Multi-agent coordination via a shared wireless spectrum

Transcription

1 217 IEEE 56t Annual Conference on Decision and Control (CDC) December 12-15, 217, Melbourne, Australia Multi-agent coordination via a sared wireless spectrum Cameron Nowzari Abstract Tis paper considers a planar multi-agent coordination problem. Unlike oter related works, we explicitly consider a globally sared wireless communication cannel were individual agents must coose bot a frequency and power to transmit teir messages at. Tis problem is motivated by te pressing need for algoritms tat are able to efficiently and reliably operate on overcrowded wireless networks or oterwise poor-performing RF environments. We develop a selftriggered coordination algoritm tat guarantees convergence to te desired set of states wit probability 1. Te algoritm is developed by using ideas from event/self-triggered coordination and allows agents to autonomously decide for temselves wen to broadcast information, at wic frequency and power, and ow to move based on information received from oter agents in te network. Simulations illustrate our results. I. INTRODUCTION Tis paper considers a multi-agent coordination problem were agents can only communicate wit one anoter by broadcasting radio signals over a sared wireless spectrum. Tis is in contrast to te overwelming majority of similar works tat assume direct point-to-point communications are possible at all times wit te possibility of independent cances of packet drops. Instead, we consider a model were agents must not only determine wen to communicate wit teir neigbors, but also wat frequency and power teir signal sould be broadcast on. Tis model is motivated by te RF spectrum already becoming overcrowded wit estimates tat tere will be 25 times more demand from te spectrum by te year 23 [1]. In te context of wireless networked cyber-pysical systems as a wole, tis requires a new generation of coordination algoritms tat can (i) efficiently utilize te scarce system wile being (ii) resilient to te fact tat communication may not always be possible. Literature review: In terms of multi-agent coordination, tere is a considerable amount of available literature, see e.g., [2], [3], [4]. A continuous-time algoritm tat acieves asymptotic convergence to average consensus for bot undirected and weigt-balanced directed graps is introduced in [5]. To incorporate te idea of reducing communication, te works [6], [7] use event-triggered broadcasting. More recently, tese works ave been extended to arbitrary directed graps, rater tan only undirected ones [8], [9], []. However, tese works all assume a fixed communication topology wit direct point-to-point communications on demand. Instead, te same type of coordination problem is considered wit a cloud communication model in [11], [12], [13], were agents are only able to communicate indirectly troug te use of a central cloud. However, in te above works, tis communication is always reliable wen needed. Instead, our communication model is based off of works on Cognitive Radio Networks (CRNs), Dynamic Spectrum Access (DSA), and NeXt Generation (xg) networks [14]. In Cameron Nowzari is wit te Electrical and Computer Engineering Department, George Mason University, Fairfax, VA 223, cnowzari@gmu.edu. particular, we are interested in works tat consider dynamic allocation of te spectrum [15], [16]. Tere are a myriad of works on tis topic tat propose various metods for sensing te spectrum in real time and selecting cannels [17], [18], [19], bot in centralized and distributed ways. However, tese works base teir decisions on real-time sensing of te spectrum rater tan troug coordination or collaboration among te users. Instead, we are interested in te coupling of tis type of communication model wit an underlying multiagent/cyber-pysical coordination problem, to find conditions to guarantee reliability of te overall system even under poor RF conditions. Tis idea of utilizing collaboration to elp overcome scarcity in te RF spectrum is already of large interest as corroborated by te currently ongoing DARPA Spectrum Collaboration Callenge [2]. Peraps most similar to te problem we consider is found in [21], were te autors consider te resource management of a single wireless cannel and determine exactly wen agents sould transmit or listen, assuming bot cannot be done simultaneously. Instead, we consider a multiple cannel communication model were agents are assumed to be listening at all times, but must decide for temselves exactly wen, wic frequency, and wat power to broadcast signals on. Our solution metods are based on ideas from event-triggered and self-triggered control, were algoritms are designed to determine precisely wen actions like broadcasting signals or updating actuators sould be carried