes to zero then average consensus is achieved. More specifically, the proposed algorith can be described as follows. Set n y =. Each coponent y i R is kept in eory and updated by node i along with its state x i. At tie t k,a given node i sends a essage containing x i (t k ) and y i (t k ) to one or ore out-neighbors (corresponding to the non-zero
entries of colun i of the adjacency atrix N). The node i does not change its state, i.e., and resets the auxiliary state to zero x i (t k+ )=x i (t k ) (8) y i (t k+ )=0. (9) Let n out (i, k) be the nuber of out-neighbors of node i to which a essage is sent at tie t k. A node j receiving this essage, updates its state x j (t) according to x j (t k+ )=( α)x j (t k )+αx i (t k )+βy j (t k )+γy i (t k ) (0) and updates its variable y j (t k ) according to y j (t k+ )= y i(t k ) n out (i, k) + y j(t k )+x j (t k ) x j (t k+ ) () so that the total state average is kept constant, i.e., (7) holds. The average consensus proble in the two scenarios considered in the present paper, i.e., gossip and broadcast algoriths, can be odeled by a linear distributed algorith taking the for (6). In fact, the following atrices U k can odel the algorith (8)-(), when applied to the counication assuptions for the gossip and broadcast algorith described in Section II-C. Gossip algorith G: The atrices U k are taken fro the set {Q ij, i, j, i j}, where each Q ij corresponds to a transission fro node i to an out-neighbor node j, and these atrices are described as follows. Let Λ i := diag(e i ) and Ω ij := I (Λ i + Λ j ). Then Aij B Q ij = ij () C ij D ij where A ij := I αλ j + αe j e i B ij := βλ j + γe j e i C ij := Λ j (I A ij ) D ij := Ω ij + Λ j (I + e j e i B ij). (3) The atrices defined in () also odel the case where a node i picks itself when there is a clock tick (with probability p ii ), in which case we arbitrate that the state update is described by and x i (t k+ )=x i (t k )+(α + β)y i (t k ) y i (t k+ )=( (α + β))y i (t k ), for which (7) is et. The atrices U k are by construction independent and identically distributed, and satisfy ( Prob[U k = Q ij ]= p ij, is the probability that node i is the one whose clock ticks at t k and p ij the probability that i picks its out-neighbor node j). The ties between transissions t k+ t k are independent and identically distributed with exponential distribution with ean (since nodes can trigger according to a rate Poisson process). Broadcast Algorith B: The atrices U k are taken fro the set {R i, i }, where each R i corresponds to a transission fro node i to every other node. Let Λ i := diag(e i ), Ω i =(I Λ i ). Then Ai B R i = i (4) C i D i A i =( α)i + α e i B i = Ω i (βi + γ e i ) C i = Ω i (I A i ) (5) D i = Ω i (I + e i B i). The atrices U k are independent and identically distributed due to our assuption that nodes access the network with the sae probability, i.e. Prob[U k = R i ]=. The ties between transissions t k+ t k are independent and identically distributed with an arbitrary distribution iposed by the network protocol. Hereafter, we denote by gossip algorith G the linear distributed algorith odeled by (6) and (), and denote by broadcast algorith B the linear distributed algorith odeled by (6) and (4). Note that, by construction, for both gossip and broadcast algoriths the atrices {U k,k 0} are such that Uk =, (6) which eans that the total average is preserved at each iteration, i.e., z(t k+) = z(t k), and U k = (7) 0 0 which eans that if consensus is achieved at iteration k, i.e., if x(t k )=c and y(t k )=0, the state reains unchanged at iteration k +, i.e., x(t k+ )=c and y(t k+ )=0. Exaple 5: In the sae setup of Exaple 4 and for α = /, β = /, and γ = /, the atrices (), for the proposed gossip algorith are given by 0 0 0 / / / / Q = / / / / 0 0 0 0, Q = 0 0 0 / / / / / / / / 0 0 0 0 and the atrices (4) for the broadcast algorith are given by R = Q and R = Q. Suppose that node i transits with probability p =/ and node transits with probability p =/. We will show in the next section (cf. Theore 6) that the necessary and sufficient conditions for convergence in expectation and for convergence in variance are r := r σ (p A + p A [ 0 ] ) <, r := r σ (p A A + p A A S) <, [ S := 4 ( ])( [ 0 0 ] [ )] ), respectively, where (A,A ) should be replaced by (Q,Q ) for the gossip algorith G and (R,R ) for the broadcast algorith
B. Coputing r and r yields, r =0.5 and r =0.54 and therefore we conclude that consensus is achieved in expectation and second oent (and a fortiori alost surely), which eans that this linear distributed algorith eets the desired requireent of the proble of Exaple 4, i.e., it achieves average consensus. A. Stability IV. MAIN RESULTS We start by presenting a general result for the converge analysis of the stochastic linear distributed algorith (6). Theore 6: Consider a linear distributed algorith (6) where {U k,k 0} are characterized by (6), (7), and are randoly chosen fro a set M := {B i, i n p }, according to n p Prob[U k = B i ]=p i, p i =. (8) i= Then, the linear distributed algorith converges in expectation to average consensus if and only if n p r σ ( p i B i [ 0 )] ) < (9) i= and converges in second oent to average consensus if and only if n p r σ ( p i B i B i S) <, (0) where S := i= [ ( ])( [ 0 0 )] ). () Note that both gossip and broadcast algorith proposed in Section III coply with the fraework of Theore 6, and therefore we can use the conditions (9) and (0) to assert convergence to average consensus of these algoriths. The