Network Discovery: An Estimation Based Approach

Transcription

1 Nework Discovery: An Esimaion Based Approach Girish Chowdhary, Magnus Egersed, and Eric N. Johnson Absrac We consider he unaddressed problem of nework discovery, in which, an agen aemps o formulae an esimae of he global nework opology using only locally available informaion. We show ha under wo key assumpions, he nework discovery problem can be cas as a parameer esimaion problem. Furhermore, we show ha some form of exciaion mus be presen in he nework o be able o converge o a soluion. The performance of wo mehods for solving he nework discovery problem is evaluaed in simulaion. I. INTRODUCTION Successful negoiaion of real world missions ofen requires diverse eams o collaborae and synergisically combine differen capabiliies. The problem of conrolling such neworked eams has become highly relevan as advances in sensing and processing enable compac disribued sysems wih wide ranging applicaions, including neworked Unmanned Aerial Sysems (UAS), decenralized balefield negoiaion, decenralized smar-grid echnology, and inerne based social-neworking (see for example [15], [11], [2], [5], and [14]). The developmen of hese sysems however, presen many challenges as he presence of a cenral conrolling agen wih access o all he informaion canno be assumed. There have been significan advances in conrol of neworked sysems using informaion available only a he agen level, including reaching consensus in neworked sysems, formaion conrol, and disribued esimaion (see for example [15], [5]). The emphasis has been o rely only on local ineracions o avoid he need for a cenral conrolling agen. However, here are many applicaions where he knowledge of he global nework opology is needed for making inelligen inferences. Inferences such as idenifying he ineracions beween agens, idenifying fauly or misbehaving agens, or idenifying agens ha enjoy high conneciviy and are in a posiion o influence he decisions of he neworked sysem. This informaion in urn, can allow agens o make inelligen decisions abou how o conrol a nework and how o build opimal neworks in real-ime. The key problem ha needs o be addressed for enabling he needed inelligence is: How can an agen use only informaion available a he agen level o make global inferences abou he nework opology? We erm his problem as Nework Discovery, and formulae he problem in he framework of esimaion heory. This work was suppored in par by NSF ECS and NASA Cooperaive Agreemen NNX08AD06A. G. Chowdhary and A. Prof. Eric Johnson are wih he school of aerospace engineering while Prof. Magnus Egersed is wih he school of Elecrical and Compuer Engineering a he Georgia Insiue of Technology, Alana GA, Girish.Chowdhary@gaech.edu, magnus.egersed@ece.gaech.edu, Eric.Johnson@ae.gaech.edu The idea of using measured informaion o make inferences abou he nework characerisics was explored by Franceschelli e al. hrough he esimaion of he eigenvalues of he nework graph Laplacian [6]. They proposed a decenralized mehod for Laplacian eigenvalue esimaion by providing an ineracion rule ha ensured ha he sae of he agens oscillae in such a manner such ha he problem of eigenvalue esimaion can be reduced o a problem of signal processing. The eigenvalues are hen esimaed using Fas Fourier Transforms. The Laplacian eigenvalues conains useful informaion ha can be used o characerize he nework, paricularly he second eigenvalue of he Laplacian conains informaion on he conneciviy of he nework and how fas i can reach agreemen. However, he knowledge of eigenvalues does no yield informaion abou oher deails of he opology, including he degree of conneciviy of individual agens and he graph adjacency marix. Agen level measuremens of oher agens saes was used by Franceschelli, Egersed, and Giua for faul deecion hrough he use of moion probes [7]. The idea behind moion probes is ha individual agens perform in a decenralized way a maneuver ha leaves desirable properies of he consensus proocol