How to Stop Consensus Algorithms, locally?

Transcription

1 1 How to Stop Consensus Algorthms, locally? Pe Xe, Keyou You and Cheng Wu arxv: v1 [cs.dc] 15 Mar 2017 Abstract Ths paper studes problems on locally stoppng dstrbuted consensus algorthms over networks where each node updates ts state by nteractng wth ts neghbors and decdes by tself whether certan level of agreement has been acheved among nodes. Snce an ndvdual node s unable to access the states of those beyond ts neghbors, ths problem becomes challengng. In ths work, we frst defne the stoppng problem for generc dstrbuted algorthms. Then, a dstrbuted algorthm s explctly provded for each node to stop consensus updatng by explorng the relatonshp between the so-called local and global consensus. Fnally, we show both n theory and smulaton that ts effectveness depends both on the network sze and the structure. Index Terms dstrbuted algorthms, strongly connected graphs, local stoppng, consensus. I. INTRODUCTION Ths work proposes local stoppng rules for dstrbuted algorthms wth a network of computng nodes, and each node updates ts states by only nteractng wth neghbors. Due to the advantages of cooperaton among nodes, they have been wdely used n many areas, ncludng dstrbuted localzaton [1], cogntve networks [2], cooperatve control of unmanned ar vehcles (UVAs) [3], flockng [4], and socal networks [5]. Snce optmalty of many dstrbuted algorthms s attaned n an asymptotc sense, t s crtcal for each node of the algorthm to decde by tself when to stop updatng, whch to the best of our knowledge, has not be well studed n the lterature. Ths problem s substantally dfferent from that of a centralzed algorthm as t lacks a centralzed coordnator to montor the network. Most of exstng stoppng rules are based on that each node wll stop updatng after an adequately large number of teratons. Clearly, ths dea s very easy to mplement, but also suffers several lmtatons. Frstly, the number of teratons depends on qute a few factors, e.g., the network topology, the ntal network state, and the convergence rate of dstrbuted algorthms, all of whch essentally requre global nformaton of the system. For nstance, the convergence rate of the celebrated consensus algorthm s quantfed by the algebrac connectvty of the nteracton graph and the ntal network state, both of whch are dffcult to evaluate n a dstrbuted way and also requre much more computatonal and communcaton overheads. Secondly, the teraton number s always derved from the worst-case pont of vew, and ts conservatveness s case-by-case dependent, whch s unable to evaluate. The last but not the least, the number of teratons Ths work was supported by the Natonal Natural Scence Foundaton of Chna ( ), and Tsnghua Unversty Intatve Scentfc Research Program. P. Xe, K. You, and C. Wu are wth the Department of Automaton and TNLst, Tsnghua Unversty, , Chna (emals: xep13@mals.tsnghua.edu.cn, {youky, wuc}@tsnghua.edu.cn). s mpossble to obtan n advance for tme-varyng networks, especally for stochastcally tme-varyng networks due to the causalty ssue. In ths work, we formally defne the local stoppng problem of a dstrbuted algorthm over networks under the followng constrants: () each node s only able to access nformaton from ts neghbors. Ths enables the rule scalable wth the network sze. () To save storage capacty, t s not allowed for nodes to store large data, e.g., the hstorcal state nformaton. () the rule can be easly checked by an ndvdual node, and (v) the communcaton overhead to support the rule s as small as possble. Then, we propose a fully dstrbuted stoppng scheme for general dstrbuted algorthms by allowng each ndvdual node n the network to mantan a vector, whch s updated teratvely by usng nformaton from ts n-neghbors to gradually learn the network. Each node wll stop updatng based on ths auxlary vector. For dstrbuted consensus algorthms, the earlest concern about ths topc has been addressed n [6], whch requres the nodes to smultaneously run mnmum and maxmum protocols along wth the consensus protocol, and both the maxmum and mnmum protocols are re-ntalzed per D- teratons where D s the dameter of the graph. The drawback of the method s that t requres to communcate three dfferent states of the same dmenson at each tme slot. Ths ncreases the communcaton burden, especally when the state of a node s a mult-dmensonal vector. Moreover, the synchronzaton problem n re-ntalzng the mnmum and maxmum protocols remans dffcult n practce. To overcome these ssues, an adaptve stoppng-recallng mechansm s desgned n [7] to stop the push-sum consensus algorthm n fnte tme. The method permts the nactve nodes to recover transmttng agan, but the problem s that a node never know whether t wll reactvate or not even though the whole network has reached consensus, and has to awat orders all the tme. In fact, ths paper combnes the advantages of the above two methods. We adopt the dea of mnmum-consensus [8] to check the unformly local consensus to verfy the global consensus. Dfferent from that n [6], our method needs not to reset mnmum-consensus protocol, and the state of the mnmum consensus s a dscrete-valued scalar even f each node mantans a vector state. To evaluate the performance of our method, we characterze the extra tme that the nodes spend to locally detect the global consensus. An outlne of ths paper s as follows. In Secton II, we propose the local stoppng problem n dstrbuted algorthms. In Secton III, a partcular local stoppng rule for the consensus algorthm s desgned. In Secton IV, we analyze the senstvty of the method. Fnally, some concludng remarks are drawn n Secton V.