out to improve efficiency wile maintaining stability and correctness [22], [23], [24], [25], [26]. Statement of contributions: First, we formulate te coordination problem wit a novel communication model tat explicitly considers te globally sared wireless spectrum. Ten, we develop a distributed self-triggered coordination algoritm tat guarantees convergence to a consensus state wit probability 1; owever, tis algoritm is not guaranteed to converge to te exact average of initial states. In particular, tis paper is concerned wit te minimum conditions required to guarantee convergence to a consensus state. Te subject of designing efficient algoritms to better utilize te wireless spectrum troug collaboration and more efficiently acieve consensus will be discussed in a separate paper. Preliminaries: Te Euclidean norm on R N is denoted by. Given two matrices A R n1 n1, B R n2 n2, A B R n1n2 n1n2 denotes teir Kronecker product. Te N-dimensional identity matrix is denoted I N. An undirected grap G = (V, E) is defined by a set of vertices V = {1,..., N} and edges E V V. Undirected means tat (i, j) E if and only if (j, i) E also. An edge (i, j) E means tat vertex j is a neigbor of i. Te set of neigbors of a given node i is given by N i. Te adjacency matrix A R N N is defined by a ij = 1 if (i, j) E and a ij = oterwise. A pat from vertex i to j is an ordered sequence of vertices suc tat eac intermediate /17/$ IEEE 6714

2 pair of vertices is an edge. A grap G is connected if tere exists a pat from all i V to all j V. Te degree matrix D is a diagonal matrix were d ii = N i. Te maximum degree is d max = max i {1,...,N} N i. Te Laplacian matrix is a symmetric, positive semidefinite matrix given by L = D A. If te grap G is connected, te associated Laplacian as exactly one eigenvalue equal to (wit associated eigenvector 1 N ) wit te rest being strictly positive, = λ 1 (L) < λ 2 (L) λ N (L). II. PROBLEM FORMULATION Consider te N agent coordination problem on te plane, were agent i s position is given by p i (t) R 2 wit singleintegrator dynamics ṗ i (t) = u i (t) (1) for all i {1,..., N}, were we are interested in reacing a configuration suc tat p i (t) p j (t) as t for all i, j {1,..., N}. We only consider planar states for simplicity, but note tat te results are extendable to arbitrary dimensions given appropriate communication models. Given a connected interaction grap G, it is well known [5] tat te distributed continuous control law u i (t) = (p i (t) p j (t)) (2) drives eac agent of te system to asymptotically converge to te average of te agents initial conditions. In compact form, tis can be expressed by ṗ = Lp, were p = (p T 1,..., p T N )T is te column vector of all agent states and L = L I 2. However, tis control law requires eac agent to ave exact information about its neigbors at all times and continuously update its control law. Tere ave been many recent works aimed at relaxing one or bot of tese requirements [8], [27], [], [6]. However, tey all require events triggered by some agent to be immediately acknowledged by neigboring agents. In oter words, wen an event is triggered by one agent, its neigbors are immediately aware and can take action accordingly. Te autors in [7] consider tis problem in te presence of unreliable communication but assume tat packet drops randomly occur independent of wat is actually going on in te network. Instead, we are interested in explicitly modeling te fact tat packet drops occur because of interference wit oter agents in te system. In all of tese works (except [7]), tey utilize a piecewise constant implementation of te controller (2) given by u i (t) = ( p i (t) p j (t)), (3) were p j (t) is te last broadcast position of agent j. Note tat altoug agent i as access to its own state p i (t), te controller (3) uses te last broadcast state p i (t). Tis is to ensure tat te average of te agents initial states is preserved trougout te evolution of te system. Tis means tat wen an agent i broadcasts its current position p i (t ) at some time t, all its neigbor j N i immediately update p i = p i (t ) so tat te average can be preserved. Instead, in tis work, we are interested in explicitly modeling wireless interference depending on ow different agents in te network are broadcasting messages. To simplify te communication model later, we consider a discrete-time implementation of te dynamics (1) given by p i (l + 1) = p i (l) + u i (l), (4) were > is te timestep. Unlike all te similar works mentioned above, we consider a new communication model tat is motivated by te overcrowding of te RF spectrum. We describe te communication model next. Communication via a Sared Wireless Spectrum In order to communicate wirelessly, te agents will ave to broadcast signals wit a cosen power and frequency among a