invarian and analyze he response of ohers o deec fauly or malicious agens. This work emphasized he imporance of exciaion in he nework saes for nework propery discovery. Muhammad and Jabdabaie have proposed using Gossiplike algorihms for minimizing communicaions overhead in discovering nework properies hrough relayed informaion [11], while Abdolyusefi and Mesbahi have proposed a node knockou procedure [12] for idenifying nework opology. These algorihms rely on he inernal communicaion in he nework o relay relevan informaion o idenify he nework opology. There are various siuaions however, where such communicaion may no be possible or canno be rused. For example, communicaions based approach canno work if some of he agens have become fauly, are unable o communicae, are maliciously relaying wrong informaion, or if he aim is o coverly discover he nework opology of a (possibly unfriendly) nework. In his paper, we do no assume access o he neworks inernal communicaion proocol, and concenrae on he developmen of nework discovery algorihms ha use only measured or sensed informaion a he agen level. Clearly he addiion of communicaions would complimen any of he presened approaches. Finally, we menion ha he problem we are concerned wih is quie differen from ha of disribued esimaion (see for example reference [9] and he references herein). In disribued esimaion he purpose is o reach consensus

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3 III. POSING NETWORK DISCOVERY AS AN ESTIMATION PROBLEM Obaining a soluion o problem 1 in he mos general case can be a quie dauning ask due o a number of reasons, including: The neighbors of he arge agen may change wih ime, The esimaing agen may no be able o sense informaion abou all of arge agen s neighbors, The arge agen may be acively rying o avoid idenificaion of is consensus proocol. In order o progress, we will make he following simplifying assumpion. Assumpion 1 Assume ha he nework edge se does no change for a predefined ime inerval (), ha is he nework is slowly varying. The above assumpion requires ha wihin a ime inerval (), W() = W, ha is he Laplacian vecor W() is ime invarian for a predefined amoun of ime, which amouns o slow variaion in he nework opology. Such slowly varying neworks can be used o model many real-world neworked sysems. This assumpion allows us o cas he problem of nework discovery as a problem of esimaing he consan Laplacian vecor of he arge agen over a ime inerval. The Laplacian vecor conains he informaion abou he degree of agen i and is adjacency o oher agens in he nework, informaion ha can be used o solve he nework discovery problem. Le x R k conain he measuremens of he saes of agens ha are available o he esimaing agen. Noe ha wihou loss of generaliy we can assume ha he agens whose saes he arge agen can measure are bounded above by he oal number of agens in he nework, i.e. k N; for if k > N, ha is when no all agens whose measuremens are available are par of he nework, hen we can always se N = k. Then, leing Ŵ Rk he following esimaion model can be used for esimaing W : ν() = ŴT () x(). (2) Recalling ha y() = W T ()x() he esimaion error can be formulaed as: ǫ() = ν() y() = ŴT () x() W T x(). (3) One way o approach he nework discovery problem, is o design a weigh law Ŵ() such ha ǫ() 0 uniformly in finie ime, ha is ǫ() is idenically equal o zero afer some ime T (ǫ() 0 > T ). The following proposiion shows ha when only a single esimaing agen i used, hen if he esimaing agen canno measure he saes of all of he arge agen s neighbors, hen ǫ() canno be idenically equal o zero. Proposiion 1 Consider he esimaion model of (2) and he esimaion error ǫ of (3), and suppose x does no conain he sae measuremens of all of he arge agen s neighbors, hen ǫ() canno be idenically equal o zero. Proof: Ignoring he irrelevan case when he arge agen has no neighbors, le ζ R m denoe he vecor conaining all of arge agen s neighbors. Then leing i denoe he idenifying subscrip for he arge