2 2 Notaton: Consder a drected graph (dgraph) G = (V, E), where V s the set of nodes and E s the set of drected edges (, j) wth, j V. For node V, ts n-neghbors and out-neghbors are respectvely defned by N := {j V (, j) E} and N + := {j V (j, ) E}. For two nodes 1, j V, the drected path from j to 1 s defned by {( 1, 2 ),..., ( j 1, j )} wth ( k, k+1 ) E for k [1, j 1]. We denote d j as the dstance from j to,.e., the length of the shortest path from j to. The dgraph s sad to contan a spannng tree f t has a root node that s connected to any other node n the graph va a drected path, and s strongly connected f each node s connected to every other node n the graph va a drected path. The dameter of the strongly connected dgraph G s defned by D = max,j V d j. Fnally, we denote x A {x A}. II. LOCAL STOPPING SCHEME FOR NETWORKED DISTRIBUTED ALGORITHMS A. Dstrbuted Algorthms Let x (k) be the state of node at tme slot k n a fxed graph 1 G = {V, E}. We are concerned wth dstrbuted algorthms of the followng form x (k + 1) = h (x N {}(k), k), (1) where h (, k) s a functon of x (k) and x N (k), both of whch are drectly known to node at tme slot k. In dfferent applcatons, h (, k) may take dfferent forms. Clearly, the update law n (1) s very generc, and contans a lot of mportant algorthms. Here we present three nterestng applcatons for an llustraton. Example 1 (Consensus algorthm). In dscrete consensus algorthms [9], h (, k) s smply a convex combnaton of x and x N,.e., x (k + 1) = α j x j (k), (2) j N {} where α j > 0 f and only f j N {}, and zero, otherwse. Moreover, j N {} α j = 1. Then, the network state asymptotcally converges to consensus (.e., lm k x (k) x j (k) = 0) f and only f G contans a spannng tree [10]. Example 2 (Opnon dynamcs). In Fredkn-Johnsen model of opnons evoluton [11], the state of node s updated as x (k + 1) = ω α j x j (k) + (1 ω )x (0), (3) j N {} where 0 ω 1 represents the couplng condton and x (0) s referred to as the node prejudce. Under certan condtons, the network state asymptotcally converges [11]. Example 3 (Dstrbuted gradent descend). The am of the dstrbuted gradent descend (DGD) s to solve the followng optmzaton problem mn V 1 It can be easly extended to tme-varyng graphs. f (x), (4) where f ( ) s a convex functon only known by node. In the DGD algorthm, h ( ) s gven as follows x (k + 1) = α j x j (k) ζ(k) d (k), (5) j N {} where ζ(k) s a sequence of approprately selected stepszes, d (k) s a (sub)-gradent of f at x (k) and α j s gven as n (2). By [12], t follows that f G s balanced and strongly connected, the state of each node asymptotcally converges to some common optmal pont of the optmzaton problem (4). In the above mportant applcatons, the convergence of the network state s shown n the asymptotc sense. Then, a natural queston s when to stop updatng (1) for an ndvdual node to acheve a desred level of optmalty? If there were a central authorty to montor the network state, the problem s obvously trval. However, n dstrbuted algorthms over networks, an ndvdual node s unable to drectly observe the whole network state, whch s qute dfferent from the centralzed algorthms, and the local stoppng problem requres further nvestgaton. A nave dea s that the node can smply stop updatng (1) after a suffcently large number of teratons. Indeed, ths s very easy to mplement. However, to ensure a desred qualty of the node state, the number of teratons usually depends on qute a few factors, most of whch are global nformaton of the system. Take the consensus algorthm n (2) as an example. Let d(k) = max,j V { x (k) x j (k) }, t follows from [9] that d(k) cρ k d(0) where c > 0 s postve and ndependent of network. The decayng rate ρ s the second largest egenvalue for a symmetrc stochastc matrx A = (α j ), and strctly less than one f the undrected graph G s connected. To ensure d(k) ɛ, the number of teratons s usually set to satsfy that k log ɛ/(cd(0)). (6) log ρ Clearly, d(0) and ρ n the lower bound respectvely depend on the ntal network state and the network structure, both of whch are unavalable to any ndvdual node. One may argue that d(0) can be obtaned by usng dstrbuted maxmumand mnmum-consensus algorthms. However, dstrbutedly estmatng ρ s nontrval [13] and agan requres an addtonal local stoppng rule. More mportantly, the lower bound s only obtanable for fxed graphs, and there does not exst such a ρ for tme-varyng graphs. Another dea s to use fast communcatons between the tme slots k and k + 1 to dstrbutedly evaluate d(k). Although ths s possble wth a fnte number of communcatons, t certanly ncreases the communcaton overhead, and even offset the beneft of dstrbuted algorthms. Overall, the problem of desgnng effectve rules to locally stop dstrbuted algorthms n (1) largely remans open, whch s the focus of ths work. B. Local Stoppng Scheme We are nterested n desgnng local stoppng rules of the followng form s (k + 1) = Q (x N {}(k), s N {}(k)), (7)

3 3 where s (k) s embeded n node to montor teratons of dstrbuted algorthms of (1). If a gven performance of (1) s acheved,.e., there exsts a performance dependent functon J ( ) such that J (s (k)) 0, (8) then node stops updatng and wll no longer broadcast any nformaton to ts out-neghbors. Clearly, s (k) can be dstrbutedly mplemented as n (1), and synchronze wth the orgnal dstrbuted algorthm, see Fg. 1 for an llustraton of mplementng a local stoppng rule n node h8.9! h8.9!! " ($)! " ($ + 1)! " ($ + 2) Observer Observer Judger Fg. 1. Local stoppng rule n node.! 0 ($), 1 0 ($)! 0 ($ + 1), 1 0 ($ + 1) Judger )(! " ) 0,!-./ )(! " ) 0,!-./ Snce s (k) needs to be broadcast, the extra communcaton overhead s now decded by s (k). The number of bts to represent s (k) should be as small as possble. Whle to ensure fast detecton, t s hghly desrable to easly compute both Q ( ) and J ( ). Usually, the more nformaton we obtan from the network topology, the smper form of these two functons are. Moreover, the stoppng rule of (7) and (8) needs to be senstve, meanng that t s able to quckly detect the attanablty of the performance qualty of dstrbuted algorthms. In the sequel, we desgn a local stoppng rule of the form (7) and (8) for the dstrbuted consensus algorthm n (2). The case of generc dstrbuted algorthms wll be studed n the full verson of ths work. III. LOCAL STOPPING RULES FOR CONSENSUS A. Notons of ɛ-consensus ALGORITHMS Obvously, the consensus algorthm n (2) can be wrtten n the followng matrx form x(k + 1) = Ax(k), (9) where A = (α j ) R N N s row stochastc. It follows from Gershgorn s dsc theorem [14] that A k 1 N v T wth 1 T N v = 1 f one s the smple egenvalue of A, whch mples that x(k) 1 N v T x(0),.e., x (k) x j (k) 0 and consensus s asymptotcally acheved. By [10], one s the smple egenvalue of A f and only f G assocated wth A contans a spannng tree. To examne the asymptotc behavor of (9), we ntroduce the concept of global ɛ-consensus. Defnton 1 (global ɛ-consensus). Gven ɛ > 0, the state vector x = [x 1,..., x N ] of the network s sad to reach global ɛ-consensus f x x j < ɛ for any, j V. The man objectve of ths paper s to provde a local stoppng rule of the form (7) and (8) to ensure that the global ɛ-consensus s acheved. However, an ndvdual