Submodular Game for Distributed Application Allocation in Shared Sensor Networks

Transcription

1 Submodular Game for Disribued Applicaion Allocaion in Shared Sensor Neworks Chengjie Wu, You Xu, Yixin Chen, Chenyang Lu Deparmen of Compuer Science & Engineering Washingon Universiy in S. Louis {wu, yx, chen, Absrac Wireless sensor neworks are evolving from singleapplicaion plaforms owards an inegraed infrasrucure shared by muliple applicaions. Given he resource consrains of sensor nodes, i is imporan o opimize he allocaion of applicaions o maximize he overall Qualiy of Monioring (). Recen soluions o his challenging applicaion allocaion problem are cenralized in naure, limiing heir scalabiliy and robusness agains nework failures and dynamics. This paper presens a disribued game-heoreic approach o applicaion allocaion in shared sensor neworks. We firs ransform he opimal applicaion allocaion problem o a submodular game and hen develop a decenralized algorihm ha only employs localized ineracions among neighboring nodes. We prove ha he nework can converge o a pure sraegy Nash equilibrium wih an approximaion bound of /. Simulaions based on hree real-world daases demonsrae ha our algorihm is compeiive agains a sae-of-he-ar cenralized algorihm in erms of. I. INTRODUCTION Tradiionally, wireless sensor neworks (WSNs) are used as specialized plaforms where only a single applicaion is deployed on each sensor. Recenly, large-scale, inegraed WSNs ha suppor muliple applicaions sar o emerge. Many applicaion domains such as urban sensing [], building auomaion and environmenal monioring [] have already adoped he inegraed WSN paradigm o suppor muliple applicaions. Compared o separae applicaion-specific sensor neworks, a shared WSN can be more cos effecive and more flexible as i enables resource sharing among applicaions and dynamic resource allocaion in response o changes in he environmen and user needs. Severe resource consrains limi he allocaion of all possible applicaions o sensors in a shared WSN. For example, he TelosB moe [3], a represenaive sensor plaform, only has KB of RAM, a 5 Kbps radio, and a 6-bi CPU running a 8 MHz. On he oher hand, he Qualiy of Monioring () of applicaions depends on applicaion allocaions. Therefore, i is imporan o opimize he allocaion of muliple applicaions among sensor nodes in order o maximize he overall, subjec o resource consrains. This problem is challenging because i is essenially a discree opimizaion wih an exponenially large soluion space. Some recen works uilize he submodulariy of he funcion o ackle his discree opimizaion problem. Submodulariy is an imporan propery of he funcions for neworked sensing applicaions. Inuiively, a funcion f ha maps a subse of a se S o a real value is submodular if i has a diminishing reurn propery, i.e., adding an elemen o a smaller subse of S makes a bigger difference o he funcion values han adding i o a larger subse of S. The submodulariy of is due o he inheren propery ha sensor readings from differen nodes are ofen correlaed. For insance, since he emperaure readings from differen nodes in he same room are correlaed wih each oher, allocaing a new node o a emperaure monioring applicaion resuls in diminishing improvemen o he as he se of nodes allocaed o he applicaion grows. Submodulariy of sensor allocaion for monioring emperaure [4] and waer qualiy [5] [7] has been observed in previous sudies of real-world daases. Many exising works cenered around submodular opimizaion have been proposed for opimizaion problems in sensor neworks. Recen heoreical works also show approximaion algorihms ha can achieve a ( /e)-approximaion bound [8]. Xu e al. [9] proposed a greedy algorihm and achieved a /3 approximaion bound. Submodular opimizaion approaches are also used in sensor selecion and placemen applicaions [6], [7]. However, all hese exising submodular opimizaion approaches are essenially cenralized soluions. For WSNs, a cenralized algorihm implies here is eiher a node or gaeway ha mainains he global informaion of he nework. A cenralized approach is no desirable for WSNs due o is limiaions in scalabiliy and faul olerance. Firs, a shared WSN is usually of large scale in erms of he number of nodes and hop couns. Hence, i is inefficien or even impossible o achieve global informaion sharing ha is required by cenralized opimizaion algorihms. Second, in a cenralized approach, much of he compuaion and communicaion happens on a single poin resuling in a single poin of failure. To address he limiaions of cenralized approaches, we sudy disribued opimizaion approaches for applicaion allocaions. Meanwhile, we sill exploi he submodulariy propery of funcions o achieve desirable approximaion bounds. In his paper, we provide several major heoreical resuls: ) We propose he covariance cover funcion as a new meric ha is amenable o disribued opimizaion; ) We show ha he opimal muli-applicaion allocaion problem wih covariance cover as objecive funcion is a submodular opimizaion problem wih muliple knapsack consrains; 3) We