agen, and deg i denoe he degree of i we have ha y() = ẋ i () = [ 1, 1,...,deg i,..., 1] T ζ() = ˇW T ζ(). Therefore he vecor ˇW R m conains only nonzero elemens. Le x R k, and assume ha k < m (he case when k > m follows in a similar manner), furhermore, le ζ = [ x,ξ], wih ξ R m k. Suppose ad absurdum ǫ() is idenically equal o zero, hen we have ha: ν() y() = [Ŵ(),0..0]T [ x() ξ() ] ˇWζ() = 0. (4) Since we claim ha ǫ() is idenically equal o zero, hen in he nonrivial case (i.e. ζ() 0) we mus have ha [Ŵ(),0..0] ˇW = 0, for all > T in order o saisfy (4). Therefore ˇW mus conain m l zero elemens, which conradics he fac ha ˇW conains only nonzero elemens. Hence, if x does no conain he sae measuremens of all of he arge agen s neighbors, hen ǫ() canno be idenically equal o zero. Remark 1 Noe ha in he above proof we ignored he case when ζ() is idenically equal o zero. If ζ() is idenically equal o zero hen he saes of all agens have converged o he origin, an unlikely prospec, considering he consensus law only guaranees x span(1). Anoher unlikely bu ineresing case arises when ζ() is such ha [Ŵ(),0..0] ˇW ζ() > T. In boh hese cases, one can argue ha ζ() does no conain sufficien exciaion, and proposiion 1 becomes irrelevan. The imporance of exciaion in he saes for solving he nework discovery problem is explored furher in secion III-A. Remark 2 Proposiion 1 formalizes a fundamenal obsrucion o obaining a soluion o he problem of nework discovery: If he esimaing agen canno measure or oherwise know he saes of he arge agen s neighbors, hen an esimaion based approach wih only one esimaing agen canno be used o solve he nework discovery problem. Fuure work will consider muliple esimaing agens. Therefore, we have shown ha in order o use he esimaion model of (2) o solve he nework discovery problem wih one esimaing agen, he following assumpion mus be saisfied: Assumpion 2 Assume ha he esimaing agen can measure or oherwise perceive he posiion of all of he arge agen s neighbors. The following heorem shows ha if a weigh updae law Ŵ() exiss such ha ǫ() can be made idenically equal o zero, hen a soluion o he nework discovery problem (problem 1) can be found.

4 Theorem 1 Suppose assumpion 2 is saisfied, and x() is no idenically equal o zero, hen finding a weigh updae law Ŵ() such ha ǫ() becomes idenically equal o zero (ha is ǫ() = 0 > T ), is equivalen o finding a soluion o he nework discovery problem for he case of saic neworks (assumpion 1). Proof: Suppose here exiss a weigh updae law Ŵ() exiss such ha ǫ() becomes idenically equal o zero. Since assumpion 2 holds, we can arbirarily reorder he saes such ha x = [ζ,ξ], where ξ denoe he saes of he agens which are no neighbors of he arge agen, hence we have: ν y = ŴT () x() [W,0..0] T [ ζ ξ Leing W = Ŵ [W,0..0], we have: ] = 0. (5) ν() y() = W() x() = 0. (6) Since x() is assumed o be no idenically equal o zero, in he nonrivial case we mus have ha W() = 0 > T. Therefore i follows ha Ŵ = [W,0..0] conains he Laplacian vecor of he arge agen, which is sufficien o idenify he degree and neighbors of he arge agen. Remark 3 As in he proof of proposiion 1, an ineresing bu unlikely case arises when W() x(). Once again his relaes o a noion of sufficien exciaion in he sysem saes and is furher explored in secion III-A. To simplify he noaion, we can le x = x. Due o heorem 1, his is equivalen o saying ha for he purpose of he nework discovery problem, he nework can be assumed o be made of only he agens ha eiher inerac wih he arge agen or are visible o he esimaing agen. Hence, his change in noaion does no affec he srucure of he problem, excep ha we now have ǫ() = ν() y() = Ŵ T ()x() W T x() = Wx, which is simpler o deal wih. In his case, he Laplacian vecor of he arge agen W will conain zero elemens corresponding o agens ha he arge agen is no conneced o. Through he above discussion,we have essenially shown ha subjec o assumpion 1 and 2 he nework discovery problem can be cas as he following simpler parameer esimaion