On Selfishness, Local Information, and Network Optimality: A Topology Control Example

Transcription

1 On Selfshness, Local Informaton, and Network Optmalty: A Topology Control Example Ramakant S. Komal, Allen B. MacKenze, and Petr Mähönen Department of Wreless Networks, RWTH Aachen Unversty, Aachen Vrgna Polytechnc Insttute and State Unversty, Blacksburg, Vrgna Abstract Topology control of ad hoc and mesh networks specfes how to assgn per-node transmsson parameters (such as power level, frequency etc.) so as to acheve energy effcency, whle mantanng certan desrable propertes such as connectvty. In autonomous networks, nodes may act n ther selfnterest and mprove ther performance, perhaps at the expense of other nodes, or even the overall network s, performance. Besdes, nodes must also contend wth lmted nformaton about the network operatng state durng ther decson-makng. We analyze the above problem usng non-cooperatve game theory and quantfy the mpact of partal network state knowledge that nodes possess on the network optmalty. We develop a local topology control algorthm that uses the dea of mantanng connectvty of 1-hop neghborhoods. Ths algorthm s frst shown to converge and be stable. We then examne the trade-off between network performance (energy effcency) and the cost of havng knowledge (by exchangng control messages): more nformaton exchange makes the nodes more network-aware, and hence leads to more effcent networks, but exchange of control nformaton tself s costly. Takng the cost of obtanng knowledge nto account, we observe that when nodes can operate along the contnuum of knowledge, from 1-hop to omnscence, the network consumes least energy when nodes have sgnfcantly less connectvty nformaton. I. INTRODUCTION Ad hoc networks are touted as autonomous, self-confgurng and decentralzed, wth an ntrnsc ablty to adapt and be reslent. These networks also embrace an open network phlosophy by dstrbutng network control to autonomous, dstrbuted, and ndependent nodes. Such a radcal networkng paradgm can foster aberrant behavor among nodes. By not complyng to protocols, nodes can act selfshly, sometmes even malcously, and game the system. Improvng wreless network performance, on the other hand, often requres crosslayer optmzaton schemes that rely on cooperaton and access to nformaton. 1 Network engneers may then be faced wth a dlemma: How can a network, whose performance reles on node cooperaton, come to terms wth (a) selfsh nodes that are tryng to optmze ther own objectves, and (b) the lack of suffcent nformaton, often ntegral to most cooperatve algorthms? In other words, how can cooperaton arse when nodes are actng selfshly, or when they do not have enough Ths materal s based upon work supported n part by the Natonal Scence Foundaton under Grant No , the European Commsson (ARAGORN), and the German Research Foundaton (DFG) through UMIC research center. 1 In fact, cooperaton can not exst wthout exchange of nformaton and vce-versa. nformaton, nformaton that can otherwse foster cooperaton. Gven the dynamcs of ad hoc networks, t seems natural to assume that sustanng such networks requres some degree of cooperaton and nformaton exchange amongst the nodes explctly (e.g., by exchangng routng tables), through publc or prvate sgnalng mechansm, or passvely (e.g., by overhearng). The focus of ths work s on nformaton-constraned ad hoc networks contanng nodes that are nherently selfsh but lack the knowledge needed to make optmal decsons. To assess the effects of selfshness and partal knowledge (taken to mean ncomplete network state nformaton that nodes possess) on network performance, we examne the dstrbuted topology control problem. Topology Control (TC) s a technque to enhance end-to-end network performance objectves such as energy effcency or throughput whle takng network connectvty nto consderaton. We develop a gametheoretc model for connectvty and energy effcency, and assess ts effcacy n constructng energy-effcent networks. Our game-theoretc model admts the study of selfshness and gnorance 2 jontly: The effects of partal knowledge and selfshness are captured by the end-to-end connectvty constrant and energy mnmzaton crteron, respectvely. Our model of partal knowledge s closely related to the concept of zones n Zone Routng Protocol (ZRP) and Independent Zone Routng (IZR) [1], [2]. These hybrd routng protocol schemes am at reducng the amount of control traffc and mprovng delay performance by explotng routng structures and elmnatng overhead traffc redundances. Our work s also naturally relevant to Cogntve Rados (CRs) and Cogntve Networks (CNs) that are expected to be able to handle the lack of nformaton when makng complex optmzaton decsons [3], [4]. In fact, gven the exorbtant costs of sgnalng overhead assocated wth achevng cooperaton n wreless networks, balancng the costs and benefts of nformaton access s a desgn decson task for CNs [5], [6], [7]. Game-theoretc analyss of adaptve power control technques for TC can be found n [8], [9], [10] that take a utltybased approach smlar to our work. Several cooperatonbased TC algorthms have been proposed to create effcent topologes; for a survey, see [11]. Gven the prohbtve costs assocated wth global nformaton access, lterature on TC 2 The terms gnorance, partal knowledge and local knowledge are used nterchangeably; lkewse, the terms nformaton and knowledge mean the same n the context of ths paper.