node cannot drectly check ts attanablty. Instead, by drectly usng states of ts n-neghbors and wthout nducng extra communcaton, each node can easly check local ɛ-consensus below. Defnton 2 (local ɛ-consensus). Gven ɛ > 0, f the state of node satsfes max j N x x j < ɛ, node s sad to acheve local ɛ-consensus. A strkng dstncton between local ɛ-consensus and global ɛ-consensus les n that the later one s attractve, meanng that once the network attans global ɛ-consensus, t wll stay n ths state afterwards. To elaborate t, let x u (k) = max {x (k)}, and x l (k) = mn {x (k)}. It follows from (2) that x u (k) x (k +1) and x j (k +1) x l (k) for all, j V. Then x (k + 1) x j (k + 1) x u (k) x l (k) = max,j x (k) x j (k). That s, once global ɛ-consensus s acheved at tme slot k, t wll contnue to hold at next tme slot. Whle global ɛ-consensus mples local ɛ-consensus, the converse does not hold. To brdge the gap of the above two notons of ɛ-consensus, we further ntroduce unformly local ɛ-consensus. Defnton 3 (unformly local ɛ-consensus). Gven ɛ > 0, f all the nodes acheve local ɛ-consensus, the network acheves unform ɛ-consensus. Agan, global ɛ-consensus mples unformly local ɛ- consensus. On the other sde, we obtan the followng result. Lemma 1. Suppose that G s strongly connected. If the state of the network s unformly local ɛ-consensus, t s also n the state of global (ɛd)-consensus, where D s the dameter of the network. Proof. Gven two arbtrary nodes, j V, then there exsts a drected path wth edges (, 1 ) E, ( 1, 2 ) E,..., ( m, j) E and m + 1 D. For convenence, denote 0 = and m+1 = j. Snce the network s n unformly local ɛ-consensus, then x k x k+1 < ɛ holds for k = 0,..., m. Therefore, x x j = m k=0 (x k x k+1 ) m k=0 x k x xk+1 < (m + 1)ɛ ɛd, whch ndcates that the network s n (ɛd)- consensus. If G s strongly connected, the above shows that unformly local ɛ-consensus s essentally equvalent to global ɛ-consensus. Then, the remanng problem s how to locally check the attanablty of unformly local ɛ-consensus. One may consder a straghtforward dea by usng a certan number of consecutvely attanng local ɛ-consensus. However, ths number cannot be settled as a pror and n fact depends on the ntal network state, whch s llustrated n the followng example. Example 4. Consder a strongly connected rng network wth 3 nodes. The assocated weghtng matrx A and ntal

4 4 network state are gven by A = , and x(0) = Once can readly verfy that x(3) = [0.1, , ] T and node 1 attans local 0.5-consensus at any tme slot. However, ether unformly local 0.5-consensus or global 0.5- consensus s not acheved before tme slot k 3. It s easy to observe that the attanablty of unformly local ɛ-consensus depends on the ntal state vector, and cannot be smply checked va a fxed number of consecutvely attanng local ɛ-consensus. In the sequel, we shall make the followng assumpton. Assumpton 1. The graph G s strongly connected. In fact, Assumpton 1 s necessary. If G s not strongly connected, there exsts a node that s not reachable from node j. Then, node s unable to receve nformaton from node j, whch renders node genercally unable to locally decde the stoppng tme. B. Desgn of Local Stoppng Methods To resolve the above mentoned ssue for unformly local ɛ-consensus, our dea s to adopt a mnmum-consensus algorthm n the form (7) so that each node can track the mnmum number of consecutve attanng local ɛ-consensus. To ths purpose, each node uses y ɛ (k) to record the latest number of consecutvely attanng local ɛ-consensus n the tme slot k,.e., y ɛ (0) = 0, (10) { y y ɛ ɛ (k + 1) = (k) + 1, max j N x (k) x j (k) < ɛ, 0, otherwse. Clearly, f y ɛ (k) s postve, node attans local ɛ-consensus at tme slot k 1. If node s further able to locally check the postveness of mn V {y ɛ (k)}, the problem s solved as mn V {y ɛ (k)} > 0 mples that unformly local ɛ-consensus s acheved! To ths end, each node may use fast communcatons between tme slot k and k + 1 to send one-bt data for D tmes to ts out-neghbors, and adopts the mnmumconsensus algorthm,.e., s (k, 0) = sgn(y ɛ (k)), and s (k, m+1) = mn s j (k, m), j N {} (11) where sgn(x) = 1 f x > 0, and zero, otherwse. Specfcally, node only needs to broadcast one-bt data to ts out-neghbors. If G s strongly connected, t s clear that s (k, D) = mn V {s (k, 0)}. That s, the mnmumconsensus s acheved after D rounds of communcatons. If s (k, D) > 0, then node stops updatng (2) at tme slot k, and reports that the network attans unformly local ɛ- consensus. However, ths approach s not applcable to tmevaryng graphs. For example, f G(k) s not strongly connected, the mnmum consensus s not achevable by usng (11). Inspred also motvated by ts lmtaton, we ntroduce another varable z ɛ n node, whch plays a smlar role of s (k, m) of (11). Let z ɛ (0) = 0, and z ɛ (k + 1) = mn {zj(k), ɛ yj(k)} ɛ + 1. (12) j N {} Clearly, both the auxlary varables y ɛ n (11) and z ɛ n (12) can be computed n a dstrbuted way. Next, we show how to use z ɛ to check the attanablty of unformly local ɛ-consensus. The followng result s straghtforward, and ts proof s omtted. Lemma 2. Consder the dstrbuted algorthms gven by (11) and (12), the followng statements hold. (a) For any k 1 k 2 0, then y ɛ(k 1) y ɛ(k 2) + k 1 k 2 and z ɛ(k 1) z ɛ(k 2) + k 1 k 2. (b) If (, j) E, then yj ɛ(k) zɛ (k + 1) 1. (c) The network s unformly local ɛ-consensus at tme slot k f and only f y ɛ (k + 1) 1 for all V. Theorem 1. Consder the dstrbuted algorthms gven by (11) and (12) under Assumpton 1. If there exsts V such that z ɛ (k) D + 1, the network attans global (ɛd)-consensus at tme slot k from any ntal network state. Proof. For any j V, t follows from Assumpton 1 that there exsts a drected path (, 1 ), ( 1, 2 ),..., ( m, j) from node j to wth m D 1. Snce z (k) D+1 and 1 N, t follows from Lemma 2(a) and (b) that z ɛ 1 (k 1) D, and y ɛ 1 (k 1) D. Smlarly, t follows that z ɛ 2 (k 2) D 1 and y ɛ 2 (k 2) D 1. Repeat the same steps, we obtan that (k m 1) D m. Note that m+1 D, t follows from y ɛ j Lemma 2(a) that yj ɛ(k D) yɛ j (k m 1)+m+1 D 1. As j s arbtrary, t follows from Lemma 2(c) that the system s n unformly local ɛ-consensus at tme k D 1. By Lemma 1, the system has already attaned global (ɛd)-consensus at tme slot k D 1. Snce the state of global (ɛd)-consensus s attractve, t s obvous that the network attans global (ɛd)- consensus at tme slot k. If the dameter D of the network s unknown, we may replace t by usng the network sze N 1 and obtan the followng result. Corollary 1. Consder the dstrbuted algorthms gven by (11) and (12) under Assumpton 1. If there exsts V such that z ɛ (k) N, then the network attans global (ɛ(n 1))- consensus at tme slot k from any ntal network state. Clearly, t s almost mpossble for all nodes n the network to smultaneously detect the attanablty of global ɛ- consensus. In practce, once the node locally determnes the attanablty of global ɛ-consensus, t stops updatng. In addton, ths node wll nform ts out-neghbors to stop updatng and there s no longer need to broadcast ts state. Thus, t maxmally takes an addtonal D tme slots for other nodes to detect the attanablty of global ɛ-consensus. Overall, the local stoppng method for node s detaled n Algorthm 1.