2 propose a game heory based disribued algorihm for solving his submodular opimizaion problem and prove ha our algorihm can achieve a /-approximaion bound when each sensor achieves opimal allocaion of applicaions. We also prove our algorihm can achieve a /( + β)-approximaion bound when each sensor achieves a /β-approximae allocaion of applicaions. Simulaions based on hree real-world daeses demonsrae ha our disribued algorihm can achieve comparable as a sae-of-he-ar cenralized algorihm [9], while scaling effecively in erms of boh execuion ime per node and he communicaion overheads. II. RELATED WORK Originaed from cenralized opimizaion, subgradien mehods have been used o opimize problems where he gradiens of he objecives are hard o obain, while he subgradiens of objecive funcions wih respec o a subse of variables are easy o obain [], []. The subgradien opimizaion mehod can be used as a disribued opimizaion algorihm for problems in WSNs, in which each sensor node opimizes he objecive funcion disribuively using is own subgradien value. However, he subgradien mehod is no suiable for he muli-applicaion allocaion problem in a large-scale, mulihop WSN, due o he fac ha in each ieraion of he algorihm, i is sill required o propagae he soluion for he subsequen subgradien calculaion. Tha is o say, alhough he opimizaion is localized o each sensor, global communicaion is sill required. Game-heoreic approaches have been proposed o address he above issue. In hese approaches, communicaions are made only beween cerain sensors in a user-defined neighborhood. Anoher unique propery of game-heoreic approaches is ha hey do no assume ha agens (in his case, sensor nodes) work cooperaively. Insead, selfish sensor nodes opimize a local version of he objecive funcions, ofen called uiliies or privae uiliies, independenly, unil none of hem can furher improve heir privae uiliies by making a differen decision. When hese uiliies are carefully designed o reflec he objecive funcion, he overall objecive funcion, also called social uiliy, is subsequenly opimized by hese noncooperaive agens []. When he social and privae uiliies are carefully designed, game-heoreic approaches guaranee a consan opimizaion bound [] [4]. Since he uiliy sysem decides he naure of he game, needless o say, for game-heoreic approaches o work for muli-applicaion allocaion problems, designing he uiliy sysem is criical. Specifically, in a disribued soluion, a uiliy sysem ha is easy o calculae and has no global informaion propagaion requiremens is desirable. In oher words, in he applicaion allocaion problem, a uiliy sysem should reflec he value based on he decisions of each sensor, while i does no require global communicaion in he nework. In Game Theory, a sae where no player can improve is uiliy is called an equilibrium sae. Previous works proposed differen formulaions [5], [9], [5], including variance reducion and muual informaion gain. However, neiher is suiable as he objecive funcion in a disribued game-heoreic approach, which requires ha a sensor s uiliy is independen of oher sensors ha are no in is neighborhood. This condiion is violaed when using variance reducion or muual informaion as, because one sensor s uiliy of allocaing an applicaion is relaed o all sensors ha carry he applicaion. We address his issue by proposing a new meric ha is submodular and suiable for game-heoreic disribued opimizaion while serving as an effecive proxy for variance reducion in opimizaion. III. PROBLEM FORMULATION In his secion, we firs review he variance reducion formulaion. Afer discussing he disadvanages of using variance reducion in disribued algorihms, we propose a new meric called covariance cover ha is amenable o disribued soluions. In he end, we formulae he applicaion allocaion problem in shared WSNs using covariance cover as meric. A. Formulaion Variance reducion is commonly used o measure in WSN applicaions [5], [9]. Assuming sensor readings follow a Gaussian Process, he variance reducion measures how much he variance of he readings from he unallocaed sensors. Variance reducion is calculaed based on covariance. Assuming K is he covariance marix for sensor nodes, and for wo subses of sensor nodes G, H V, he covariance marix of G and H is denoed by K GH, where is rows corresponding o G and columns corresponding o H exraced from K. For a given se G wih applicaion allocaed, he variance of he unassigned se Ḡ = V \G is σ Ḡ G = r(kḡḡ) r(k ḠG K GG K GḠ), where r() is he race of a marix. In his applicaion allocaion problem, he goal is o minimize he variance of Ḡ given G such ha he qualiy of sensing is maximized. Namely, we wan o maximize he negaion of he variance. Given r(k) = r(k GG ) + r(k ḠḠ), variance reducion for one applicaion is: Q V R = r(k GG ) + r(k ḠG K GG K GḠ) () Variance reducion is jus one of he many possible ways o formulae. There is an inheren disadvanage of using variance reducion for our problem. I is no feasible for a sensor node wih limied memory resource o sore a kernel marix ha is quadraic o he size of he nework, and i is expensive o compue variance reducion since i involves marix muliplicaion and inversion. To overcome his inheren disadvanage, we decompose he variance reducion and propose a new formulaion which is more amenable o disribued approaches. We begin wih inroducing he nework model. A nework consiss of a group of sensors {,, n}. Each sensor node

