Dstrbuted Topology Control of Dynamc Networks Mchael M. Zavlanos, Alreza Tahbaz-Saleh, Al Jadbabae and George J. Pappas Abstract In ths paper, we present a dstrbuted control framework for controllng the topology of dynamc mult-agent networks. Agents are equpped wth local sensng and wreless communcaton capabltes, however, due to power constrants, they are requred to swtch between two modes of operaton, namely actve and sleep. The control objectve nvestgated n ths paper s to determne dstrbuted coordnaton protocols that regulate swtchng between the operaton modes of every agent such that the overall network guarantees mult-hop communcaton lnks among a subset of so called boundary agents. In the proposed framework, coordnaton s based on a vrtual market where every request to swtch off s assocated wth a bd. Combnatons of requests are verfed wth respect to connectvty and the one correspondng to the hghest aggregate bd s fnally served. Other than nearest neghbor nformaton, our approach assumes no knowledge of the network topology, whle verfcaton of connectvty reles on notons of algebrac graph theory as well as gossp algorthms run over the network. Integraton of the ndvdual controllers results n an asynchronous networked control system for whch we show that t satsfes the connectvty specfcaton almost surely. We fnally llustrate effcency of our scalable approach n nontrval computer smulatons. I. INTRODUCTI Dstrbuted control of networked mult-agent systems has recently receved consderable attenton. Such systems typcally consst of large numbers of nexpensve agents equpped wth ntegrated sensng and wreless communcaton capabltes. Whle the agents prmary task s detecton of certan physcal changes wthn ther proxmty, ther communcaton capabltes enable them to share the ndvdually collected data wth ther peers, n order to acheve a global coordnated objectve. Consequently, connectvty of the underlyng network s a crtcal requrement. In the presence of moble agents or agents that can swtch between actve and sleep operatng modes, mantanng connectvty of the underlyng network becomes a challengng task due to the contnuous changes n the network topology. In the former class of problems belongs [], where a measure of local connectvty of a network s ntroduced that under certan condtons s suffcent for global connectvty as well as [2], where a controllablty framework for state-dependent dynamc graphs s developed. Dstrbuted mantenance of nearest neghbor lnks n formaton stablzaton s addressed n [3], whle n [4] topology control of a moble network Ths work s partally supported by ARO MURI SWARMS Grants W9NF-5--29 and W9NF-5--38 as well as DARPA DSO SToMP and NSF ECS-347285. Mchael M. Zavlanos, Alreza Tahbaz-Saleh, Al Jadbabae and George J. Pappas are wth the Department of Electrcal and Systems Engneerng, Unversty of Pennsylvana, Phladelpha, PA 94, USA. {zavlanos,atahbaz,jadbaba,pappasg}@seas.upenn.edu s acheved by means of gossp algorthms and marketbased coordnaton. In [5], the authors address the problem of maxmzng the second smallest egenvalue of the graph Laplacan, whle a decentralzed approach to ths problem based on supergradent methods and dstrbuted egenvector computaton s consdered n [6]. Network connectvty for double ntegrator agents s nvestgated n [7], where exstental as well as optmal controller desgn results are dscussed. Equally challengng problems arse when the changes n the network topology are due to power constrants that requre agents to occasonally swtch off. In ths context, duty cyclng of sensor networks s nvestgated n [8]. Smlarly, cone based topology control for ad-hoc sensor networks [9] as well as dstrbuted connectvty control algorthms n the absence of exact locaton nformaton [] are among other varants of the problem nvestgated n the lterature. Inspred by the problems of the latter class, n ths paper, we propose a dstrbuted strategy for topology control of a network of statonary agents, each one capable of swtchng between on and off operaton modes. In partcular, gven a subset of so called boundary agents, we desgn local coordnaton protocols that allow agents to ndvdually swtch on and off, whle mantanng a connected network among the boundary agents. Our approach assumes no knowledge of the network