5 5 Algorthm 1 Local stoppng method for unformly local ɛ- consensus Intalzaton: For each node V, let y ɛ 0, zɛ 0. whle true do Local nformaton exchange: Every node broadcasts ts state x, and mn{y ɛ, zɛ } to ts out-neghbors. Local varables update: Each node receves x j and mn{yj ɛ, zɛ j } from ts n-neghbors,.e., j N, and let z ɛ mn j N {}{yɛ j, zɛ j } + 1. f z ɛ D + 1 then node reports unformly local ɛ-consensus and stops updatng. break end f f max j N y ɛ yɛ + 1. else y ɛ 0. end f end whle x x j < ɛ then Remark 1. Gven any δ > 0, let ɛ = δ/d n Algorthm 1. Then, we are able to obtan global δ-consensus by usng the stoppng method n Algorthm 1. IV. SENSITIVITY ANALYSIS Theorem 1 confrms feasblty of Algorthm 1 for locally detectng both unformly local and global consensus. If the network attans unformly local or global ɛ-consensus, a node usually s unable to locally detect t mmedately due to the lack of the whole network state. However, the detecton task can be completed by every node under Algorthm 1 after a fnte number of extra tme slots, whch s defned as response tme n ths work. Clearly, a short response tme s desrable n applcatons, and s zero f the node s able to montor the whole network state smultaneously. In ths secton, we are nterested n evaluatng the response tme of Algorthm 1. To ths purpose, we use adopt the concept of ergodc coeffcent. Defnton 4 (Ergodc coeffcent). Gven a nonnegatve matrx A = [a j ] R N N, ts ergodcty coeffcent s defned as τ(a) = mn j N mn{a k, a jk }. (13) k=1 For a row-stochastc matrx A, t clearly hods that 0 τ(a) 1, and ergodc coeffcent s mportant to quantfy the convergence rate of consensus algorthm n (9). Snce x(k + j) = A j x(k) for any k, j N, t follows from [4] that d(k + j) (1 τ(a j ))d(k), where d(k) = max,j { x (k) x j (k) }. By [4], we obtan the followng lemma. Lemma 3. Under Assumpton 1, there exsts a mnmum h N such that 0 < τ(a h ) < 1 and h D. Theorem 2. Under Assumpton 1, consder the local stoppng method n Algorthm 1. Gven any δ > 0, the response tme T r for achevng global δ-consensus s upper bounded by log D T r h log (1 τ(a h + D + 1, (14) )) where h s gven n Lemma 3. Proof. Under Assumpton 1, t s obvous that the network state dctated n (9) wll asymptotcally reach consensus. Then, there exsts a fnte tme slot k 0 > 0 such that the network wll frstly reach global δ-consensus. Clearly, k 0 depends on both the network structure n the form of A and the ntal network state. Thus, t s genercally unavalable to an ndvdual node. If the network state attans the unformly local (δ/d)- consensus, each node s able to locally detect t before tme slot D + 1 by usng Algorthm 1. In vew of Lemma 1, the network certanly reaches global δ-consensus. Snce x(k 0 + lh) = A h x(k 0 + (l 1)h), t follows from Lemma 3 that d(k 0 + lh) (1 τ(a h ))d(k 0 + (l 1)h) (1 τ(a h )) l δ. Let l be selected such that (1 τ(a h )) l δ δ/d,.e, log D l = log (1 τ(a h )), the network reaches unformly local (δ/d)-consensus at tme slot tme k 0 + lh. Fnally, we obtan that T r lh + D + 1, whch s explctly gven by log D T r h log (1 τ(a h )) + D + 1. That s, the proof s completed. Remark 2. In (14), the frst quantty n the rght hand sde (RHS) s appled to overcome the conservatveness between global δ-consensus and unformly local δ/d-consensus. The addtonal D+1 tme slots are for an ndvdual node to detect the attanablty of unformly local δ/d-consensus. The followng example llustrates the senstvty of the stoppng method va the response tme. Example 5. Consder the consensus algorthm (9) wth A = , and x(0) = In ths example, the goal s to acheve global 0.5-consensus. The smulaton result n Fg. 2 shows that the frst tme when stoppng condton satsfes n some node s k = 24, whle the global ɛ-consensus actually arrves at k = 18. The result clearly llustrates the effectveness of Algorthm 1 snce each node stops, locally. The response tme for dfferent levels of global consensus are lsted n Table.1. Va calculaton, we obtan h = 2, D = 3, and τ(a h ) = It follows from Theorem 2 that T r 40. As shown n Table.1, the largest response tme s 10, whch s smaller than 40 and verfes the correctness of Theorem 2.