3 can be presened as a verex. For a pair of sensors i and j, if each of hem is in he oher s communicaion range, i and j are defined as a pair of neighbors. The nework can be presened as a graph G = (V, E), where V = {,,, n} and (i, j) E if and only if i and j are a pair of neighbors. Now we will decompose he variance reducion based on wo assumpions. Theorem. Variance reducion formulaion () is equivalen o K ii + Kij, (i,j) E, or j G if (I) he covariance of any wo nodes is nonzero if and only if hey are a pair of neighbors, and (II) any wo allocaed nodes are no a pair of neighbors. Proof: Le us firs simplify he variance reducion formulaion. Since any wo allocaed nodes are no neighbors of each oher, and only neighbors have nonzero variance, we can prove K GG is an ideniy marix. I immediaely follows ha Q V R = r(k GG ) + r(k ḠG K = K ii + Kij.,j Ḡ GG K GḠ) Since only a pair of neighbors have nonzero covariance, Q V R = K ii +,j Ḡ,(i,j) E K ij. We assume wo allocaed nodes are no neighbors, which means if (i, j) E, i G, hen j Ḡ. I follows Q V R = K ii + Kij. (i,j) E, or j G One quesion raises naurally: how realisic are he assumpions? We argue ha he proposed wo assumpions, alhough someimes violaed, provide good approximaions of he real-world scenarios. Firs, i is reasonable o assume a pair of nearby sensors have larger covariance. For example, he emperaure measuremens of wo differen sensors in he same office room are more correlaed han wo sensors in differen rooms. Second, since our applicaions have he inheren propery of submodulariy, allocaing wo neighboring nodes simulaneously ypically does no give much gain in erms of. To maximize, a good soluion should naurally allocae nodes ha have unallocaed nodes as neighbors. Acually, our submodular game algorihm is no limied by hese wo assumpions, i can handle siuaions when assumpions do no hold. We name he new formulaion covariance cover. Denoing τ ij = Kij as he weigh of edge (i, j), and τ ii = K ii as he weigh of node i, we define he covariance cover formulaion as Q CC = τ ii + τ ij. () (i,j) E, or j G B. Applicaion Allocaion Problem Formulaion Given meric as covariance cover, we wan o furher formulae he applicaion allocaion problem in shared sensor neworks. When here are muliple applicaions P = {,,, p} wih weighs {w, w,, w p }, we wan o maximize he summaion of all applicaions p = w Q, where Q is he covariance cover for applicaion. Q = τii + (i,j) E, or j G τ ij This problem is challenging because of criical resource consrains, e.g., CPU and memory consrains. For each sensor, he oal memory and CPU consumed by all applicaions can no exceed is limis. Therefore, suppose each node has m resource consrains R = {,,, m}, he capaciy of node i on resource k is C i,k, and applicaion consumes c i,k unis of resource k on node i, he consrained opimizaion problem can be formulaed as: max s.. = w Q P Q = τii + c i,k C i,k, (i,j) E, or j G τ ij i V, k R here G is he se of nodes which are assigned applicaion. I is easy o see ha all resource consrains here are knapsack consrains. This ype of consrain formulaion also can be used o characerize various communicaion paerns among nodes, such as he paern in a daa collecion applicaion ha collecs daa from every node on he rouing ree. IV. SUBMODULAR GAME In his secion, we will formulae a non-cooperaive game based on he covariance cover formulaion discussed in he previous secion, which leads o a compleely disribued algorihm. We inroduce ypical erminologies in game heory a firs. A. Submodular Game Formulaion Suppose we have n sensor nodes, and each sensor node i in he nework is an agen i in he game. For each sensor, is sraegy a i is he subse of applicaions ha can run on i. a i = { applicaion runs on sensor i} = { i G, P }. Under he resource consrain we discussed earlier, he sraegy se A i of player i is A i = {a i a i c i,k C i,k, k R}, A pure sraegy is one in which each agen decides o carry ou a specific sraegy. In game heory, mixed sraegy is also widely discussed. However, we only discuss pure sraegy in