topology, other than nearest neghbor nformaton, whle the proposed coordnaton scheme depends on the operaton status of every agent. In partcular, off agents can only swtch on f by dong so they do not create clusters of actve agents, dsconnected from the man network. On the other hand, every request to swtch off s assocated wth a bd and gossp algorthms run over the network allow agents to verfy combnatons of requests wth respect to the connectvty specfcaton and serve the one correspondng the hghest aggregate bd. Connectvty s captured by the graph Laplacan matrx and can be checked n a dstrbuted way by comparng the asymptotc values of a randomly ntalzed consensus run by all actve agents n the network that do not request to swtch off. Integraton of the ndvdual controllers results n a dstrbuted networked mult-agent system for whch we show that connectvty of the network nvolvng the boundary agents s guaranteed almost surely. The rest of ths paper s organzed as follows. In Secton II we defne the problem of topology control of networked mult-agent systems and develop the necessary graph theoretc background to capture connectvty. In Secton III we develop the proposed dstrbuted coordnaton protocols and dscuss ther propertes. Fnally, nontrval computer smulatons llustratng the effcency of our approach are presented n Secton IV.
II. PRELIMINARIES AND PROBLEM FORMULATI Consder a group of N statonary agents wth ntegrated sensng and wreless communcaton capabltes, deployed n a p-dmensonal space R p. Let x R p denote the poston of agent and assume that each agent s subject to energy constrants and as a result, has two modes of operaton, namely (actve) and (sleep). When, agent s capable of communcatng wth other agents located n a dsk of radus r c centered at x. When, all communcaton lnks wth other agents n the network are dsabled. Fnally, we assume a set of so called boundary agents, whch are agents that reman permanently. Such agents would be placed n key locatons where contnuous sensng and communcaton s requred. The setup descrbed so far can be captured usng an undrected dynamc graph G(t) = (V b V(t), E(t)), where V b denotes the fxed set of boundary agents, V(t) denotes the set of agents that are at tme t and E(t) denotes the set of communcaton lnks between all agents n V b V(t) at tme t. Any par of agents and j at tme t such that x x j < r c are called neghbors or adjacent and the assocated communcaton lnk s denoted by (, j) E(t). In partcular, we can defne the set of neghbors of agent at tme t by N (t) = {j V b V(t) (, j) E(t)}. Then, the objectve nvestgated n ths paper can be stated as follows. Problem (Dstrbuted Topology Control): Gven a set of N agents, consstng of boundary and non-boundary ones, determne a dstrbuted control framework that regulates the operaton status of all non-boundary agents such that a communcaton path s mantaned between any two boundary agents at all tmes. The exstence of a communcaton path between any two boundary agents s closely related to the noton of connectvty of the graph G(t). In partcular, we say that G(t) s connected at tme t, f there exsts a path,.e., a sequence of dstnct nodes such that consecutve nodes are adjacent, between any two nodes n G(t). Hence, Problem equvalently mples that we want G(t) to reman connected for al tme. The desred connectvty objectve can be captured usng the algebrac representaton of the dynamc graph G(t). In partcular, the structure of any dynamc graph G(t) can be equvalently represented by a dynamc Laplacan matrx, L(G(t)) = D(G(t)) A(G(t)) where A(G(t)) = (j (t)) denotes the V b V(t) V b V(t) adjacency matrx of the graph G(t), such that (t) = and j (t) = f and only f (, j) E(t), and D(G(t)) = dag ( j V b V(t) j(t) ) denotes ts correspondng degree matrx. 2 The followng lemma relates graph connectvty to the spectral propertes of the Laplacan matrx L(G(t)) []. Clearly, communcaton paths between any two boundary agents may exst even f G(t) conssts of multple connected components that are dsconnected from each other, as long as all boundary agents belong to the same connected component. We wll not deal wth ths case here. Instead we wll requre that G(t) conssts of a sngle connected component. 