On the Convergence of Distributed Random Grouping for Average Consensus on Sensor Networks with Time-varying Graphs

Transcription

1 On the Cnvergence f Distributed Randm Gruing fr Average Cnsensus n Sensr Netwrks with Time-varying Grahs Jen-Yeu Chen and Jianghai Hu Abstract Dynamical cnnectin grah changes are inherent in netwrks such as eer-t-eer netwrks, wireless ad hc netwrks, and wireless sensr netwrks. Cnsidering the influence f the frequent grah changes is thus essential fr recisely assessing the erfrmance f alicatins and algrithms n such netwrks. In this aer, we analyze the erfrmance f an eidemic-like algrithm, DRG (Distributed Randm Gruing), fr average aggregate cmutatin n a wireless sensr netwrk with dynamical grah changes. Particularly, we derive the cnvergence criteria and the uer bunds n the running time f the DRG algrithm fr a set f grahs that are individually discnnected but jintly cnnected in time. An effective technique fr the cmutatin f a key arameter in the derived bunds is als develed. Numerical results and an alicatin extended frm ur analytical results t cntrl the grah sequences are resented t exemlify ur analysis. I. INTRODUCTION Dynamical grah changes are inherent in netwrks such as eer-t-eer netwrks, wireless ad hc netwrks, and wireless sensr netwrks. Take the examle f the wireless sensr netwrks, which have attracted tremendus research interests in the recent years. In a ractical r even hstile envirnment, the cnnectin grah f a sensr netwrk may vary frequently in time due t varius reasns. Fr instance, the cmmunicatin links (edges f the grah) may fail fr being interfered, jammed r bstructed; sensr ndes may be disabled r relcate in the field; t save energy, sme sensr ndes may slee r adjust their transmissin ranges, thus altering the cnnectin grah. Algrithms and rtcls develed n these netwrks need t take this nature int cnsideratin. In this setting, distributed and lcalized algrithms requiring n glbal data structure and centralized crdinatin are referable fr their scalability and rbustness t the frequent grah changes [], [2], [3], [4], [5], [6]. Althugh varius distributed algrithms have been designed t wrk well n a time-varying grah, their erfrmances are usually analyzed under the assumtin f a fixed cnnectin grah [2], [3], [4]. In this aer, as a articular examle, we analyze the erfrmance f a distributed randmized algrithm, namely, the DRG (Distributed Randm Gruing) algrithm rsed in [2], fr average aggregate cmutatin n sensr netwrks with randmly changing grahs. The analysis techniques intrduced in this aer can als be alied t ther algrithms whse erfrmances deend n the netwrk cnnectin grah. This wrk was artially surted by the Natinal Science Fundatin under Grant CNS The authrs are with the Schl f Electrical and Cmuter Engineering, Purdue University, USA. {jenyeu,jianghai}@urdue.edu. Average aggregate cmutatin is an essential eratin in sensr netwrks. One tyical categry f algrithms t cmute the average aggregate is the tree-based algrithms [7], [8]. In these algrithms, sensr ndes values are gathered alng an aggregatin tree t the tree s rt, the sink nde. The ther class f algrithms is the eidemic-like algrithms, such as gssi and the DRG algrithm. Unlike tree-based araches where nly the sink nde btains the glbal average value, eidemic-like algrithms btain an distributed average cnsensus amng all sensr ndes. Distributed average cnsensus has been an imrtant rblem with many alicatins in distributed and arallel cmuting. Recently it als finds alicatins in the crdinatin f distributed dynamic systems and multi-agent systems [9], [0], [], [2], as well as in distributed data fusin in sensr netwrks [2], [3], [4], [5]. In analyzing the erfrmance f the rsed distributed schemes fr average cnsensus, authrs f [2], [3], [4] bund the running time f their