4 his paper, because in realiy of sensor neworks, i is hard o implemen sraegies wih probabiliy disribuion. Also, we prove ha our game has a leas one Nash equilibrium wih pure sraegies. We denoe he sraegy space of he game as A = A A A n. A game is always defined on a uiliy sysem. To build he uiliy sysem, we need o define he uiliy funcion a firs. Given a sraegy profile A = (a, a,, a n ) A, le A a i denoe he sraegy profile obained if agen i changes is sraegy from a i o a i. Formally, A a i = (a,, a i, a i,, a n). The goal of our game is o maximize he social uiliy γ : V R defined on pure sraegy profile A = {a,, a n } as p γ(a) = γ (A) = = = p w ( = = a i or a j,(i,j) E τ ij + a i,i V p w ( τij + τii). (i,j) E, or j G τ ii) Remind G is he se of nodes who are assigned applicaion. For each agen i, we define a privae uiliy φ i : V R as: φ i (A) = a i φ i(a) = a i w (τ ii + j N i τ ij ) + δ j G where N i = {j (i, j) E} is sensor i s neighborhood. For edge (i, j), if no only i, bu also j runs applicaion, (i, j) s edge weigh τij need o be equally shared by boh i and j. Oherwise, sensor i will accoun all (i, j) s edge weigh ino is privae uiliy. The goal of each sensor is, herefore, o selec a sraegy in order o maximize is privae uiliy under resource consrains. Clearly, such sraegies may no produce a good soluion wih respec o he social uiliy γ. However, we will show ha he sraegies sensors finally selec will resul in a reasonable good social uiliy γ in nex secion. To localize he opimizaion problem o each sensor, given sraegies of is neighbors fixed, we redefine sensor i s privae uiliy φ i (A) as is uiliy funcion u i (x i ), which is a funcion of is own decisions x i. Is decision x i = {x i,, xp i } is redefined from is sraegy a i, where x i = means a i. u i (x i ) = p w [τii + = j N i τ ij + δ j G We denoe Ω i = [τii + τ ij j N i +δ ] as a consan, j G assuming sraegies of i s neighbors are given. To maximize ]x i (3) (4) is uiliy funcion, sensor node i needs o solve a ineger programming problem: p Max u i (x i ) = w Ω ix i where s.. = x i {, }, P p c i,k x i C i,k k R = Acually, his is a ypical mulidimensional knapsack problem. There is a rich package of lieraure o solve his problem. We propose wo algorihms based on p, which is he number of applicaions. If p is no larger han T p, our soluion will adop a naive enumeraion algorihm. Basically, i enumeraes all possible applicaion assignmens and reurns an opimal soluion. Oherwise, our soluion will adop a polynomial ime approximae algorihm, which is proven o have a +m approximaion bound (secion 9.4. of [6]), where m = R is he number of resource consrains. Here T p is he hreshold for p, we se i o 5 in our implemenaion. We show he skech of our soluion for problem (5) in Algorihm. Algorihm : Algorihm for knapsack problem (5) Se ˆx = {,, }; if p T p hen Adop he Enumeraion Algorihm; Enumerae x {, } p, reurn opimal soluion ˆx. else Adop he Approximaion Algorihm; Relax problem (5) o a linear programming problem and compue an opimal soluion x LP of he LP-relaxaion. Se I = { x LP = } Se F = { < x LP < } Reurn ˆx = max{ I w, max{w F }} end Now we analyze compuaional cos of our algorihm. If p T p, he ime complexiy is O( ( p d) ) where d is he maximum number of applicaions ha can be allocaed on one node. And if p is larger han T p, he relaxed linear programming (LP) problem is significanly simplified due o he small numbers of resource consrains as well as applicaions. The number of resource consrains is usually no more han 3 (e.g., memory, CPU, and bandwidh). The number of applicaions is also small due o he limied resources available per node. Our algorihm employs an efficien and pracical soluion as follows. We solve he dual problem of he aforemenioned LP problem which only has hree variables (he shadow prices of memory, CPU and bandwidh consrains) and p consrains (p is he number of applicaions). Even a naive LP solver ha enumeraes all possible exreme poins (each of he hree consrains deermines one exreme poin) and finds he bes feasible one has he compuaional complexiy O(p ( p 3) ) = O(p 4 ), and he memory requiremen (5)

5 of he naive enumeraion algorihm is O(). Eiher way, he cos of each individual mulidimenional knapsack problem is O(p d ), where d is a small ineger. B. Submodular Game Algorihm Now we discuss our disribued submodular game algorihm. In he beginning sage, sensor nodes in he nework need o ge wo key parameers abou applicaions se P : ) ypes of required sensor readings; ) he frequency of each sensor reading. These wo key parameers are disribued o he nework from a cenral faciliy like base saion. Afer his sage, no cenral faciliy is needed in he algorihm, so our algorihm is fully disribued. Algorihm : Game algorihm for sensor node i iniializaion; i measures sensor readings for each applicaion ; i broadcass all sensor readings in is neighborhood; i calculaes τii and τ ij for every neighbor j; if Timer Λ i fires hen if receiving sraegy changes from neighbors hen i runs algorihm, oupu ˆx sraegy; if sraegy changes hen i broadcass is sraegy in neighborhood; end end end In he iniializaion sage, each node measures sensor readings for a cerain inerval and broadcass sensor readings in is neighborhood. Based