2 We denote the cardnalty of the set V by V. Lemma 2.: Let L(G) be the Laplacan matrx correspondng to the graph G and let λ (L(G)) λ 2 (L(G))... be ts ordered egenvalues. Then, λ (L(G)) = wth correspondng egenvector,.e., the vector of all entres equal to. Moreover, λ 2 (L(G)) > f and only f G s connected. Lemma 2. mples that the second smallest egenvalue, also called the Fedler egenvalue, of the postve semdefnte Laplacan matrx L(G) s strctly postve, or equvalently, that ker L(G) = span{}. As a result, we have the followng well-known result. Theorem 2.2: Consder a fxed graph G on N nodes assocated wth state varables θ (t) R each, that are updated accordng to the set of lnear dfferental equatons θ(t) = L(G)θ(t), where θ(t) = [ θ (t) θ N (t) ] T. Then the network G s connected f and only f, lm θ(t) = α span{}. () t for all ntal condtons θ() R N. In other words, Theorem 2.2 says that all nodes n G wll eventually reach a consensus on ther state values θ (t), for all ntal condtons, f and only f the graph G s connected. Ths theorem s n fact a specal case of the well-known dstrbuted consensus schemes over graphs dscussed n [2] [6]. In the case that G s dsconnected, let G c = (V c, E c ) denote ts c-th connected component. Then, ker L(G) = span{ Gc G c } where Gc s an N vector wth ts -th entry equal to f V c and equal to, otherwse. Consequently, for random ntalzaton of the states θ(), lm t θ(t) = c α c Gc such that α c R are dfferent for dfferent connected components G c almost surely. Therefore, connectvty of a network G can be verfed almost surely by comparng the asymptotc state values () of all agents, for any random ntalzaton. III. DISTRIBUTED COORDINATI The man objectve n ths secton s to derve a dstrbuted coordnaton mechansm that allows agents to swtch and wthout volatng the desred connectvty specfcaton. Clearly, agents swtchng can, n general, only ncrease connectvty of the network, f by dong so they do not create new connected components. On the other hand, swtchng whle preservng connectvty becomes possble by means of a dstrbuted vrtual market, where agents that are bd n order to swtch. Combnatons of swtchng requests can be consdered smultaneously. Each such request s verfed wth respect to the connectvty specfcaton and among the ones that are safe, those correspondng to the hghest aggregate bds are eventually processed. Other than knowledge of the nearest neghbors of every agent, no further nformaton regardng the topology of the network s requred. Nevertheless, correctness of our approach reles on knowledge of all agents partcpatng n every aucton, whch can be obtaned n a dstrbuted multhop fashon, as well as on some noton of synchronzaton
Algorthm Intalzaton Phase for Agent. () =, phase() = ; Requre: Self-Bd b () ; : f I I I then 2: Fnd neghbors N := {j N 3: Exchange status and bds wth neghbors,.e., := max j N {, s [j] b := max j N 4: Fnd agents, I 5: else f I {b, b [j] }; := {j I s I = I then 6: f wat() = then 7: Set wat() := wat() + and repeat steps 2-3; 8: else f wat() = then 9: Set R := 2 : Set I {j I j=argmax(k) l I {b a (l)}} := I\I ; : Swtch to Verfcaton Phase,.e., phase() = 2; 2: end f 3: end f of all agents to the same aucton. Snce, dstrbuted systems, ncludng the one proposed n ths paper, are n general asynchronous, the desred synchronzaton s event-based and s obtaned by labelng every aucton n the set {, 2, 3} and requrng that any nformaton exchange takes place only among neghbors that are n equally labeled auctons. Effectvely, all agents that are are always synchronzed n the sequence of auctons {, 2, 3,, 2, 3,... }. Dependng on the status of an agent, the coordnaton mechansm can be decomposed nto an ntalzaton phase, a verfcaton phase and a decson phase for agents as well as a swtchng phase for agents. A. Intalzaton Phase for Agents As dscussed above, any new aucton {, 2, 3} that s ntalzed requres knowledge of all the agents partcpatng n that aucton. Ths nformaton s encoded n a status vector {, } N such that (j) = f agent j s and (j) = f agent j s. We further defne the sets I and I that consst estmates of the and agents n the network, respectvely. Intally, every agent s aware of ts own operaton status only,.e., I = {} and I =. On the other hand, the operaton status of the whole network s obtaned when I I = I. The ntalzaton phase