sared spectrum. We denote by S = {s 1,..., s M } te set of M different frequency bands, or cannels, available for wireless communication. Since signals at te same frequency can still be successfully transmitted and received wen considering spatial reuse, te wole sared spectrum state at any given timestep l is given by te Cartesian product of pysical space and frequency S = R 2 S. In order for tese signals to be successfully received, tey will ave to be received wit a sufficient level of power witout interference. We now define exactly wat subset of te spectrum S is considered occupied at a given time l given full broadcast information about te network. At any given time l Z, we denote by B i (l) = (s i (l), P i (l)) S R te current broadcast state of agent i, were s i (l) S and P i (l) R are te cosen frequency and power of te signal, respectively. For simplicity, we may drop te explicit dependence on time l. Given an agent i s current position p i and broadcast state B i, we assume tat te signal can be successfully received by any agent j if it is witin r(p i ) of agent i, were r : R R is a nondecreasing function of te broadcast power, and tere is no signal interference at node j s location. Given p i and B i, we can ten compute te portion of te sared spectrum S tat is being used by O i (p i, B i ) = {(p, s i ) S p p i (l) r(p i ), }, Wit tis, we can compute te entire occupied subset of te spectrum O S at any given time by O = O i (p i, B i ). Similarly, we denote by i {1,...,N} O i = j i O j (p j, B j ) te portion of te spectrum being occupied by any agent besides i. More specifically, O i S is te set of pysical locations and frequencies at wic a signal broadcast by agent i cannot be received due to RF interference. Wit tis, given an agent i s current position p i, broadcast state B i, and te current spectrum occupancy O, te set of locations in R 2 tat can successfully receive tis transmission is given by R(p i, B i, O) = B(p i, r(p i )) \ loc(o i, s i ), }{{}}{{} points tat signal can reac points wit RF interference 6715

3 p i = (x i, y i ) R 2 position u i R 2 control input B i = (s i, P i ) S R broadcast state s i S frequency of signal P i R power of signal O i (p i, B i ) S occupied portion of spectrum p i j R 2 last received position of agent j TABLE I AGENT i MODEL DEFINITIONS. were B(p i, r(p i )) = {p R 2 p i p r(p i )} denotes te closed ball centered at p i wit radius r(p i ), and loc(o i, s i ) = {p R 2 (p, s i ) O i } extracts te pysical locations from O i were interference exists at frequency s i. Wit a sligt abuse of notation, we let R(p i, B i, O) = wenever P i =. In oter words, if a signal is broadcast wit zero power, te set of points tat can successfully receive tis signal is empty. Tus, in order for agent j to successfully receive a message from agent i at time l, we need p j (l) R(p i (l), B i (l), O(l)), wic means agent j is close enoug to agent i to receive te signal and tere is no RF interference from any oter agent at tat time. Note tat we assume all agents can receive signals at any frequency s S. Equipped wit tis model, we define p i j (l) as te last successfully received position of agent j by agent i up to time l. More specifically, for neigbors j N i, { p i p j (l) if p i (l) R(p j (l), B j (l), O(l)), j(l) = p i j (l 1) oterwise. (5) Tus, p i j (l) is updated wit agent j s true position p j(l) only if agent i is in a location tat was able to successfully receive te signal B j (l) broadcast by agent j. Based on te information p i j (l) available to agent i at time l, it must ten determine a control input for te dynamics (4). Tese objects are summarized in Table I. Te full problem is formalized below. Problem II.1 Let p i j () = p j() for all i {1,..., N}, j N i. Given te discrete-time dynamics (4) wit te communication model (5), find a distributed communication and control strategy {u i (l), B i (l)} for all agents suc tat lim l p i (l) p j (l) = for all i, j {1,..., N}. To summarize, in eac timestep l Z, eac agent i must determine weter to broadcast a signal or not, wat frequency and power to broadcast at, and wat local control input to use, suc tat te closed-loop system converges. Remark II.2 (Interaction vs. communication) It sould be noted tat our communication model allows agents to communicate independent of te fixed interaction grap G, so an agent i migt ave information about any subset of te agents rater tan only te neigbors in G. Remark II.3 (Multi-agent formation control) We note ere tat for simplicity, we formulate te multi-agent 6716 coordination problem as a consensus problem. Te formal treatment can easily be modified to andle a formation control problem by letting p i (t) = p i (t) b i, were p i (t) is te actual position of agent i and b i is te desired displacement from