on neighbor j s readings, node i can calculae he covariance beween i and j as well as τ ij. Algorihm shows he deailed decision-making procedure for each sensor. In each round of he game, nodes share he same ime inerval T. Each node generaes a random number Λ i (Λ i < T ) as a imer using a unique seed, such ha wo imers will no fire a he same ime. Each imer Λ i will fire once and only once during each ime inerval T for sensor node i o solve he allocaion problem (5) locally. If a new sraegy is generaed, node i will broadcas i in he neighborhood. Oherwise i will keep quie. Each sensor node i also receives messages from is neighbors abou heir updaed sraegies. The algorihm erminaes when no sraegy changes are made in a round. Here we analyze he efficiency of our game algorihm. From he compuaional cos perspecive, we already give he compuaional cos of each node in each round as O(p d ). Since boh p and d are small inegers, i is reasonable o say he compuaional cos is accepable on a sensor node wih limied resources. From a nework perspecive, we wan o analyze he communicaion cos. In each round, sensor node i needs o receive messages from all is neighbors and broadcas is own sraegy in is neighborhood if necessary. We denoe he Expeced Transmission Coun (ETX) of link e is ν e. Since sensor node i broadcass in he neighborhood, in he wors case, he number of messages i needs o send is he maximum of all he ETXs in is neighborhood ζ i = max j Ni ν (i,j). So he overall number of packes sensor i sends in he game is lower han κζ i, given κ is he number of overall number of rounds. Because κ is always a small number (less han ) based on our evaluaion, he communicaion cos is relaively small. V. CONVERGENCE AND APPROXIMATION BOUND In his secion, we firs show he social uiliy (3) is submodular. Then we prove our submodular game (γ, i V φ i ) defined in (3) and (4) can converge o a pure sraegy Nash equilibrium wih an approximaion bound of, if sensors use he enumeraion algorihm o solve he mulidimensional knapsack problem (5). If sensors use he +m -approximae soluion for he knapsack problem, he game can converge o a (+m)-approximae pure sraegy Nash equilibrium and he approximaion bound is +m, where m is number of resource consrains. A. Submodulariy Definiion. (Submodulariy) Le V be a finie se, a funcion f : V R is submodular if for any A B V and x V B, f(b {x}) f(a {x}). Recall ha we defined he social uiliy (3) as a funcion of pure sraegy profiles in he las secion. We redefine he social uiliy here as a funcion of he se of sensors ha we allocae applicaions on: γ = p = w Q (G ), where Q (G ) = τij + {(i,j) or j G } τ ii. This definiion is equivalen o he one we defined in (3), bu i is now defined on he se of sensors. Based on his se based definiion, we can prove he social uiliy γ is submodular. Theorem. The social uiliy γ is submodular. Proof: Since γ = p = w Q (G ), we only need o show Q (G ), P is a submodular funcion. By definiion, we need o prove: if A B V and x V B, f(b {x}) f(a {x}). If x B, i is obvious ha Q (B {x}) Q (B) = Q (A {x}) Q (A). If x / B, i follows: A B {(i, k) i = x & k / B} {(i, k) i = x & k / A} τik i=x & k / B i=x & k / A τ ik Q (B {x}) Q (B) Q (A {x}) Q (A) B. Convergence and Pure Nash Equilibrium Now we will discuss he convergence of our game. By defining a poenial funcion, we show he increase of each agen s privae uiliy will lead o he increase of he poenial funcion. Then we can prove our game will converge a a pure sraegy Nash equilibrium. Here we assume our Submodular

6 Game Algorihm (Algorihm ) is using he enumeraion algorihm. Definiion. (Pure Sraegy Nash Equilibrium) A pure sraegy profile A A is a pure sraegy Nash equilibrium if no agen has an incenive o change is sraegy. For any agen i, a i A i, φ i (A a i) φ i (A). Equivalenly, given he oher agens sraegies, a i is he bes response of agen i. Theorem 3. A pure sraegy Nash equilibrium always exiss for he uiliy sysem (γ, i φ i ) we defined in (3) and (4). And Submodular Game Algorihm (Algorihm ) converges o a pure sraegy Nash equilibrium. Proof: The proof sars from defining he poenial funcion of he game. We define he poenial funcion ψ for a sraegy profile A as ψ(a) = p w ( τii + i V, a i = (i,j) E, a i or a j n (i,j) l= τ ij l ) where n (i,j) is he number of agens which are assigned applicaion as well as he end poins of edge (i, j). n (i,j) is if boh i and j are assigned applicaion, and i is if only one of i and j is assigned he applicaion. Assume sensor i changes is sraegy from a i o a i, as a resul he sraegy profile of he game changes from A o A. Here a i is he se of applicaions which are assigned o sensor i in original sraegy profile A, and a i is ha in new sraegy profile A. Le G = a i a i and H = a i a i. We use E i o denoe he se of edges which coincide wih sensor i. Since he change only happens on node i and edges si on i, we will ignore oher nodes and edges in following