for every agent, denoted by phase() =, conssts of updatng I, gven an estmate of I. Durng ths process, agent also collects bds b (j) R from all agents j that desre to swtch and forms the set of requests to be verfed wth respect to connectvty. The ntalzaton phase s descrbed n Algorthm. For every agent enterng the ntalzaton phase we requre that () =, (j) = for all j, I = {} as well as a self-bd b a () R, such that b () >, f agent desres to swtch, and b () =, otherwse. Note that b () = for all boundary agents. Whle the operaton status of the whole network has not yet been obtaned (lne, Algorthm ), agent dentfes ts neghbors N (lne 2, Algorthm ) and updates ts status and bds vectors and b, respectvely (lne 3, Algorthm ), as well as the set I (lne 4, Algorthm ). Once the condton I I = I ndcatng full knowledge of the network status s satsfed (lne 5, Algorthm ), one more update of the status and bds vectors s requred for agent to also obtan accurate knowledge of the status and bds of all agents. Ths s due to the requrement that agents can only swtch f they have neghbors, n order to avod clusters of agents that are dsconnected from the man network. Clearly, collectng the status and bds of the new addtons to the network requres no more than one communcaton cycle once the condton I I = I, ndcatng that nformaton from ther neghbors has been receved, s satsfed (lne 7, Algorthm ). To model ths extra communcaton cycle, we ntroduce a dummy varable wat(), ntalzed at and set to once the extra update has been done. The fnal step of the ntalzaton phase conssts of dentfyng the agents assocated wth the k largest bds n b and formng the set of requests R consstng of all 2 k combnatons of these agents (lne 9, Algorthm ). Havng accurate knowledge of the agents I, agent can also update I and then swtch to the verfcaton phase (lnes and 2, Algorthm ). Note that, snce updatng of the status and bds vectors of any agent nvolves nformaton provded by neghbors that are n the same aucton (lne 3, Algorthm ), all agents mplementng Algorthm wll eventually converge to the same values for and b. B. Verfcaton Phase for Agents The verfcaton phase for every request r R conssts of every agent r runnng a consensus update, x (r) := x ( (r) x (r) x [j] (r) ) (2) j N \{r} where x (r) R s a randomly ntalzed scalar, on the reduced network obtaned by assumng that all agents n r are. If the reduced network s not connected, the assocated consensus wll converge to equal values x [j] (r) for all agents j I \{r} wth probablty zero. Ths can be checked n a dstrbuted fashon, by means of a maxmum and mnmum consensus on the solutons of the correspondng consensus (2). For ths, every agent updates varables M a, m R R contanng the current values for all requests for the maxmum and mnmum consensus, respectvely. The verfcaton phase s descrbed n Algorthm 2. Runnng consensus (2) for request r conssts of two stages, namely ntalzaton and updatng of consensus (2) and determnng whether t has converged. Wth each one of these stages and every agent we assocate varables u, c {, } N R, respectvely, such that u (j, r) = (smlarly, c a (j, r) = ) ndcates that agent s aware that agent j has begun updatng (smlarly, has determned convergence) of
Algorthm 2 Verfcaton Phase for Agent. () =, phase() = 2; Requre: mn j I,r R c (j, r) = ; : Fnd neghbors N := {j N 2: Exchange nformaton wth neghbors,.e., u := max j N {u, u [j] c := max j N {c, c [j] M a := max j N {M a, M a [j] m := mn j N {m, m [j] }; 3: Fnd set of requests that are not beng updated, R u := {r R u (, r) = }; 4: for all requests r R u do 5: Set u (, r) = ; 6: Randomly ntalze a scalar x a (r); 7: end for 8: Fnd set of requests that are beng updated but have not converged, R u c := {r R u (, r) =, c 9: for all requests r R u c do : f r then : Update x (r) accordng to (2); 2: f {x (r)} and mn j I 3: Set c (, r) :=, 4: end f 5: end f 6: end for (, r) = }; u M (r) := max{x (r), M m (r) := mn{x (r), m (j, r) = then (r) (r)}; consensus (2) for request r. Durng the ntalzaton stage of the verfcaton process, every agent dentfes requests R u that are not beng updated yet (lne 3, Algorthm 2) as well as requests R u c that are beng updated but the correspondng consensus (2) has not yet converged (lne 8, Algorthm 