schemes n fixed cnnected grahs. In [9] Olfati- Saber and Murray characterize the cnvergence seed f their scheme ver a set f ssible directed grahs assumed t be balanced and strngly cnnected by the minimal algebraic cnnectivity f the mirrr (undirected) grah f each ssible netwrk grah. Authrs f [5], [0], [], [2] rvide criteria fr their schemes t cnverge n a dynamical changing grah which is ssibly discnnected during sme time erid but d nt characterize the cnvergence seed. Different frm these wrks, the gal f this aer is t nt nly determine the cnvergence criteria but als bund the running time f the DRG algrithm n a randmly changing grah which esecially culd be discnnected all the time. This aer is rganized as fllws. In Sectin II we briefly intrduce the DRG algrithm and review its erfrmance n a fixed grah. Then in Sectin III and Sectin IV, we analyze the erfrmance f the DRG algrithm n a netwrk grah randmly switching amng a set f individually discnnected but jintly cnnected grahs. Numerical results n fur tyical sets f such grahs are rvided in Sectin V. An alicatin based n ur analytical results t timize and cntrl the sensrs slee/awake schedule is resented in sectin VI. Finally, we cnclude ur wrk in Sectin VII. II. DISTRIBUTED RANDOM GROUPING We resent in [2] a distributed, lcalized, and randmized algrithm called the Distributed Randm Gruing (DRG) algrithm t cmute the aggregate statistics such as average, sum r maximum amng all sensr ndes measurements. The DRG algrithm is similar t the gssi algrithm [3],

2 [4] but with a better erfrmance (shwn in [2]) unlike the gssi algrithm which frms airs f ndes t exchange infrmatin, the DRG algrithm cnstructs grus each f which tyically includes mre than tw ndes t exedite the infrmatin mixing rcess. (Hence, in this aer, we fcus n the DRG algrithm rather than the renwned gssi algrithm.) In the fllwing, we will briefly describe the DRG algrithm, which will be the fcus f this aer in a generalized setting f time-varying netwrk grahs. Each sensr nde i is assciated with an initial bservatin r measurement value dented by v i (0) R. The values ver all ndes frm a vectr v(0). The gal is t cmute (aggregate) functins such as the average, sum, max, min, etc f the entries f v(0) in a distributed manner. Thrughut this aer we use v i (k) t dente the value f nde i and v(k) = [v (k), v 2 (k),...] T the value distributin vectr after running the DRG algrithm fr k runds. The main idea f the DRG algrithm is as fllws. In each rund f the iteratin, each nde indeendently becmes a gru leader with a rbability g and then invites its neh neighbrs t jin its gru by wireless bradcasting an invitatin message. A neighbr wh successfully 2 receives the invitatin message then jin its gru. Nte that unlike the cncet f a cluster in the sensr netwrk literature, a gru cntains nly the gru leader and its ne-h neighbrs. Several disjint grus are thus frmed ver the netwrk. Next, in each gru, all members ther than the gru leader then send the leader their values s that the leader can cmute the lcal aggregate (e.g., average in the gru) and bradcast it back t the members t udate their values. Since in each rund, grus are frmed at different laces f the netwrk, thrugh this randmized rcess, the values f all ndes will diffuse and mix ver the netwrk and cnverge t the crrect aggregate value asymttically almst surely, rvided that the grah is cnnected. DRG iteratins st when certain accuracy criteria are satisfied. A high-level descritin f a rund (iteratin) f the DRG algrithm t cmute the average aggregate, is shwn in Fig.. Aggregates ther than the average can be btained by an easy mdificatin f this algrithm [2]. Fr simlicity, in the aer we will fcus n the average aggregate nly. In [2], a Lyaunv functin called the tential (functin) is defined t assess the cnvergence f the DRG