next Theore establishes that the gossip algorith G converges in expectation to consensus when the counication topology is syetric. The proof is postponed to Subsection IV-C. Theore 7: For a graph with a syetric adjacency atrix N, described by (), there always exists a syetric doubly stochastic weighted adjacency atrix P (possibly containing self-loops). Moreover, if we set p ij = [P ] ij, α = β = γ =/, the gossip algorith G converges in expectation to consensus. Note that the probabilities p ij can be any as long as P : [P ] ij = p ij is a syetric atrix. An interesting direction for future work is to investigate how to choose the p ij for exaple, to optiize the convergence rate to average consensus, in a siilar way to the distributed algorith proposed in []. For broadcast wireless networks, we have the following result: Theore 8: The broadcast algorith G converges in expectation to consensus, when α = β = γ =/. B. Convergence rates The following theore provides the second oent convergence rate according to Definition of the general stochastic iteration (6), (8) when the tie intervals t k+ t k are independent and identically distributed. Theore 9: Consider the stochastic iteration (6), (8) and suppose that the intervals t k+ t k are independent and identically distributed following a distribution ρ. Then, provided that the linear algorith converges in second oent to average consensus, the second oent convergence rate is given by the unique α that satisfies 0 e αt ρ(dt) = r () where r := r σ ( n p i= p ib i B i S). The proof uses siilar arguents to the ones provided for results for a class of syste known as ipulsive renewal systes [3], and is oitted due to space liitations. It is possible to prove that r is the convergence rate at which the discrete variable z k := x(t k ) converges in second oent to consensus, i.e., (zi k x av) r k(z0 i x av) i, for every coponent i of z. Thus, () relates the convergence rate in absolute-tie t with the convergence rate in discretetie k which denotes the nuber of transissions. If we specialize the Theore 9 to the two scenarios that we consider in the present paper, we obtain the following results. Corollary 0: For the gossip algorith G the second oent convergence rate is given by α = +r where r = r σ ( i= j N out(i) p ij Q ij Q ij S). Corollary : For the broadcast algorith B, the second oent convergence rate is given by the unique α > 0 such that 0 e αt µ(dt) = r where r = r σ ( i= R i R i S). C. Proof of Theore 7 Let R := E[U k ] = i= j N p out(i) ijq ij. Since E[z(t k+ )] = RE[z(t k )], we have that x(tk+ ) E[z(t k+ )] = E[ y(t k+ ) and therefore it suffices to prove that li k Rk = ]=R k z(0) = R k [ x(0) 0 0 ] (3)
fro which we conclude that li k E[x(t k+ )] = x av, x av = x(0). Fro (),[ (3) we] notice that we can R R partition R into blocks R = where each block is R 3 R 4 a linear cobination of the following three atrices X = p ij Λ j, Y = p ij Λ i, Z = i= j N out(i) i= j N out(i) i= j N out(i) p ij e j e i. (4) It is easy to see that Z = P = P (since we assue that P is syetric) and Y = I. Moreover, X = p ij Λ j = Λ j = I, j= i N in(j) j= where we used the fact that i N in(j) p ij =, due to the key assuption that P : P ij = p ij is a doubly stochastic atrix. Therefore, each R i is a linear cobination of the atrices P and I and we can write R = P I + P P. where for α = β = γ =, P = 3, P = [ ]. (5) We denote an eigenvalue of a atrix A by λ i (A) and the set of eigenvalues by {λ i (A)}. Let P S (δ) := P +δp. Then one can obtain that λ i (P S (δ)) = + δ ± δ, i {, }. (6) Let w Pi be the two eigenvector of P S (δ), and v Pj denote the eigenvectors of P (note that P is syetric and therefore it has eigenvectors). Then R has eigenvectors w Pi v Pj, since one can show that R(w Pi v Pj )=λ l (R)w Pi v Pj where the set of eigenvalues of R is given by {λ l (R), l } = {λ i (P S (η j )) : η j {λ j (P )}, i, j } (7) Since P is syetric and doubly stochastic, and it is a weighted adjacency atrix of a strongly connected and aperiodic graph, the eigenvalues of P are real, P has a siple eigenvalue at, and all the reaining eigenvalues belong to the set (, ). Corresponding to the siple eigenvalue of P, R has two eigenvalues at {λ i (P + P )} = {, /}. Corresponding to the eigenvalues of P that belong to the set (, ), the eigenvalues of R are inside the unit circle. This can be shown by noticing that (6) is a strictly increasing function when < δ < for each i and, using this fact, it is easy to conclude that r σ (P + δp ) < for < δ <. Thus R has a single eigenvalue at, all the reaining eigenvalues are inside the unit disk, and the [ ] vectors v R := and w 0 R := [ ] are left and right eigenvalues of R, respectively, associated with this eigenvalue. This iplies that li k R k = w Rv R v R w R, which is (3). V. CONCLUSIONS AND FUTURE WORK We addressed average consensus and randoized gossip algoriths with the constraint that only one node transits data at a given tie. Our ain result is to provide a linear distributed gossip algorith, that provably converges to consensus in expectation. Directions for future work include: (i) extending the results to the stronger stability notion of convergence in second oent; (ii) considering general digraphs, i.e., not requiring the counication topology to be syetric; (iii) optiizing the probability paraeters of the gossip algoriths with respect to soe quantity of interest as the speed of convergence to consensus. VI. 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