2). For all requests r R u, agent ntalzes consensus (2) by settng u (, r) = and randomly ntalzng a scalar x (r) (lnes 5 and 6, Algorthm 2). On the other hand, for every request r R u c, f agent r, t updates x (r) accordng to consensus (2) (lne 5, Algorthm 2). Determnng whether consensus (2) for request r R u c has converged depends not only on convergence of the sequence {x (r) but also on the condton that all other agents I have ntalzed the correspondng consensus (lne 2, Algorthm 2). In ths way, false convergence alarms due to delays n nformaton propagaton n the network, are avoded. When consensus (2) for a request r R u c has converged, agent sets c (, r) = and performs a maxmum and mnmum update on the varables M a (r) and m (r), respectvely (lne 3, Algorthm 2). The verfcaton phase lasts as long as there exst requests r R for whch agent s awatng a convergence message c (j, r) by agents j I, whch translates to the requrement that Algorthm 3 Decson Phase for Agent. () =, phase() = 2; Requre: mn j I,r R c (j, r) = ; : Reset, b := a old a old N, u a old M a old := 3 R, m a old = (mod 3);, c := a old N R and := 3 R, where a old 2: Fnd safe requests, S := { r R M a (r) m (r) < ɛ } { 3: f S and argmax r S j r b (j) } then 4: Set := and () := ; 5: else 6: Set := + (mod 3); 7: Set () :=, I := {} and phase() := ; 8: Set I := I { { argmax r S j r b (j) }} ; 9: end f mn j I,r R c (j, r) =. Note that, wth every cycle of Algorthm 2, the varables u, c, M, m are locally updated wth nformaton from agent s neghbors N that are n the same aucton (lne 2, Algorthm 2), whch guarantees that eventually all varables converge to the same values for all agents. C. Decson Phase for Agents Once agent has obtaned convergence messages for all requests from all agents,.e., once mn j I,r R c (j, r) =, t enters the decson phase (Algorthm 3). Upon enterng the decson phase both varables M a, m have converged to ther global mnmum and maxmum values, respectvely, over the whole network, whch s due to smultaneous updatng of all c, M a, m (lne 2, Algorthm 2). The decson phase conssts of comparng the values of M a (r) and m (r) for every request r R and decdng safety dependng on whether these values are equal or not. In partcular, every agent dentfes the set of safe requests S (lne 2, Algorthm 3) and wth every request r S, t assocates a cost correspondng to the aggregate bd values j r b (j) of all agents partcpatng n ths request. 3 If requests that are safe wth respect to connectvty exst,.e., f S, then the one assocated wth the hghest cost s fnally served. In other words, f agent belongs to the request wth the hghest cost (lne 3, Algorthm 3) t swtches (lne 4, Algorthm 3). Otherwse, t ntalzes a new aucton (lne 6, Algorthm 3) and updates ts set of agents I by addng the agents n the hghest-cost request that s to be served (lne 8, Algorthm 3). Ths new set of agents I s to be used n the followng ntalzaton phase (Algorthm ). Whether agent stays or swtches, t resets all varables correspondng to the prevous aucton 3 Other cost functons could also be consdered.
Algorthm 4 Agent swtchng () = ; : f pend() = and agent wakes up then 2: Fnd neghbors N := {j N s [j] 3: Set := max j N a j {a j } + (mod 3); 4: Set pend() = ; 5: else f pend() = then 6: Fnd neghbors N := {j N s [j] 7: f N and max j N 8: Set a j {a j } = then () := and phase() := ; 9: Set I := {} and I := I[l] \{ where l := argmax j N {a j }; : else f N = then : Set pend() = ; 2: end f 3: end f (lne, Algorthm 3), so that ths old nformaton can not be used n future equally labeled auctons. D. Agents Swtchng Swtchng needs to satsfy two man objectves. Frst, no clusters of agents that are dsconnected from the man network should be created. Second, all agents swtchng should synchronze themselves wth the ntalzaton phase of the current aucton of ther neghbors. The frst objectve s acheved by requrng that no agent can swtch f t has no neghbors. The second, on the other hand, reles on observng the sequence of auctons of the neghbors and jonng when possble. In partcular, when an agent wakes up, t dentfes ts neghbors N (lne 2, Algorthm 4) and, f such neghbors exst, t prepares to jon the aucton followed by ts neghbors current aucton by approprately ntalzng (lne 3, Algorthm 4). Note that, due