algrithm. Definitin : Cnsider a grah G(V, E) with V = n ndes. Given a value distributin v(k) = [v (k),..., v n (k)] T where v i (k) is the value f nde i after k runds f the DRG algrithm, the tential φ k f rund k is defined as φ k = v(k) v 2 2 = x T (k)x(k), where the cnstant i V v i(k) is the glbal average value ver the v = n netwrk; the vectr is the vectr with all entries ne and x(k) = [v (k) v,..., v n (k) v] T is the errr vectr. Prvided the grah is cnnected and unchanged in time, it is A wireless bradcast transmissin by the gru leader can be received by all its ne-h neighbrs. 2 Cllisins amid multile invitatin messages frm different gru leaders may ccur at sme ndes which hence will nt jin any gru in that rund. Als, a gru leader will ignre invitatins frm its neighbrs. Alg: DRG: Distributed Randm Gruing fr Average. Each nde in the idle mde indeendently riginates t frm a gru and becmes the gru leader with a rbability g..2 A nde i that decides t becme a gru leader enters the leader mde and bradcasts a gru call message, GCM (gru id = i), t all its neighbrs and waits fr JACK message frm its neighbrs. 2. A neighbring nde j, in the idle mde and successfully receiving a GCM, resnds t the gru leader by a jining acknwledgment, JACK (gru id = i,v j,jin(j) = ), with its value v j included. It then enters the member mde and waits fr the gru assignment message GAM frm its leader. 3. The gru leader, nde i, gathers the received JACKs frm its neighbrs; cunt the ttal number f gru members, J = jin(j) + ; and cmute the average value f the gru, j gi Ave(i) = ( v k gi k)/j. 3.2 The gru leader, nde i, bradcasts the gru assignment message GAM (gru id = i,ave(i)) t its gru members and then returns t the idle mde. 3.3 A neighbring nde j, in the member mde and un receiving receiving GAM frm its leader nde i, udates its value v j = Ave(i) and then returns t the idle mde. Fig.. A rund f DRG algrithm t cmute average aggregate shwn in [2] that the tential φ k will strictly[ decrease]} by a φk φ minimal cnvergence rate γ := inf v v {E k+ φ k > 0 frm its initial value t zer, i.e., v = v, asymttically almst surely. Further, by γ we btain the uer bunds f running time and the ttal number f transmissins fr the DRG algrithm t reach the average cnsensus n the fixed cnnected grah. III. TIME-VARYING NETWORK GRAPH It is nntrivial t extend ur results f the DRG algrithm n a fixed cnnected grah in [2] t the general case f timevarying grahs. Fr a fixed grah, we have shwn in [2] that all the nde values will eventually reach cnsensus by cnverging t the glbal average, starting frm an arbitrary initial value if and nly if the grah is cnnected. Hwever, in the case when the grah is time-varying, even if the grah is discnnected in sme time erids, it is still ssible that cnsensus can be reached, rvided that the unin f the grahs aearing infinitely ften is cnnected. (This cnvergence criterin has been shwn in [5], [0] fr their schemes. Later, we will briefly rve that it is als valid fr the DRG algrithm. Fr the detail rf, lease refer t [3]) Mrever, characterizing the cnvergence rate f the DRG algrithm in this case is a challenging task, as it deends n the ssible grahs f the netwrk, as well as the rules fr the (randm) evlutin f the netwrk grah in time. If sme (at least ne) f the ssible grahs are cnnected and visited infinitely ften, then ur results in [2] can be directly extended. We can lwer bund the cnvergence rate fr the whle executin by the minimum amng thse nnzer cnvergence rates n cnnected and infinitely-ftenvisited grahs, thugh this bund is cnservative. If these grahs are visited accrding t sme articular frequency, e.g., the statinary distributin f an ergdic discrete-state