problem: Problem 2 Le an esimaion model for he nework discovery problem be given by (2), and he esimaion error be given by (3). Design an updae law Ŵ such ha Ŵ() W as. Various approaches have been proposed for online parameer esimaion, in he following we will highligh hree such approaches. A. Insananeous Gradien Descen Based Approach In his simples and mos widely sudied approach for parameer esimaion Ŵ is updaed in he direcion of maximum reducion of he insananeous quadraic cosv(ǫ()) = 1 2 ǫ2 (). Tha is, leing Γ be a posiive learning rae we have Ẇ = Γ V. This resuls in he following updae law: Ŵ Ŵ() = Γx()ǫ(). (7) The convergence properies of he gradien descen based approach have been widely sudied, i is well known ha for his case a necessary and sufficien condiion for ensuring Ŵ W as exponenially is a condiion on Persisency of Exciaion (PE) in x() [1],[13],[16]. Various equivalen definiions of exciaion and he persisence of exciaion of a bounded signals exis in he lieraure [1],[13]; we will use he definiions proposed by Tao in [16]: Definiion 1 A bounded vecor signal x() is persisenly exciing if for all > 0 here exiss T > 0 and γ > 0 such ha +T x(τ)x T (τ)dτ γi. (8) Noe ha definiion 1 requires ha he marix +T x(τ)x T (τ)dτ be posiive definie over all predefined finie ime inervals. If a signal saisfies his condiion over only one such inerval, i is called as exciing, bu no persisenly exciing. As an example, consider ha in he wo dimensional case, vecor signals conaining a sep in every componen are exciing, bu no persisenly exciing; whereas he vecor signal x() = [sin(),cos()] is persisenly exciing. Hence, in order o ensure ha W 0, we mus ensure ha he sysem saes x() are persisenly exciing. However, here is no guaranee ha he nework sae vecor x() would be exciing if he nework is only running he consensus proocol of (1). For example, he following fac shows ha if he iniial sae of he nework happens o be an eigenvecor of he graph Laplacian, hen he sysem saes are no persisenly exciing. Fac 1 The soluion x() o he differenial (ẋ() = Lx()), where L is he graph Laplacian, need no be persisenly exciing for all choices of x(0). Proof: Lex(0) andλ R be such halx(0) = λx(0), ha is le x(0) be an eigenvecor of L. Then we have x() = e λ x(0), hence +T +T x(τ)x T (τ)dτ = e 2λ x(0)x T (0), (9) which is a-mos rank 1, and hence no posiive definie over any inerval. Therefore, an exernal forcing erm will be needed o enforce PE in he sysem. The consensus proocol can hen be wrien as: ẋ i () = j N i [x i () x j ()+f(x i (),)], (10)

5 wheref(x i (),) is a known bounded mappingr 2 R used o inser exciaion ino he sysem. In is mos simples form f(x i (),) can simply be a random sequence of numbers, or i could be an elaborae periodic paern (such as in [7]) which is known over he nework. We evaluae he performance of his algorihm hrough simulaion on a nework conaining 9 nodes wih each of he nodes updaed by (10), for solving he nework discovery problem. I is assumed ha f(x i (),) is a known Gaussian random sequence wih an inensiy of 0.01 and ha y i () = ẋ i () f(x i (),) can be measured. Noe ha he chosen f(x i (),) does inroduce persisen exciaion. The agens are arbirarily labeled, and he hird agen is picked as he esimaing agen, and i esimaes he consensus proocol for he second agen (which is he arge agen). The Laplacian vecor for he arge agen is given by W = [0, 3,1,0,0,1,1,0,0], and is consensus proocol will have he form y i = W T x. The arge agen has 3 neighbors (i.e. degree of i is 3), hey are agen 3,6, and 7. Figure 2 shows he performance of he gradien descen algorihm for he nework under consideraion wih Γ = 10. I can be seen ha he algorihm is unsuccessful in esimaing he Laplacian vecor for W by he end of he simulaion, even when persisen exciaion is presen. Increasing he learning rae Γ may slighly speed up he convergence, however he key condiion required is ha he x() remain persisenly exciing such ha he scalar γ in definiion 1 is large. Tha is, he