to synchronzaton of all agents to the same aucton, at the tme when agent wakes up, all ts neghbors are ether n the same aucton or n two consecutve ones. The latter case occurs f wakng up of agent concdes wth a transton of ts neghbors to a new aucton. Hence, the update n lne 3 of Algorthm 4 s well defned. Once agent has dentfed the aucton t plans to jon, t enters a pendng phase denoted by pend() = (lne 4, Algorthm 4). Durng ths phase, t keeps observng the auctons of ts neghbors N and as soon as one of ts neghbors enters the aucton agent plans to jon (lne 7, Algorthm 4), 4 t swtches and prepares to jon the ntalzaton phase by updatng the varables I, I and s () (lnes 8 and 9, Algorthm 4). Note that, f durng the tme that agent s pendng, all ts neghbors decde to swtch, then agent remans and wats for ts neghbors to swtch back (lnes and, Algorthm 4). Note also that, n ths latter case 4 The maxmum here s taken modulo 3....2.4.6.8 2 (a) Intal Confguraton.2.4.6.8 2 (c) Swtchng / Phase Fg.....2.4.6.8 2 (b) Swtchng Phase.2.4.6.8 2 (d) Fnal Confguraton Dstrbuted Topology Control for N = 5 agents. none of ts neghbors wll swtch to the aucton that agent wshes to jon due to the update n lne 4 of Algorthm 3, whch guarantees that the condton n lne 7 of Algorthm 4 wll not be enabled. E. Correctness of Dstrbuted Coordnaton Correctness of the proposed dstrbuted coordnaton framework s obtaned by constructon and s dscussed n detals n the prevous subsectons. Those deas are summarzed n the followng result. 5 Proposton 3. (Correctness): Assume N agents, ntally formng a connected network, each one of whch s able to swtch or accordng to Algorthms -4. Then, a connected network ncludng all boundary agents s guaranteed almost always. IV. SIMULATIS In ths secton we llustrate the proposed topology control algorthm n a nontrval connectvty task and show that t has the desred lveness, safety and scalablty propertes. In partcular, we consder ten boundary agents symmetrcally postoned on the upper and lower faces of a rectangle regon and 4 other agents randomly dstrbuted n the nteror of the regon such that all N = 5 agents ntally form a connected network (Fg. (a)). Boundary agents are denoted wth green squares, whle the remanng agents wth blue or red dots, dependng on whether they are or, respectvely. The communcaton range s taken r c =.2 and neghborng relatons are denoted wth lnes drawn between them. The goal of ths task s to let agents swtch and, always mantanng a connected network among the boundary agents. To best llustrate our approach, we decompose the proposed connectvty task n three stages. Frst we only allow agents to swtch amng at ntroducng some sparseness n the network (Fg. (b)). Then, we enable agents to also swtch (wth probablty.) and study how they are able to synchronze wth the man network (Fg. (c)). 5 Due to space lmtatons, the proof of ths result s omtted.
5 5 2 25 3 35 4 45 5 5.9 3 4.8 2.8 3.7 2.6 2 9.6.5.4 2.4 2.2 2.8 2 2 3 8.3.6 7.2.4.2 6. 5 5 5 2 25 3 35 4 45 5 75 8 85 9 95 5 5 2 25 Fg. 2. (a) Number of Agents (b) Fedler Egenvalue Performance of Dstrbuted Topology Control for N = 5 agents. The fnal stage conssts of pushng our algorthm to ts lmt and checkng whether t s able to always mantan a connected network even f we never allow agents to swtch agan. In partcular, we see that the fnal network s almost a tree structure (Fg. (d)). The requests verfed wth respect to connectvty durng every aucton consst of all 2 k combnatons of the k = 6 hghest bds. Fgs. 2(a) and 2(b) llustrate the number of agents that are and the Fedler egenvalue of the overall network, respectvely. Note the three stages of our task as well as the fact that our approach succeeds n mantanng a connected network among the boundary agents. Fg. 3 shows a snapshot of the sequence of auctons durng the transton tme from the swtchng stage of the task to the swtchng / stage. Vertcal lnes ndcate transtons from one aucton to another. Horzontal lnes, on the other hand, ndcate ether the last aucton of agents before they swtched (thn blue lnes) or the tme spent n every aucton by agents (thck red lnes). Observe that the sequence of auctons s of the form {, 2, 3,, 2, 3,... } as predcted, as well as that all agents are synchronzed n equally labeled