2 underscores the need for dstrbuted solutons to the problem. Most studes cope wth the lack of complete nformaton ndrectly, ether by developng local algorthms such as [12], [13] or by adoptng probablstc models that are often argued as beng robust to nformaton constrants, e.g. [14]. Despte these efforts, a thorough evaluaton and analyss of the mpact of partal nformaton on network optmalty s stll mssng from most TC work n lterature. The remander of the paper s organzed as follows: We begn by dscussng the system model and assumptons n Secton II; here we also provde a detaled descrpton of the topology control algorthm. We then analyze the algorthm and ts outcomes n Secton III. We present the smulaton results n Secton IV and conclude n Secton V. A. System Model II. FRAMEWORK AND ASSUMPTIONS We model the network topology by a graph G =(N, E) where N s a set of nodes and E s a set of drected arcs that represent the undrectonal connectons. The set of connectons can be wrtten as: } E = e j p ω(j) (1) where ω(j) s the transmsson power requred to form a connecton from node to node j. We assume that ω s symmetrc,.e. ω(j) =ω(j), j. Thus a connecton exsts f the transmsson power (p ) s no less than the mnmum power requred for to reach j (that depends on the underlyng channel model and condtons, transmtter recever separaton etc.). We dsregard nterference by assumng that there exsts some Medum Access Control (MAC) protocol that regulates channel access and prevents transmsson conflcts. We assume that topologcal connectvty comes from bdrected connectons, whch occur when there exst connectons between nodes n both drectons. The set of b-drected connectons conssts of members of E that have ther reverse also n E. The nduced topology G s sad to be connected f and only f there exsts a b-drected path a collecton of contguous b-drectonal lnks between every node par, j N. B. Game-Theoretc Model We formally descrbe the topology control process as a normal form game. Indvdual nodes form the player set, N = 1, 2,..., n} of the game. Each node can autonomously set ts transmt power level p A = [0,p max ]. The ndvdual power levels can be collected nto a power vector p =(p 1,p 2,..., p n ), and the acton space, A, for the game s gven by the cartesan product A = n =1 A. The power vector nduces a topology G that s a collecton of feasble lnks defned by (1). When every node transmts at ts maxmum power level p max, the resultng topology s denoted by G max. Each node perceves a tradeoff between the beneft t derves from a connected topology G, and the cost t ncurs n establshng G. These tradeoffs are captured by a utlty functon that models the preferences of ndvdual nodes. Our utlty functon s based on the dea of a k-hop neghborhood (also referred to as k-neghborhood elsewhere n ths paper). The k-hop neghborhood n a topology s defned as the set of nodes reachable wthn k hops va b-drectonal connectons. The members of each successve k-hop neghbor of can be descrbed recursvely: } k =0 N k = N k 1 j e jl,e lj E, l N k 1 },j k>0 (2) We suppose that nodes possess k-hop knowledge, meanng that they are aware of ther k-hop neghborhood. Specfcally, each node knows how to reach (and what route to take to) all destnatons that are at most k hops away and no more. Defnng k-hop knowledge n ths manner lmts the sgnalng overhead assocated wth route request control messages that are broadcast perodcally to obtan such knowledge. In addton, the assumpton also allows k to be an expermental parameter that can be tuned to study the mpact of partal knowledge on effcent network desgn. We pont out that the concept of k-hop neghborhood s not new and s smlar to the concept of zones (where k s the equvalent of zone radus) n hybrd routng protocols such as ZRP and IZR proposed to lmt the overhead control traffc. Wth ths noton of k-neghborhood, the