te average position of te fleet. Alternatively, te interaction grap G could also be used to encode a desired formation into te consensus problem. III. SELF-TRIGGERED ALGORITHM DESIGN Having fully defined te problem, we first point out tat if we remove te problem of determining a frequency and power to broadcast a signal on, te problem becomes very similar to tose studied in many different works considering event-triggered consensus. In particular, tese works study wen agents sould broadcast state information to eac oter and ow to update teir control signals. More specifically, te motion control algoritm used in many event-triggered consensus problems [6], [7], [9], [8], [] is given by u i (l) = ( p i (l) p j (l)), (6) were p i (l) is te last state broadcast by agent i at time l. Note tat tis motion control algoritm implicitly assumes tat any message broadcast by an agent i will be successfully received by its neigbors j N i, e.g., p i j (l) = p j(l) for all i {1,..., N} and l Z. Under tis assumption, we first extend te main results in [27], [7] to discrete time and te iger order space R 2. Lemma III.1 Given te dynamics (4) wit control law (6), if te sequence of broadcast times {l i k } k Z for all agents i {1,..., N} guarantees tat p i p i 2 D i p i p j 2, ( a for all l wit D i = σ i N i 1 2 a N i ) for some a ( ), 2d max and σ i (, 1), ten te system will asymptotically converge to te average consensus state. In oter words, p i (l) Avg(p()) as l for all i {1,..., N}. Lemma III.1 provides a sufficient event-trigger for convergence tat determines wen agents sould broadcast teir states assuming no packet drops or oter wireless interference. Tis result is useful as tese prior works study exactly wen broadcasting state information is necessary in order to reduce te amount of communication required by te network to acieve consensus. Unfortunately, te communication and control strategy given in Lemma III.1 assumes tat communication is always available wen necessary. Instead, under our new communication model and assuming no prior information is sared among te agents, avoiding packet drops completely will be impossible. More specifically, it is very possible tat a signal broadcast by an agent j is not received by some neigbor i for two different reasons. Te first is tat te power used by agent j to send te signal was simply not strong enoug to reac agent j, and te second is tat a tird agent s transmission is interfering wit agent j s signal at agent i s location. Consequently, te control law (6) is no longer implementable because tis assumes no packet drops or wireless interference occurs.

4 Motion Control Law Instead, we consider te control input for agent i at any given time to be u i (l) = p i (l) p i j(l), (7) were p i j (l) is te last successfully received position of agent j by agent i given in (5). In oter words, p i j (l) only gets redefined as p j (l) if agent i was in te set of points R(p j (l), B j (l), O(l)) R 2 tat could successfully receive te broadcast B j (l) from location p j (l) wit no interference. Anoter difference ere wit respect to (6) is tat agent i is now using its exact position p i (l) rater tan its last broadcast position p i (l). Te reason for tis is due to te nature of ow communication occurs. In particular, agents are only able to decide for temselves wen to broadcast signals to oter agents, but tere is no mecanism for tem to request information from tem. Tis means tat wen an agent i doesn t know ow to move because its information is too outdated, it must wait for additional information to arrive before it can begin moving again. Tus, wile te agent is waiting, te control law (7) ensures tat tis agent will eventually stop moving until new information is received. Tis type of control law is desirable in our scenario because it is guaranteed to not drive te system to instability even if te communication network is completely sutdown. Interestingly, trajectories of te dynamics (4) wit control (7) beave very similarly wit consensus algoritms wit randomized gossiping [28], [29] or switcing topologies [5]. Consequently, we are able to provide a result to guarantee convergence as long as enoug successful transmissions occur. Wit a sligt abuse of notation, we define {l i j k } k Z {l i k } k Z as te sequence of times tat a message broadcast by agent i was successfully received by agent j. Ten, given te communication model (5), {l i j k } k Z = {l i k p j(l i k ) R(p i(l i k ), B i(l i k ), O(li k ))}, is te subset of times at wic agent i broadcast a message tat was successfully received by anoter agent j. A condition for convergence is ten formalized in Teorem III.2. Teorem III.2 Given te dynamics (4) wit control law (7), if l i j k as k for all i {1,..., N}, j N i, ten lim l p i (l) p j (l) = for all