3 (grahs) stchastic rcess, then we can btain a nrmalized cnvergence rate γ = ( γ G ) G, where γ G is the cnvergence rate n a candidate grah G and G is the frequency that the grah G is visited. If all grahs are discnnected, the results in [2] can nt be directly alied t find the cnvergence rate since the cnvergence rate n each grah is zer (γ = 0) fr all grahs. Hwever, if the unin f all grahs visited infinitely ften is cnnected, the DRG algrithm can still cnverge. In the fllwing sectins we will shw the cnvergence rate f the DRG algrithm running n a netwrk grah randmly switching amng a set f individually discnnected but jintly cnnected grahs. IV. INDIVIDUALLY DISCONNECTED BUT JOINTLY CONNECTED GRAPHS A. Cnvergence criteria In this sectin, we cnsider a mdel where all the ssible grahs are individually discnnected but jintly cnnected. We assume that each grah ccurs with sme sitive rbability in a rund f the DRG algrithm and is visited infinitely ften in the whle stchastic grah sequence. In such a mdel, the exected tential decrement in a single rund in the wrst case is unifrmly zer. Hwever, even thugh all ssible grahs are discnnected, if their unin grah is cnnected, the DRG algrithm can still cnverge t glbal average. We can shw this cnvergence criterin, by extending the rf f Therem in [5]. Suse there are r < ssible discnnected grahs {G i } each f which is visited infinitely ften. On each grah G i there are a set f ssible gru distributins {D Gi } fllwed frm the randmized gruing rule f the DRG algrithm. Each is assciated with a duble stchastic, symmetric and D Gi aracntracting 3 matrix W DG i (w.r.t. Euclidean nrm) s that the value vectr is udated by v(k + ) = W DG iv(k) when D Gi ccurs at rund k. (The value udating matrix W i f [5] deends nly n the chsen netwrk grah G i but ur W DG i is determined by the gru distributin D Gi which in turn deends n the grah G i and the randmized gruing strategy f the DRG algrithm.) Each G i is visited i.., s is D Gi. Hence, there exists at least a set f udating matrices Γ = {W DG i } fr i = r such that M = r r i= W DG i is stchastic, symmetric and irreducible if the unin f ssible grahs is cnnected. This imlies that M s fix-int subsace, i.e., the eigensace assciated with the eigenvalue, H(M) = san(). Therefre, by [5], Γ H(W DG i) = san(), which by [4] leads t the cnclusin that the DRG algrithm will asymttically cnverge t the unique fixed int ( n T v ), i.e., the status f the average cnsensus. B. Cnvergence rate and the uer bund f running time T characterize the cnvergence rate, fr illustratin urse, we analyze the simlest case where the netwrk grah switches randmly (infinitely ften) between tw grahs that are individually discnnected but jintly cnnected. 3 A matrix W is aracntracting w.r.t. a vectr nrm if Wx x Wx < x. [5] Our analysis can be easily extended t the general case f switching amid mre than tw grahs. As an examle, see Fig. 4(a). In each rund, the netwrk grah can be either G r. The grah changing attern can be characterized by a tw-state Markv chain shwn in Fig. 3(a) by setting each grah as a state f the Markv chain. (The dynamics f the executin f the DRG algrithm n a time-varying grah can be mdeled by a stchastic hybrid system [3].) Since G and are each discnnected, the lwer bund n the cnvergence rate fr each f them in a single rund is zer. Hwever, in tw runds, the netwrk may switch between these tw jintly cnnected grahs with sitive rbability, resulting in a sitive exected tential decrement. Thus t lwer bund the exected tential decrement rate, we need t cnsider tw runds f DRG iteratins. Since this is a wrst-case analysis, we assume the wrst scenari: nly ne gru is frmed in each rund. The DRG algrithm can have mre grus in a rund and hence cnverge faster than the uer bund derived here. Als, we assume every nde has equal rbability t becme a leader. Withut lss f generality, define x(k) = v(k) v, which is rthgnal t the vectr, i.e., x(k). Then each