objectve functon for each node can then be expressed as: ū (p) =M f (k) (p) p (3) where f (k) s the number of nodes n the orgnal k-hop neghborhood of node n G max to whch there exsts a path, and M p max s a scalar beneft multpler. Notce that wth each power adaptaton, the k-hop neghborhood of nodes keep changng. We represent the orgnal k-hop neghborhood of node n G max by Ñ k. The objectve functon captures the fact that nodes would prefer to mantan connectvty wth ther orgnal k-hop neghbors over a power decrease, but get more reward for mantanng such connectvty at lower power levels. (In the next secton, we formally show that the connectvty of G max s always preserved.) Hence, preferences are cast n a lexcographc order, where nodes regard connectvty as more mportant than power consumpton. Network connectvty s a basc requrement n TC as t provdes the means for nodes to communcate wth each other. The beneft term n (3) sgnfes the reachablty of nodes. It mplctly assumes that each node has some traffc for every other node n the network, necesstatng that the underlyng topology be connected. Ths s a reasonable assumpton as traffc load and selecton of destnatons are typcally not avalable durng topology formaton. C. A Dstrbuted Algorthm Under the aforementoned game-theoretc model, each node seeks to selfshly maxmze ts objectve (3) by choosng an approprate power level. Naturally, dfferent selfsh power control strateges wll lead to dfferent outcomes (dfferent topologes). It has been shown n [10], for nstance, that greedy best

3 response strateges lead to socally undesrable outcomes topologes that are neffcent from a network vewpont. On the other hand, when nodes have complete knowledge about network connectvty, a better response strategy proposed n [10] s shown to result n topologes that are sgnfcantly close to the globally optmal ones (those that mnmze the sum power n the network). In real systems, however, nodes are expected to have only a partal and ncomplete vew of the network. Wth these consderatons mproved performance of better response algorthms and the need to contend wth lmted nformaton durng decson-makng we develop a dstrbuted and local algorthm that s functonally smlar to the one proposed n [10]. We propose Routng Asssted Topology Control (RATC) a local algorthm that uses the dea of k-hop knowledge and explots the routng state nformaton to optmze the overhead cost of topology constructon. We study the Nash Equlbrum (NE) topologes that emerge n steady state when nodes employ RATC strategy. RATC s descrbed n Algorthm 1 where each node ntalzes ts transmt power to the maxmum power level p max. Nodes, chosen accordng to some random permutaton 3 π, then determne a power level one step lower than ther prevous level, such that the change mproves ther objectve functon. RATC operates on a fnte set of transmsson powers, elements of whch are denoted by p (m), for node (m s the element ndex). Followng (1), t s suffcent to consder an acton space for each node that corresponds to the set of transmsson levels needed to reach each neghbor (.e. ω(j) s). We defne the search space for each node as the followng set of ordered powers: } Ā = p max = p (0),p (1),p (2),... (4) The power levels n Ā are chosen such that the followng condtons are met: C.1 At most one connecton s dropped n the 1-hop neghborhood of as t adapts ts powers from p (m) to p (m+1). C.2 For a gven k, each node only ensures connectvty wth all ts current 1-hop neghbors, such that they are wthn k hops after each power adaptaton (from p (m) to p (m+1) ). By restrctng the amounts by whch power levels can be reduced (to selfshly mnmze power consumpton), condton C.1 ams to acheve some degree of farness n the steady state power level dstrbuton, and avod the frst mover advantage. Condton C.2 s nvoked to reduce the overhead cost of control messagng