6 6 x <0.5 Dstrbuted Stoppng Rule for Global 0.5-consensus k=18, reach global 0.5-consensus k=24, dstrbutedly stop k Fg. 2. Smulaton result of Example 1 as ɛ = 0.5. Table 1: Response tme for dfferent levels of global consensus Level of consensus Consensus tme Stoppng tme Response tme V. CONCLUSION In ths paper, we formally defned the local stoppng problem for dstrbuted algorthms. Focusng on the consensus algorthm, a dstrbuted stoppng method was desgned for each nodes to locally check consensus, and ts senstvty s quantfed. The method can be extended to tme-varyng network and s robust to communcaton delays, whch wll be presented n the journal verson. [7] N. E. Mantara and C. N. Hadjcosts, Dstrbuted stoppng for average consensus n drected graphs va a randomzed event-trggered strategy, n 6th Internatonal Symposum on Communcatons, Control and Sgnal Processng. IEEE, 2014, pp [8] J. Cortés, Dstrbuted algorthms for reachng consensus on general functons, Automatca, vol. 44, no. 3, pp , [9] R. Olfat-Saber and R. M. Murray, Consensus problems n networks of agents wth swtchng topology and tmedelays, IEEE Transactons on automatc control, vol. 49, no. 9, pp , [10] W. Ren and R. W. Beard, Consensus seekng n multagent systems under dynamcally changng nteracton topologes, IEEE Transactons on automatc control, vol. 50, no. 5, pp , [11] S. E. Parsegov, A. V. Proskurnkov, R. Tempo, and N. E. Fredkn, Novel multdmensonal models of opnon dynamcs n socal networks, IEEE Transactons on Automatc Control, [12] A. Nedć and A. Ozdaglar, Dstrbuted subgradent methods for mult-agent optmzaton, IEEE Transactons on Automatc Control, vol. 54, no. 1, pp , [13] P. D Lorenzo and S. Barbarossa, Dstrbuted estmaton and control of algebrac connectvty over random graphs, IEEE Transactons on Sgnal Processng, vol. 62, no. 21, pp , [14] R. A. Horn and C. R. Johnson, Matrx analyss. Cambrdge Unversty Press, REFERENCES [1] J.-P. Sheu, P.-C. Chen, and C.-S. Hsu, A dstrbuted localzaton scheme for wreless sensor networks wth mproved grd-scan and vector-based refnement, IEEE transactons on moble computng, vol. 7, no. 9, pp , [2] J. A. Bazerque and G. B. Gannaks, Dstrbuted spectrum sensng for cogntve rado networks by explotng sparsty, IEEE Transactons on Sgnal Processng, vol. 58, no. 3, pp , [3] A. Rchards, J. Bellngham, M. Tllerson, and J. How, Coordnaton and control of multple uavs, n AIAA gudance, navgaton, and control conference, Monterey, CA, [4] J.-G. Dong and L. Qu, Flockng of the cucker-smale model on general dgraphs, IEEE Transactons on Automatc Control, [5] D. Acemoglu and A. Ozdaglar, Opnon dynamcs and learnng n socal networks, Dynamc Games and Applcatons, vol. 1, no. 1, pp. 3 49, [6] V. Yadav and M. V. Salapaka, Dstrbuted protocol for determnng when averagng consensus s reached, n 45th Annual Allerton Conference, 2007, pp