convergence is dependen no only on he exisence of exciaion, bu also on is magniude True adjecency evoluion of esimaes rue adjecency values esimaes rue degree for esimaion. Chowdhary and Johnson have noed ha if he updae law uses specifically seleced and recorded daa concurrenly wih curren daa for adapaion, and if he recorded daa were sufficienly rich, hen inuiively i should be possible o guaranee Ŵ W as wihou requiring persisenly exciing x(). This resuls in a Concurren Learning gradien descen algorihm [4], [3]. Le j {1,2,...p} denoe he index of a sored daa poin x j, le ǫ j = W T x j, le Γ > 0 denoe a posiive definie learning rae marix, hen he concurren learning gradien descen algorihm is given as: Ŵ() = Γx()ǫ() Γx j ǫ j. (11) i=1 The parameer error dynamics W() = Ŵ() W for his case can be expressed as follows: W() = Γx()ǫ() Γ x j ǫ j = Γx()x T () W() Γ = Γ[x()x T ()+ j=1 x j x T W() j j=1 x j x T j ] W(). j=1 (12) The concurren use of curren and recorded daa has ineresing implicaions, as he exciing erm f(x i,) will no need o be persisenly exciing, bu only exciing over a finie period over which rich daa can be recorded. In fac, Chowdhary and Johnson have also shown ha he recorded daa need only be linearly independen in order o guaranee weigh convergence [4]. This condiion on sufficien richness of he recorded daa is capured in he following saemen: Ŵ True degree ime seconds Condiion 1 The recorded daa has as many linearly independen elemens as he dimension of x. Tha is, if Z = [x 1,...,x p ], hen rank(z) = N. This condiion is easier o monior online and essenially requires ha he recorded daa conain sufficienly differen elemens o form he basis of he sae space. The following heorem is proven in [4]: Fig. 2. Evoluion of he esimae of he Laplacian vecor (Ŵ ) for nework discovery using gradien descen. Noe ha he esimaes do no converge o he acual values depiced by doed lines. The resuls go o show ha he convergence of he gradien descen mehod is dependen no only on he presence of persisence exciaion bu also on is magniude. B. Concurren Gradien Descen Based Approach The gradien descen algorihm of Secion III-A is suscepible o being suck a local minima, and requires PE o guaranee convergence. For many neworked conrol applicaions he condiion on PE is infeasible o monior online, paricularly since he rajecories of individual agens are no known a-priori. On examining (7) we see ha he updae law uses only insananeously available informaion (x(),ǫ()) Theorem 2 If he recorded daa poins saisfy condiion 1, hen he zero soluion of parameer error dynamics W 0 of (12) is globally exponenially sable when using he concurren learning gradien descen weigh adapaion law of (11). We now evaluae he performance of he concurren learning gradien descen algorihm on he neworked sysem simulaion described in secion III-A. Figure 3 shows he performance of he concurren gradien descen algorihm for he nework wih Γ = 10. The simulaion began wih no recorded poins, a each ime sep, he sae vecor x() was scanned online, and poins saisfying he condiion Z T x() < 0.5 or y() ν() > 0.3 were seleced for

6 sorage. Condiion 1 was found o be saisfied wihin 0.1 seconds ino he simulaion. I can be seen ha he algorihm is successful in esimaing he Laplacian vecor for W, and hus in esimaing he degree of he hird agen and he ideniy of is neighbors. Hence, he algorihm ouperforms he radiional gradien descen based mehod (secion III-A) wih he same level of enforced exciaion. In general, he speed of convergence will be dependen on he minimum eigenvalue of he marix ZZ T and o a lesser exen, he learning rae Γ. Tha is, ideally we would like he sored daa o no only be linearly independen, bu also be sufficienly differen in order o maximize he minimum singular value of Z. A he end of he simulaion he minimum singular value was found o be Ŵ True adjecency True degree evoluion of esimaes real adjecency values esimaes real degree ime seconds Fig. 3. Evoluion of he esimae of he Laplacian vecor (Ŵ ) for nework discovery using concurren gradien descen. Noe ha he esimaes converge o he acual values (shown using doed lines) wihin 2 seconds of he simulaion. The converged esimaes of he Laplacian vecor direcly yield he degree and he neighbors he arge agen. C. Leas Squares based Approaches Recursive leas squares, or equivalenly a Kalman filer based implemenaion, can be used o solve he esimaion problem. In his approach a recursive law is developed such ha a quadraic cos of he inegral of he esimaion error is minimized [8], [1], [10]. To achieve his using assumpion 1 an updae model for he esimae of he Laplacian vecor Ŵ as Ŵ = 0, and a Kalman filer is designed o esimae Ŵ using a measuremen model ν = ŴT x and he esimaion error y ŴT x. The benefi of his approach is ha he soluion can be shown o be opimal in he leas squares sense, and noise in measuremens can be handled. The downside is ha he mehod is compuaionally expensive as he covariance marix mus also be propagaed. Furhermore, PE is required o guaranee convergence [1], [16]. IV. CONCLUSION In his paper we considered he problem of nework discovery, in which, an agen uses locally available informaion o esimae he global opology of a neworked sysem aemping o reach consensus. We showed ha subjec o wo key assumpions, he nework discovery problem can be cas as a parameer esimaion problem and he elemens of he graph Laplacian can be esimaed in real-ime. The graph Laplacian conains he adjacency and degree informaion for a given agen, and is sufficien o form an esimae of he nework opology. The firs assumpion requires ha he nework is slowly varying, ha is, i requires he nework opology o remain saic over a predefined ime inerval. The second assumpion requires ha he esimaing agen can measure (or oherwise know) he saes of all of arge agen s neighbors. In fac, we showed ha if no saisfied, his assumpion forms a major obsrucion o solving he nework discovery problem using only one esimaing agen. We discussed hree mehods for solving he nework discovery problem in he parameer esimaion framework, and compared he performance of wo in simulaion. We noed ha he concurren gradien descen mehod requires far less exciaion han he radiional gradien descen mehod, and has improved convergence. In conclusion, we noe ha regardless of wha parameer esimaion mehod is used o solve he nework discovery problem, some amoun of exciaion mus be insered ino he neworked sysem for converging o a soluion. V. ACKNOWLEDGMENTS REFERENCES [1] Karl Johan Asröm and Björn Wienmark. Adapive Conrol. Addison- Weseley, Readings, 2 ediion, [2] F. Bullo, J. Corés, and S. Marínez. Disribued Conrol of Roboic Neworks. Applied Mahemaics Series. Princeon Universiy Press, Elecronically available a hp://coordinaionbook.info. [3] Girish Chowdhary. Concurren Learning for Convergence in Adapive Conrol Wihou Persisency of Exciaion. PhD hesis, Georgia Insiue of Technology, Alana, GA, [4] Girish Chowdhary and Eric N. Johnson. Concurren learning for convergence in adapive conrol wihou persisency of exciaion. In 49h IEEE Conference on Decision and Conrol, [5] Magnus Egersed and Mehran Mesbahi. Graph Theoreic Mehods in Muliagen Neworks. Princeon Universiy Press, [6] M. Franceschelli, A. Gasparri, A. Giua, and C. Seazu. Decenralized laplacian eigenvalues esimaion of he nework opology of a muliagen sysem. In IEEE Conference on Decision and Conrol, [7] M. Franceschelli, Egersed M., and A. Giua. Moion probes for faul deecion and recovery in neworked conrol sysems. In American Conrol Conference, [8] Arhur Gelb. Applied Opimal Esimaion. MIT Press, Cambridge, [9] Vijay Gupa. Disribued Esimaion and Conrol in Neworked Sysems. PhD hesis, California Insiue of Technology, [10] Ravindra V. Jaegaonkar. Fligh Vehicle Sysem Idenificaion A Time Domain Approach, volume 216 of Progress in Asronauics and Aeronauics. American Insiue of Aeronauics and Asronauics, Reson, [11] A. Muhammad and A. Jadbabaie. 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