rund f DRG iteratin can be exressed as x(k+) = W(k)x(k) fr sme randm matrix W(k) deending n the chice f the gru leader. Fr examle, if in rund k, the netwrk grah is g k {G, } and nde i becmes the gru leader, then W(k) = W gk,i = [w g k,i ης ] where w g d, if η, ς {N i+ g k (i) i}; k,i ης =, if η, ς / {N gk (i) i} and η = ς; () 0, therwise. Here N gk (i) is the set f neighbrs f nde i in grah g k. Frm the cntinuus dynamics, in tw runds, we have x(k + 2) = W(k + )W(k)x(k) = Wx(k). The rati f tential decrement after tw runds is φ k φ k+2 = x(k) 2 x(k + 2) 2 φ k x(k) 2 = x(k)t T W Wx(k) x(k) T. (2) x(k) Define the tw-rund cnvergence rate γ 2 as the lwer bund n the exected cnvergence rate after tw cnsecutive runds: γ 2 := Frm (2), we have { inf E x(k) ; x(k) =0 [ ] φk φ k+2 E = φ k = x(k)t Kx(k) x(k) T x(k) [ φk φ k+2 φ k ]}. ] x(k) T T E [ W W x(k) x(k) T x(k) λ 2 (K) > 0, (3) where λ 2 (K) is the secnd largest eigenvalue ] T f the cmund matrix K = E [ W W =

4 P gk,g k+ (g k,g k+ ) i,j n (W gk+,j W gk,i ) T (W gk+,j W gk,i ). 2 In the abve, P gk,g k+ = P(g k )P(g k+ g k ) is the rbability that the grah f rund k is g k and the grah f rund k + is g k+. S, the lwer bund n the exected cnvergence rate after tw cnsecutive runds is γ 2 = λ 2 (K) > 0. The largest eigenvalue f K is always ne s the secnd largest eigenvalue λ 2 (K) < since the tw ssible grahs are jintly cnnected. Therem 2: On a set f individually discnnected but jintly cnnected grahs which tgether have a cmund matrix K, with an arbitrary initial value distributin v(0) and the initial tential, with high rbability (at least ( ε2 ) σ ; σ > 2), the average cnsensus rblem can be reached within an ε > 0 accuracy, i.e., v i v ε fr all i, by running the DRG algrithm in ) O (σ lg ( ε2 λ2(k) ) runds. Prf: T meet the accuracy criterin after 2τ runds f the DRG algrithm, by (3), we need E[φ 2τ ] ( γ 2 ) τ = (λ 2 (K)) τ ε 2, frm which we get τ lg ( ε2 λ2(k) ). Chse τ = σ lg λ2(k)( ε2 ) where the σ 2. Then because ( ε2 ) and (σ ), the Markv inequality P(φ 2τ > ε 2 ) < λ 2 (K) σ lg λ 2 (K) ( ε2 ) ( ε 2 ) = ( ε2 ) σ. (4) Thus, P(φ 2τ ε 2 ) ( ε2 ) (σ ) is arbitrarily clse t by chsing large σ. (Since tyically ε 2, taking σ = 2 is sufficient t have high rbability at least O( n ); in case > ε 2, then a larger σ is needed t have a high rbability). ( Hence ) running the DRG algrithm fr 2τ = O σ lg λ2(k)( ε2 runds,, with high rbability ) ( ε2 ) (σ ), the DRG algrithm cnverges t an ε accuracy. Similar rcedures can be carried ut t btain the cnvergence rate fr netwrk grahs randmly switching amng a set f individually discnnected but jintly cnnected grahs cnsisting f mre than tw grahs. In the fllwing sectin, we rvide an effective way t cmute the cmund matrix, K, fr a family f individually discnnected but jintly cnnected grahs. C. Cmutatin f the matrix K As we see frm the abve, the cmund matrix K is a key element in the uer bund f the running time f the DRG algrithm. Here we briefly intrduce an effective way t cmute the cmund matrix K. Fr the detail f this methd we refer readers t [3]. As an examle, we cnsider a set f h = n grahs each with n ndes sitined as a linear array and with nly ne edge cnnecting tw cnsecutive ndes. Shwn in Fig. 4(c), G,, and G 3 are ssible grahs with n = 4. Because f the extreme sarsity f each grah, a time-varying netwrk grah randmly switching amng these grahs is amng the wrst cases fr the DRG algrithm t cnverge. Hwever, since these grahs are jintly cnnected, the DRG algrithm will cnverge t the glbal average. Dente the stchastic i a b c d e g j f G q q (a)tw mde SHS ( )/3 G ( )/3 G 3 ( )/2 (c)fur mde SHS G 3 G 4 (b)three mde SHS ( )/3 ( )/3 G ( )/2 ( )/2 Fig. 2. The un-slid lines are Fig. 3. The Markv chain fr jintly the ssible aths (multilicatin cnnected switching grahs each f cmbinatins) fr K Λ (2, 