i, j {1,..., N}. Communication Strategy Given te result of Teorem III.2 and te fact tat we are only concerned wit stability, it is easy to imagine tere are many different ways to design communication strategies tat satisfy te conditions of Teorem III.2. Due to space restrictions, we only propose one possible strategy and discuss a few of its key features aimed at minimizing ow muc communication is required wile still guaranteeing convergence. Coosing a communication strategy to optimize different metrics are instead studied in a separate paper. Te tree components of te communication algoritm we need to design are ow agents sould determine wen to communicate, wit wat power, and wat frequency. To answer te question of wen agents sould decide to 6717 broadcast, inspired by Lemma III.1, we define te selftriggered broadcasting time as l i k+1 l i k = τ 1 τ 2 + p i p i j 2 were τ 1, τ 2 > are design parameters. Equipped wit a metod for determining wen agents sould communicate, tey also need to coose a power and frequency. In general, it would be ideal for an agent i to use te minimum power required to reac its neigbors, P i = r 1 (max p i p j ), were r 1 (D) is te power required to transmit a message a distance D away. Unfortunately, agent i will not know te positions of its neigbors in general. However, since te goal of te algoritm is to acieve consensus, te power required over time sould be decreasing. Unfortunately, we cannot do tis blindly since we do not assume any acknowledgement messages, it would be impossible for agent i to know if any message as successfully been received by any agent. To address tis, we define an integer d >, were if an agent i as not successfully received any message from a neigbor j N i in te time tat agent i as broadcast its position d times, it will increase its communication power. Tis guarantees tat, wile a neigbor j cannot get a message to i, agent i will keep increasing its communication power until eventually te broadcast can reac agent j. In turn, agent j will receive agent i s position and set its power accordingly. Altoug tis guarantees tat a message can eventually be sent wit enoug power, we still need to coose a frequency. To guarantee tat all messages will eventually succeed (even if wit a small probability), we utilize a simple jump-stay cannel opping metod were an agent i will stick to its last used frequency wit probability α and randomly cange its frequency wit probability 1 α. Te self-triggered coordination algoritm is formally described in Table II. For eac agent i {1,..., N} at times l Z, performs: 1: receive p i j = p j from all neigbors j were p i R(p j, B j, O) and set l j i = l 2: if l = l i k (tis is a sceduled broadcast time) ten 3: if j suc tat l j i l i k d ten 4: set P i (l) = 1.1P i (l 1) 5: else 6: set P i (l) = min{p i (l), r 1 (max j p i p i j ) 7: end if 8: set s i (l) = s i (l i k 1 ) w.p. α, or randomly from S w.p. 1 α 9: broadcast position p i (l) and next sceduled broadcast time l i k+1 at frequency s i(l) and power P i (l) : set k = k : end if 12: update control u i (l) according to (7) TABLE II SELF-TRIGGERED COORDINATION ALGORITHM. Te main convergence result, formally stated below, is ten a direct consequence of Teorem III.2 and te fact tat te self-triggered coordination algoritm ensures all,

5 agents will attempt to communicate wit teir neigbors wit sufficient power an infinite number of times. Teorem III.3 Te self-triggered coordination algoritm guarantees asymptotic convergence to te set of consensus states wit probability 1, i.e., p i (l) p j (l) for all i, j {1,..., N} as l wit probability 1. It is clear and known tat randomly selecting a cannel is not efficient [3] and efficiently selecting cannels based on real-time spectrum sensing is currently an active area of researc [31], [17], [18]. Instead, we are interested in using collaboration between agents to better utilize te spectrum; owever, as tis paper is only concerned wit stability tis will be discussed in detail in a separate paper. IV. SIMULATIONS Due to te limited space and te fact tat we ave not yet discussed communication strategies designed to optimize various efficiency metrics, we only sow two executions of te proposed algoritm for a system of N = 2 agents wit d = 5, α =.5, =.1, τ 1 =.5, and τ 2 =.5 to verify te main convergence result of te paper. We compare executions of te algoritm wit M = 3 and M = 5 different cannels. Figures 1-2(a) sow te orizontal component of agent positions over time and Figures 1-2(b) sow te power being utilized (capped based on initial condition) by eac agent to broadcast signals. It is interesting to note tat te plots (b) reveal tat te agents are struggling to establis communication to begin wit before tey are able to well coordinate temselves. Giving