auctons. On the other hand, agents that swtch are always synchronzed wth the network n the desred aucton (arrows and numbers ndcatng the aucton). Fnally, note that n the presence of more agents (/ stage) the tme spent n every aucton s slghtly shorter than before due to the fact that nearest neghbor consensus updates converge faster n denser networks. V. CCLUSIS In ths paper, we presented a dstrbuted control framework to regulate swtchng between the and operaton modes of a group of agents such that the overall network guaranteed mult-hop communcaton lnks among a set of boundary agents. Coordnaton was based on a vrtual market where requests to swtch off were assocated wth bds. Gossp algorthms were used to verfy safety of combnatons of these requests wth respect to connectvty. Among the safe requests, the one correspondng to the hghest aggregate bd was fnally served. Connectvty was captured by the graph Laplacan matrx and was verfed n a dstrbuted way by comparng the asymptotc values of a randomly ntalzed consensus run by all actve agents n the network. Other than nearest neghbor nformaton, our approach requred no knowledge of the network topology. We showed that the Fg. 3. Sequence of Auctons ntegrated networked control system satsfes the connectvty specfcaton almost surely. We fnally llustrated effcency of our scalable approach by nontrval computer smulatons. REFERENCES [] D. P. Spanos and R. M. Murray, Robust connectvty of networked vehcles, n Proceedngs of the 43rd IEEE Conference on Decson and Control, Bahamas, Dec. 24, pp. 2893 2898. [2] M. Mesbah, On state-dependent dynamc graphs and ther controllablty propertes, IEEE Transactons on Automatc Control, vol. 5, no. 3, pp. 387 392, 25. [3] M. J and M. Egerstedt, Dstrbuted formaton control whle preservng connectedness, n Proceedngs of the 45th IEEE Conference on Decson and Control, San Dego, CA, Dec. 26, pp. 5962 5967. [4] M. M. Zavlanos and G. J. Pappas, Dstrbuted connectvty control of moble networks, n Proceedngs of the 46th IEEE Conference on Decson and Control, New Orlens, LA, pp. 359 3596. [5] Y. Km and M. Mesbah, On maxmzng the second smallest egenvalue of a state-dependent graph laplacan, IEEE Transactons on Automatc Control, vol. 5, no., pp. 6 2, Jan. 26. [6] M. C. DeGennaro and A. Jadbabae, Decentralzed control of connectvty for mult-agent systems, n Proceedng of the 45th IEEE Conference on Decson and Control, San Dego, CA, Dec. 26, pp. 3628 3633. [7] G. Notarstefano, K. Savla, F. Bullo, and A. Jadbabae, Mantanng lmted-range connectvty among second-order agents, n Proceedngs of Amercan Control Conference, Mnneapols, MN, 26, pp. 224 229. [8] C. F. Hsn and M. Lu, Network coverage usng low duty-cycled sensors: random & coordnated sleep algorthms, n IPSN 4: Proceedngs of the thrd nternatonal symposum on Informaton processng n sensor networks, Berkeley, CA, 24, pp. 433 442. [9] L. L, J. Y. Halpern, P. Bahl, Y. M. Wang, and R. Wattenhofer, A cone-based dstrbuted topology-control algorthm for wreless multhop networks, IEEE/ACM Transactons on Networkng, vol. 3, no., pp. 47 59, Feb. 25. [] R. Wattenhofer and A. Zollnger, XTC: A practcal topology control algorthm for ad-hoc networks, n Proceedngs of the 8th IEEE Internatonal Parallel and Dstrbuted Processng Symposum, Santa Fe, NM, 24, pp. 26 223. [] N. Bggs, Algebrac Graph Theory. Cambrdge Unversty press, 993. [2] A. Jadbabae, J. Ln, and A. S. Morse, Coordnaton of groups of moble autonomous agents usng nearest neghbor rules, IEEE Transactons on Automatc Control, vol. 48, no. 6, pp. 988, 23. [3] R. Olfat-Saber and R. M. Murray, Consensus problems n networks of agents wth swtchng topology and tme-delays, IEEE Transactons on Automatc Control, vol. 49, no. 9, pp. 52 533, Sept. 24. [4] H. G. Tanner, A. Jadbabae, and G. J. Pappas, Flockng n fxed and swtchng networks, IEEE Transactons on Automatc Control, vol. 52, no. 5, pp. 863 868, May 27. [5] W. Ren and R. Beard, Consensus of nformaton under dynamcally changng nteracton topologes, n Proceedngs of the Amercan Control Conference, 24, pp. 4939 4944. [6] J. Cortes, S. Martnez, and F. Bullo, Robust rendezvous for moble autonomous agents va proxmty graphs n arbtrary dmensons, IEEE Transactons on Automatc Control, vol. 5, no. 8, pp. 289 298, Aug. 26.