3). which is discnnected. grah sequence ver h runds as Λ = {g k } h k=. We rewrite K = ] Λ [ W P(Λ)E T W Λ = Λ P(Λ)K Λ. The grah sequence Λ and P(Λ) deend n the grah changing attern. It turns ut ] that we need t effectively cmute T K Λ = E [ W W Λ. T achieve this, we cascade cmutatin blcks each f which reresents a W(k) f K Λ in the way in Fig. 2 fr the examle sequence Λ = {g = G, g 2 =, g 3 = G 3 } in Fig. 4(c). Fr an arbitrary n, frm these cascading cmutatin blcks, we btain the analytical exressin f K Λ as fllws. Let r = 2n, ζ = ( 3r) and I = (2r) x (r + rζ rn x i r + r n+ x i ); K Λ (i, j) = ζ rh r + rh, i = j = ; r + ζ 2 rn i r + ζr n i, < i = j n; I, i =, j = + x, x n ; ( 2r)I, < i n, j = i + x, x n i; K Λ (j, i), i > j. Nte that K Λ is a duble stchastic matrix which can be easily verified frm the abve exressin. V. NUMERICAL RESULTS We nw resent sme numerical results n the cnvergence rate f the DRG algrithm n the randmly grahchanging mdels studied in Sectin IV. Recall that, in this case, there are a ttal f h ssible grahs fr the netwrk, each f which is discnnected. As a family, hwever, the h grahs are jintly cnnected. A lwer bund γ h n the h- ste cnvergence ] rate is given by γ h = λ 2 (K), where T K = E [ W W and W = W(k+h ) W(k+)W(k). In Fig. 4, fur cases under study are ltted. In case I and case III, the unin f ssible grahs frms a linear array with three and fur ndes, resectively. In case II and case IV, the unin f ssible grahs frms a ring with three and fur ndes, resectively. The Markv chains (the randmly grahchanging mdel) describing the transitins amng ssible grahs are shwn in Fig. 3: Fig. 3(a) fr case I; Fig. 3(b) fr case II and case III; and Fig. 3(c) fr case IV. We cmute γ 2 fr case I, γ 3 fr case II and III, and γ 4 fr case IV, under different transitin rbabilities and q.

5 (a) Case I G G G2 G3 (b) Case II Fig. 4. (c) Case III Examle cases. G G2 G3 G G 4 G 3 (d) Case IV Fig. 5(a) lts the cmuted λ 2 (K) f case I as a functin f the transitin rbabilities and q. It can be seen that, as and q bth arach 0, λ 2 (K) achieves its minimum; hence γ 2 = λ 2 (K) achieves its maximum, imlying the fastest cnvergence rate f the DRG algrithm. This is understandable as, in this case, the transitins between the tw ssible grahs are the mst frequent and ccur in each rund, remedying the slw cnvergence caused by the individual discnnected grah. On the ther hand, by requiring that + q =, λ 2 (K) becmes a functin f nly, and is ltted in Fig. 5(b). Nte that the lt in Fig. 5(b) is a slice f the lt in Fig. 5(a) alng the line + q =. As can be seen frm the lt, the minimum λ 2 (K), hence the maximal cnvergence rate γ 2, ccurs at = q = when the tw grahs have identical statinary rbability. Fr all ther chices f and q satisfying +q =, the transitins have a tendency f staying in ne grah lnger, which slws dwn the cnvergence f the DRG algrithm. Fig. 5(c) cmares the cnvergence rates f the DRG algrithm fr these fur cases as well as an additinal case f the ring tlgy that is f five discnnected grahs, h = 5. The cmuted cnvergence rate γ h fr these five cases are ltted in Fig. 5(c) as functins f the transitin rbability (in case I, we set q = ). We bserve that, the larger the number f ndes, the slwer the cnvergence rate. In additin, with the same number f ndes, the case whse unin grah is a linear array has the slwer cnvergence rate than the crresnding case whse unin grah is a ring. This is because n a linear array each f the tw end ndes has nly ne directin t sread ut its value whereas all ndes in a ring have tw. VI. AN APPLICATION: SLEEP/AWAKE SCHEDULING One f the effrts t save energy cnsumtin in sensr netwrk is t let sme sensr ndes slee (in wer saving mde) frm time t time withut affecting the crrectness f the executin f the algrithm but ssibly with sme accetable degradatin n