proof. = = ψ(a ) ψ(a) w τii w τii + a i a i w τij n j N i H (i,j) + w τij n j N i G (i,j) φ i (A ) φ i (A) w τii w τii + a i a i w H j N i τ ij n (i,j) + G w τ ij ; n j N i (i,j) Obviously, ψ(a ) ψ(a) = φ i (A ) φ(a), we prove ha he increase of he privae uiliy of i is exacly he same as increase of he poenial funcion of he game. Once each individual sensor improves is privae uiliy, he poenial funcion ψ of he game also ges increased. Since he maximum value of his poenial funcion is finie, he algorihm will converge in finie rounds. C. Valid Uiliy Game and Approximae Nash Equilibrium Now we wan o prove our submodular game (γ, i V φ i ) is a valid uiliy sysem. Definiion 3. (Uiliy Sysem) [] A game is called a uiliy sysem if and only if he privae uiliy of an agen is a leas as grea as he loss in social uiliy resuling from he agen dropping ou of he game. Tha is, he game (γ, i φ i ) is a uiliy sysem if and only if i has he propery φ i (A) γ a i (A i ). Definiion 4. (Valid Uiliy Sysem) [] A uiliy sysem is said o be valid if and only if he sum of privae uiliies of he agens is a mos he social uiliy. Tha is, he uiliy sysem (γ, i φ i ) is a valid uiliy sysem if and only if i has he propery n i φ i(a) γ(a). We wan o prove ha he game we defined in (3) and (4) is a valid uiliy sysem. A firs, we prove i is a uiliy sysem. Theorem 4. The game (γ, i φ i ) defined in (3) and (4) is a uiliy sysem. Proof: γ a i (A i ) = γ(a) γ(a i i ) = w ( τij + τii) a i j N i / a j w ( τij + a i j N i / a j = φ i (a i ). j N i a j τ ij + τ ii) Theorem 5. The uiliy sysem (γ, i φ i ) defined in (3) and (4) is valid. Proof: We need o prove he uiliy sysem (γ, i φ i ) has he propery i V φ i(a) γ(a). Firs, we define he se of covered edges for applicaion as E = {(i, j) i G or j G }. Equaion (3) shows ha each e = (i, j) E conribues τij o γ(a). We use a vecor (ξ i, ξ j ) o denoe e s conribuion o φ i (A) and φ j (A). There are hree cases here: (τij, ), if i G, j / G (ξ i, ξ j ) = (, τij ), if i / G, j G ( τ ij, τ ij ), if i G, j G Since e s conribuion o γ(a) equals o he sum of is conribuion o φ i (A) and φ j (A), afer we sum up all e E, P w τij = ( w (i,j) E, or j G i V a i τij n j N e i Now we consider conribuion of nodes, i is obviously ha w τii = w τii. P i V a i ).

7 Combining boh conribuion of edges and nodes, γ(a) = i V φ i (A). We cie below an imporan resul on valid game []. Lemma. Le γ be a non-decreasing, submodular se funcion. If (γ, i φ i ) is a valid uiliy sysem hen for any pure sraegy Nash equilibrium A A, we have γ(a ) OP T, where OP T is he opimal social uiliy. Combining Theorem, Theorem 3, Theorem 5 and Lemma, we ge following heorem. Theorem 6. For he submodular game (γ, i φ i ) we defined in (3) and (4), here exiss a leas one pure sraegy Nash equilibrium. And for is any pure sraegy Nash equilibrium A A, we have γ(a ) OP T. Now we consider he case in which each sensor runs he approximaion algorihm and can only ge a β approximae soluion insead of opimal soluion for he mulidiminuional knapsack problem. β approximae soluion (β > ) means he soluion is no less han β of he opimal soluion. We can prove ha our algorihm can achieve a β-approximae Nash Equilibrium. Definiion 5. (β-approximae Nash Equilibrium) A pure sraegy profile A A is a β-approximae Nash equilibrium if no agen can find a beer alernaive pure sraegy in which is privae uiliy is more ha β imes beer han is curren privae uiliy. Tha is for any agen i, a i A i, φ i (A a i) ( + β)φ i (A) Theorem 7. For he submodular game defined in (3) and (4), Submodular Game Algorihm (Algorihm ) wih he β - approximaion algorihm converges o a β-approximae Nash equilibrium A A. Proof: The proof follows he same way of heorem 3. By bounding he value of he poenial funcion, we can prove ou Submodular Game Algorihm can reach a β-approximae Nash equilibrium. We cie he following imporan resul on approximae Nash equilibria []. Lemma. Le γ be a non-decreasing, submodular se funcion, and (γ, i φ i ) be a valid uiliy sysem. In any β- approximae Nash equilibrium A A we have γ(a) +β OP T, where OP T is he opimal social uiliy. Theorem 8. For he submodular game defined in (3) and (4), Submodular Game Algorihm (Algorihm ) wih a β- approximae soluion converges o a β-approximae Nash equilibrium A A, and we have γ(a) +β OP T, where OP T is he opimal social uiliy. Theorem 8 follows Theorem 5, Theorem 7 and Lemma. In our implemenaion, we use a +m-approximaion algorihm, so our Submodular Game Algorihm can converge o a ( + m)-approximae Nash equilibrium wih an +m approximaion bound. VI. EVALUATION In his secion, we evaluae our Submodular Game Algorihm (SG) by comparing i agains a sae-of-he-ar cenralized opimizaion algorihm Fracional Relaxaion Greedy (FRG) [9]. We conduc simulaions on hree real world daases: Inel daase is colleced in Inel Berkley lab [7]. 