tem more cannels allows tem to establis successful communications early on to begin coordinating temselves better. As tis paper is only concerned wit stability, discussions about performance will be presented in a separate paper. x i (a) r(p i ) Fig. 1. Plots of (a) orizontal component of positions and (b) power allocations over time wit M = 3 cannels and N = 2 agents. x i (a) r(p i ) 15 5 (b) Fig. 2. Plots of (a) orizontal component of positions and (b) power allocations over time wit M = 5 cannels and N = 2 agents. (b) V. CONCLUSIONS Tis paper formulates and solves a coordination problem were agents communicate troug a sared wireless spectrum. We propose a self-triggered coordination algoritm tat combines a communication strategy wit a motion control law tat guarantees convergence wit probability 1. Te focus of tis paper is on minimum conditions for stability under te proposed communication model motivated by te need for coordination algoritms tat need to communicate via crowded or oterwise unreliable wireless networks. Future work will be devoted to actually quantifying te efficiency of te network and formulating appropriate optimization problems to make better use of te scarce spectrum troug tigt collaboration among te agents. In addition, we also plan to furter develop te communication model to capture more realistic scenarios. For instance, we ave only considered a syncronous discrete-time TDMA scedule wic we plan to relax in te future. We also currently only consider a ard collision model were bot packets are dropped wen collisions occur. However, by more explicitly considering real-time Signal-Interference-to-Noise Ratio (SINR) wen determining collisions, it is possible for interference to only cause one of te two interfering packets to be dropped. Tis will give agents te ability to essentially sout teir ig-priority messages to increase te cance of it being successfully received at te cost of drowning out communications to nearby agents operating at tat frequency. 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Lin, Efficient cannel-opping rendezvous algoritm based on available cannel set, arxiv preprint arxiv: , 215. [31] M. Zeng, J. Lin, W. Liang, and H. Yu, A priority-aware frequency domain polling MAC protocol for OFDMA-based networks in cyberpysical systems, IEEE/CAA Journal of Automatica Sinica, vol. 2, no. 4, pp , 215. APPENDIX Proof of Lemma III.1: For convenience, we define (x i, y i ) = p i and ( x i, ŷ i ) = p i. Let V (l) = 1 2 (xt Lx + y T Ly). Given te dynamics (4) wit te motion control law (6), defining V (l) = V (l + 1) V (l), we ave V (l) = (x T LL x + y T LLŷ) ( L x 2 + Lŷ 2 ). Letting V x (l) expand V (l+1) V (l) and e x = x x, we can V x (l) = (1 12 ) L x 2 + e T x LL x. Defining V y (l) similarly, note tat te sign of V (l) is equivalent to te sign of V x (l) + V y (l). Given te similarity of V x and V y, we now focus on V x to simplify te presentation. Letting ẑ L x, we can expand out V x (l) = (1 12 ) ẑi 2 + ẑ i (e x,i e x,j ) = (1 12 ) ẑi 2 + N i ẑ i e x,i ẑ i e x,j. Using Young s inequality, we can bound N i ẑ i e x,i 1 2 N i ẑ 2 i a + 1 ẑ i e x,j 2a N i e 2 x,i, 1 2ẑ2 i a + 1 2a e2 x,j, for any a >. Since te grap is undirected, we ave a e2 x,j = 2a e2 x,i = 2a N i e 2 x,i. Combining tis wit te inequalities above yields V x (l) (a N i + 12 ) 1 ẑi 2 + N i a e2 x,i. Coosing a small enoug suc tat a < {1,..., N}, if e 2 x,i σ i ( a N i ) ( 1 ) 2 a N i ẑi 2, 2 N i for all i is satisfied at all times l, ten ( V (l) (σ i 1) 1 ) 2 a N i ẑi 2. Combining tis wit te condition of te Lemma reveals tat V (l) < as long as tere exists i, j suc tat p i p j. Te proof is finally concluded by noticing tat Avg(p(l)) is in invariant quantity under (6). Proof of Teorem III.2: Due to space restrictions we only present a simple sketc ere. Rater tan analyzing p i directly, we will instead analyze its components (x i, y i ). Let x + (l) = max i x i (l) and x (l) = min i x i (l). Take any i suc tat x i = x +, ten according to (7) ẋ i = x i x i j. (8) Noticing tat x i j is simply a random, nonuniform sampling of x j, we can always bound x i j [x, x + ], wic means we can upper-bound (8) by. Tus, x + (l) x + () for all l. Similarly, x i(l) x () for all l. To sow tat eventually x + (l) = x i (l), notice tat if x + (l) x i (l), tere exists at least one i wit x i = x+ i wit a neigbor j suc tat x j < x i, because te grap is connected. Using tis, we see tat (8) becomes strictly negative assuming updated information is available. Finally, it is important to note tat under (7), in te case tat p i j are assumed constant, te convergence of x i to a fixed point is asymptotic. Tis rules out te possibility of two agents being able to fully swap places in between successful communications, wic concludes te result. 6719