the erfrmance f the algrithm. In ur examle, the netwrk grah may becme discnnected when sme ndes slee. This is esecially true fr sarse netwrk grahs. Hwever, frm the results f revius sectins, we knw that the DRG algrithm still cnverges as lng as the time-varying netwrk grah is jintly cnnected. In this sectin we discuss the DRG s erfrmance n several slee/awake scheduling sequences and try t find the best cntrlled grah sequence in terms f bth energy saving and cnvergence time. We cnsider a sarse grah: a linear array with 4 sensr ndes in a rw. The netwrk grah e f Fig. 6(e) is the (a) (b) (c) (d) (e) (f) Fig. 6. Cnnectin grahs fr slee/awake scheduling cnnected grah while all fur ndes are awake. Other grahs in Fig. 6 are discnnected because sme sensr ndes are in slee mde, e.g., in grah a (Fig. 6(a)) nde 3 and nde 4 are in slee mde. When a nde slees, its CPU is at wer saving mde and its radi cmnents are deactivated. We cmare nine different eridic grah sequences (i.e., different sleeing schedules fr sensr ndes): Λ = {e}, Λ 2 = {abc}, Λ 3 = {abcb}, Λ 4 = {abce}, Λ 5 = {dc}, Λ 6 = {df}, Λ 7 = {adefc}, Λ 8 = {edef}, Λ 9 = {eaebec}, where {abc} means that the netwrk grah reeats in abc attern eridically, i.e., in first rund the netwrk grah is Fig. 6(a), the secnd rund Fig. 6(b), the third rund Fig. 6(c) and the furth rund Fig. 6(a) again... etc. Since the grahs in each sequence are jint cnnected, the DRG algrithm will cnverge fr these grah sequences. Running the DRG algrithm n an n-ndes linear array with m k awake ndes, we mdel the exected energy cnsumtin in the rund k f the DRG algrithm as fllws. E DRG = 2 n (3E tx + 2(E r/w + E CPU active )) + m k 2 (4E tx + 3(E r/w + E CPU active )) n + m k n (E CPU idle + E rx ) = m k n (3E 0 + E tx + E ) 2 n E 0, where E 0 = E tx + E r/w + E CPU active and E = E CPU idle + E rx ; E tx is the energy fr transmitting a message and E rx is fr ndes t listen t MAC channels and receive messages. In a rund f the DRG algrithm, the CPU f an awake nde cnsumes E CPU active r E CPU idle when it is busy r idle, resectively. E r/w is the ttal energy required fr an awake nde t read/ write infrmatin frm/t its EEPROM in a rund f the DRG algrithm. Excet the number f awake ndes m k, all the ther arameters in the abve equatin are cnstants in runds f the DRG algrithm. (Fr detail energy quantities cnsumed by a sensr nde, we refer t [5].) The ttal energy cnsumtin E DRG in a rund f the DRG algrithm is a linear functin f the number f awake ndes m k which may vary by runds. T cmare the average energy cnsumed in a rund fr different grah sequences, we take the average number f awake ndes EDRG (Λ) = k Λ m k/ Λ, where Λ is the number f grahs f a sequence attern, as the nrmalized energy index fr the sequence Λ, e.g., fr {dc} the nrmalized energy index is EDRG ({dc}) = (3 + 2)/2 = 2.5. Frm Therem 2, given and ǫ, the running time is rrtinal t Λ lg (λ 2 (K Λ )). Thus, we define the nrmalized index fr running time f the DRG algrithm Time (Λ) = lg (λ 2(K Λ )) = Λ lg (λ 2 (K Λ )). Fig. 7 shws ur simulatin results fr the nine grah sequences mentined reviusly. The x-axis is the nrmalized

6 λ 2 The secnd largest eigenvalue q 0 0 (a) λ 2 (K) fr the tw-mde mdel. λ 2 & γ (+q)= (b) λ 2 (K) and γ 2 fr the twmde mdel when + q =. λ 2 γ 2 γ Linear array γ 4 γ 3 3 ndes 4 ndes 0 0 γ γ 5 Ring γ 4 3 ndes 4 ndes 5 ndes γ (c) Lwer bund γ fr different tyes f grahs. γ 3 3 ndes linear array ring 0 γ ndes linear array ring 0 0 (d) γ fr varius grahs with the same number f ndes. Fig. 5. Numerical results nrmalized energy index: E * DRG (Λ) {e} {edef} {df} {eaebec} {adefc} {abce} {dc} {abc} {abcb} netwrk with randmly changing grahs. Criteria are given fr the cnvergence f the DRG algrithm n a randmly changing grah. Particularly fr a family f grahs which are individually discnnected