54 MicaDo sensor nodes wih weaher boards were used o collec opology informaion along wih humidiy, emperaure and ligh values. The daa collecion las for more han one monh a a sampling period of 3 seconds. In our evaluaion, we generae he covariance marices using daa colleced from nodes in one day. DARPA daase is colleced in he DARPA SensIT vehicle deecion experimens [8]. 75 WINS NG. nodes are deployed o deec vehicles driving hrough several inersecing roads in 9 Palms, CA. Each WINS NG. node is equipped wih hree sensing modaliies: acousic (microphone), seismic (geophone) and infrared (polarized IR sensor). All nodes are deployed in an area of approximaely 9 3m. In our evaluaion, we use acousic and seismic readings from 3 nodes in he daase o generae covariance marices. BWSN daase is acquired by running simulaions on a 9-node sensor nework used in Bale of he Waer Sensor Neworks (BWSN) [9]. We use he bwsn-uiliies [] program o simulae random injecion evens o his nework for a duraion of 96 hours and use he generaed even deecion daa o calculae he covariance marices. We use wo even injecion sraegies o build wo ses of daa as wo applicaions. For each daase, we can calculae he covariance marices based on he sensor readings. The Packe Recepion Raio (PRR) of each link is included in he Inel daase. We generae PRR for DARPAR and BWSN daases based on locaion informaion of sensors following he way proposed in []. We hen generae differen nework opologies by assigning differen PRR bounds. Only links wih PRR higher han he PRR bound is used for communicaion. In our simulaions, we repea Algorihm in Secion IV imes for each nework opology. Because he number of applicaions is a mos 3 in our simulaions, we employ naive enumeraion o solve he mulidimensional knapsack problem (5) on each sensor node. As we proved in Theorem 6, SG will erminae a a pure sraegy Nash equilibrium and he approximaion bound is no less han. We implemene our SG algorihm in Malab. All resuls are gahered on a Macbook Pro machine wih CPU frequency a.4ghz and 4GB memory. Figure analyzes he behavior of SG. Here we define covariance cover raio as he raio beween covariance cover achieved by he algorihm and he maximum covariance cover in he nework. Since searching an opimal soluion is oo compuaional expensive, o assess he ighness of our bound, we compare our soluion wih he maximum covariance cover, i.e., he sum of all edge weighs and node weighs in he nework. Figure (a) shows ha he covariance cover of our

8 Covariance Cover Raio Inel DARPA BWSN Number of Rounds Inel DARPA BWSN Number of Messages Inel DARPA BWSN (a) Covariance Cover Raio (b) Number of Rounds (c) Number of messages per node Fig.. Game Behavior Analysis SG VR FRG VR SG CC FRG CC (a) Inel daase SG VR FRG VR SG CC FRG CC (b) DARPA daase 5 SG VR FRG VR SG CC FRG CC (c) BWSN daase Fig.. Performance Analysis 3 5 Variance Reducion Covariance Cover 4 3 Variance Reducion Covariance Cover 5 Variance Reducion Covariance Cover Number of allocaed nodes (a) Inel daase Number of allocaed nodes (b) DARPA daase Number of allocaed nodes (c) BWSN daase Fig. 3. Comparison beween VR and CC soluion is consisenly no less han half of he maximum covariance cover, which means our soluion is no less han half of he opimal soluion. Resuls in Figure (a) indicaes he ighness of our bound. Figure (b) shows he maximum number of rounds for SG o converge is below 6 across all cases in all hree real-world daa ses. The communicaion cos of our algorihm is evaluaed in Figure (c). I shows he average number of messages sen per node. As each sensor node has more neighbors wih a lower PRR hreshold, he average number of messages sen by each node increases from o. Our resuls show ha he communicaion cos required by SG in erms of number of packes is moderae. Figure compares he performance of SG wih FRG [9] in erms of variance reducion (VR) and covariance cover (CC). The variance reducion delivered by SG is always above 98% of ha achieved by FRG in all hree daases. The covariance cover of SG is consisenly higher han FRG. For BWSN daase, he difference beween differen mehods is wihin, which makes four curves difficul o ell. This resul indicaes he decenralized approach employed by SG is compeiive wih he cenralized soluion in erms of. Figure 3 invesigaes he correlaion beween covariance cover and variance reducion. We increase he number of allocaed sensor nodes ϱ under same PRR hreshold.5. I is difficul o disinguish VR and CC in BWSN daase, because he difference of hem is wihin. Resuls in he oher wo daases show variance reducion and covariance cover are very close when ϱ is less han n/, where n is oal number of sensors. This is because when ϱ is small, allocaed nodes are no neighbors of each oher, which coincides wih our assumpion in Theorem. The difference increases when ϱ exceeds n/, bu a higher covariance cover is always associaed wih a higher variance reducion. This resul shows ha covariance cover can be used as an effecive proxy o opimize he variance reducion of a node allocaion.