but jintedly cnnected, bunds n the cnvergence rate and the running time are resented. Numerical results are rvided t illustrate ur analytical results. An alicatin built n ur analytical results t timize and cntrl the slee/awake schedule fr sensr ndes is als intrduced. The ntins and techniques resented in this aer can be alied t ther schemes fr distributed average cnsensus in sensr netwrks with time-varying grahs * nrmalized running time: /lg(λ (KΛ )) 2 Fig. 7. The nrmalized energy er rund and nrmalized cnvergence time fr different grah sequences. index fr running time; the y-axis is the nrmalized energy index E DRG (Λ). The sequence Λ = {e} where ndes never slee cnsumes the mst energy but cnverges fastest. In cntrast, the sequence Λ 3 = {abcb} where tw ndes slee in turn rund by rund cnsumes least energy but cnverges the slwest. There will be always a tradeff between these tw erfrmance indices. T incrrate bth energy cnsumtin and cnvergence time int the erfrmance assessment, we can minimize the cmbined indices: min(α E DRG(Λ) + β Time (Λ)), where α + β =. Tw extreme cases are (α =, β = 0) and (α = 0, β = ), reresenting the cnsideratin f nly energy cnsumtin and nly cnvergence time crresndingly. By linear rgramming n the cnvex hull f Fig 7, we can find the rer grah sequence fr a desired air f α and β. Fr examle, the sequence Λ 5 = {dc} shuld be used when we set < β α < 926. VII. CONCLUSION T guarantee the crrectness and recisely bund the running time f an algrithm develed n a sensr netwrk, we need t cnsider ne f the senr netwrk s salient nature: frequently changing grahs. In this aer, we mdel the executin f the Distributed Randm Gruing (DRG) algrithm fr cmuting the average aggregate n a sensr REFERENCES [] D. Estrin, R. Gvindan, J. S. Heidemann, and S. Kumar, Next century challenges: scalable crdinatin in sensr netwrks, in Mbicm, 999, [2] J.-Y. Chen, G. Pandurangan, and D. Xu, Rbust aggregate cmutatin in wireless sensr netwrk: distributed randmized algrithms and analysis, in IEEE Trans. n arallel and distributed system,, Se. 2006, [3] D. Keme, A. Dbra, and J. Gehrke, Gssi-based cmutatin f aggregate infrmatin, in Prc. FOCS, [4] S. Byd, A. Ghsh, B. Prabhakar, and D. Shah, Gssi algrithms: Design, analysis and alicatins, in INFOCOM 05. [5] L. Xia, S. Byd., and S. Lall, A scheme fr rbust distributed sensr fusin based n average cnsensus, in Prc. IPSN, [6] B. Krishnamachari, Netwrking Wireless Sensrs. Cambridge University Press, [7] C. Intanagnwiwat, D. Estrin, R. Gvindan, and J. Heidemann, Imact f netwrk density n data aggregatin in wireless sensr netwrks, in Prc. ICDCS, [8] S. Madden, M. Franklin, J. Hellerstein, and W. Hng, Tag: a tiny aggregatin service fr ad-hc sensr netwrks, in Prc. OSDI, [9] R. Olfati-Saber and R. M. Murray, Cnsensus rblem in netwrks f agents with switching tlgy and time delays, IEEE Trans. n Autmatic Cntrl, vl. 49, n. 9, , se [0] L. Fang and P. Antsaklis, Infrmatin cnsensus f asynchrnus discrete-time multi-agent systems, in Prc. ACC 05. [] W. Ren and R. W. Beard, Cnsensus seeking in multi-agent systems under dynamically changing interactin tlgies, IEEE Trans. n Autmatic Cntrl, vl. 50, n. 5, , [2] L. Mreau, Stability f multi-agent systems with time-deendent cmmunicatin links, IEEE Trans. n Autmatic Cntrl, vl. 50, n. 2, , [3] J.-Y. Chen and J. Hu, Perfrmance analysis f distributed randmized algrithms n randmly changing grahs by stchastic hybrid systems, in Tech. Rert, Purdue Univ., 2006, submitted fr ublicatin. [4] L. Elsner, I. Kltracht, and M. Neumann, On the cnvergence f asynchrnus aracntractins with alicatins t tmgrahic recnstructin frm incmlete data, Linear Algebra Al., vl. 30, , 990. [5] V. Shnayder, M. Hemstead, B. Chen, G. Allen, and M. Welsh, Simulating the wer cnsumtin f large-scale sensr netwrk alicatins, in Prc. SenSys, 2004.