9 Variance Reducion SG FRG Nework Size Execuion Time (s).5.5 SG FRG Nework Size No. of Rounds No. of Messages Nework Size (a) Variance Reducion (b) Execuion Time (c) Cos Analysis Fig. 4. Scalabiliy Analysis We evaluae he scalabiliy of our algorihm by selecing differen subses of sensor nodes from he BWSN daase. Figure 4(a) shows SG is highly compeiive agains FRG in erms of variance reducion. Noe he difference beween SG and FRG is consisenly wihin, hence he SG and FRG curves are almos indisinguishable here. The execuion imes of SG and FRG for varying size of neworks are compared in Figure 4(b). Since he SG is a disribued algorihm, we show he average execuion ime per node. For FRG, we show is overall execuion ime because i is a cenralized algorihm. Our resuls show ha SG remains fas as he number of nodes increases, wih he run ime remaining below. second. While he Macbook machine used in our simulaion is more powerful han ypical sensor nodes, he shor execuion imes neverheless indicaes ha SG is pracical on sensors. More imporanly, he soluion scales effecively wih nework size. In comparison, he run ime of FRG increases significanly as he number of nodes increases. I is imporan o noe ha SG brings significan advanages han a cenralized algorihm in several imporan ways. I does no incur he communicaion overhead for collecing he opology informaion of he enire nework. Furhermore, i is robus agains nework disconnecion as i does no depend on a single base saion. In Figure 4(c), we analyze he number of rounds and communicaion cos of SG. Boh he number of rounds and messages per node increase moderaely as he nework size increases. The number of rounds remains wihin, indicaing he scalabiliy of our decenralized algorihm. VII. CONCLUSIONS This paper presens a disribued game-heoreic approach o applicaion allocaion in shared sensor neworks. We firs ransform he opimal applicaion allocaion problem o a submodular game and hen develop a decenralized algorihm ha only employs localized ineracions among neighboring nodes. We prove ha he nework can converge o pure sraegy Nash equilibrium wih a approximaion bound. Simulaions based on hree real-world daases demonsrae ha our algorihm is compeiive agains a sae-of-he-ar cenralized algorihm while scaling effecively wih nework size. VIII. ACKNOWLEDGMENT This work is suppored by NSF grans CNS-77 (NeTS), CNS (CPS), CNS-4455 (NeTS) and Microsof Research New Faculy Fellowship. REFERENCES [] Ciysense, hp:// [] hp://research.cens.ucla.edu/areas/5/nims/. [3] hp:// wireless-modules.hml. [4] F. Bian, D. Kempe, and R. Govindan, Uiliy-based sensor selecion, in IPSN, 6. [5] C. Guesrin, A. Krause, and A. P. Singh, Near-opimal sensor placemens in gaussian processes, in ICML, 5. [6] A. Krause, J. Leskovec, C. Guesrin, J. VanBriesen, and C. Falousos, Efficien sensor placemen opimizaion for securing large waer disribuion neworks, Journal of Waer Resources Planning and Managemen, vol. 34, no. 6, pp , 8. [7] A. Krause, B. McMahan, C. Guesrin, and A. Gupa, Robus submodular observaion selecion, JMLR, vol. 9, pp. 76 8, Dec 8. [8] A. Kulik, H. Shachnai, and T. Tamir, Maximizing submodular se funcions subjec o muliple linear consrains, in SODA, 9. [9] Y. Xu, A. Saifullah, Y. Chen, C. Lu, and S. Bhaacharya, Near opimal muli-applicaion allocaion in shared sensor neworks, in MobiHoc,. [] A. Nedic and A. Ozdaglar, Disribued subgradien mehods for muliagen opimizaion, IEEE Transacions on Auomaic Conrol, vol. 54, no., pp. 48 6, Jan. 9. [] M. Rabba and R. Nowak, Disribued opimizaion in sensor neworks, in IPSN, 4. [] A. Vea, Nash equilibria in compeiive socieies, wih applicaions o faciliy locaion, raffic rouing and aucions, in FOCS,. [3] R. Johari and J. N. Tsisiklis, Efficiency loss in a nework resource allocaion game, Journal Mahemaics of Operaions Research, vol. 9, no. 3, pp , 4. [4] O. Ben-zwi and A. Ronen, The local and global price of anarchy of graphical games, in SAGT, 8. [5] S. Bhaacharya, A. Saifullah, C. Lu, and G. C. Roman, Muliapplicaion deploymen in shared sensor neworks based on qualiy of monioring, in RATS,. [6] H. Kellerer, U. Pferschy, and D. Pisinger, Knapsack problems. Springer, 4. [7] hp://db.csail.mi.edu/labdaa/labdaa.hml. [8] M. F. Duare and Y. H. Hu, Vehicle classificaion in disribued sensor neworks, Journal of Parallel and Disribued Compuing, vol. 64, no. 7, pp , Jul. 4. [9] A. Osfeld and e al., The Bale of he Waer Sensor Neworks (BWSN): A Design Challenge for Engineers and Algorihms, Journal of Waer Resources Planning and Managemen, vol. 34, no. 6, pp , 8. [] hp:// wsp/ abou/ bwsn/. [] M. Zuniga and B. Krishnamachari, Analyzing he ransiional region in low power wireless links, in SECON, 4.