necessary to mantan route nformaton. The ratonale behnd C.2 s further explaned n the next paragraph. We argue that t s suffcent for nodes to mantan only a 1-hop neghborhood table contanng the (nexthopid, destid) entres,.e., the next hop node d for each of ther 3 Each node n the network may be assgned a random backoff wthn a fxed wndow. The backoff perods nduce an orderng that represents ths random permutaton. destnaton d entres. Note that wth k-hop knowledge, nodes are aware upto ther k-hop neghborhood: nodes know who ther k-neghbors are, and the d of the next hop node en route to these k-neghbors (ths s provded by a routng algorthm). In order to evaluate ther utltes (3), nodes must ensure that these next hop nodes are stll reachable wthn k hops after each power decrement (the cost of obtanng the reachablty knowledge s accounted for n our analyss). If these 1-hop neghbors are reachable wthn k hops, the reduced power level s retaned for further optmzaton, else the node reverts to ts prevous transmt level. If every node reasons accordng to ths strategy, we clam that all nodes wll preserve connectvty wth all ther orgnal k-neghbors, Ñ k. (We formally prove ths asserton and the correctness of our algorthm n the next secton.) We pont out that whle the overall connectvty wth nodes n Ñ k s preserved, some nodes n the orgnal Ñ k may not be present n the subsequent k-hop neghborhoods that result at reduced power levels. Note that preservng Ñ k s a utlty maxmzaton crteron. In some sense, the aforementoned power adaptaton strategy represents a fath based model: each node adjusts ts power level accordng to (4), ensurng connectvty wth only 1-hop neghbors, n the belef that overall connectvty wth orgnal k-hop neghbors wll always be preserved, gven that other nodes follow the same strategy. Fgure 1 s a smple vsual llustraton of the RATC algorthm. m s j p q m s m updates teraton 1 j p q m m updates s teraton 2 Fg. 1. Illustratng the RATC process: Wth k =2, node m s ntally aware of j, q, p, s} n ts 2-neghborhood. In teraton 1, m can reduce ts power and stll mantan connectvty wth p (n 2 hops), wthout losng connectvty wth s. In teraton 2, m can further reduce ts power wth the knowledge that t can reach q (n 2 hops), thus mantanng connectvty wth p and s (the orgnal 2-neghborhood). In general, though, m may not be aware of the exact routes to p and s, except that they are reachable through q. j p q

4 To summarze, nodes only mantan a lst of 1-hop neghbors and ensure that they are stll connected to these neghbors (n at most k hops) after each power decrement. In each subsequent round, the 1-hop neghborhood decreases (or remans the same). Mantanng only 1-hop connectvty makes RATC truly localzed. Algorthm 1 RATC(G) ˆp 1: m =0 2: ˆp = p (m) Ā N 3: whle ˆp s not a NE do 4: m = m +1 5: for π do 6: choose p = p (m) Ā 7: ˆp = arg max p p,ˆp } ū (p, ˆp ) 8: end for 9: end whle III. GAME-THEORETIC ANALYSIS The acton set (4) specfes how the local TC game mght evolve: each ndvdual node starts by transmttng at ts maxmum power, reasons accordng to a better response process, and chooses the next power level n the set f t observes an mproved utlty. In ths secton we show the correctness of the RATC algorthm. Frst, RATC s shown to converge usng the theory of potental games. 4 Then, Ñ (k) (and hence the overall network connectvty) s shown to be preserved n every round of the algorthm. Theorem 1: The RATC game Γ = < N, Ā, ū } >, where the actons and utltes are specfed by (3) and (4) respectvely, s an Ordnal Potental Game (OPG). The Ordnal Potental Functon (OPF) of the game s gven by V (p) = ū (p). Proof: We prove by applyng the defnton of an OPG. Frst we have, ū = ū (p,p ) ū (q,p ) [ ] = M f (k) (p,p ) f (k) (q,p ) Smlarly, (p q )(5). V = V (p,p ) V (q,p ) [ ] = M f (k) (p,p ) f (k) (q,p ) (p q ) + )} M j f (k) j (p,p ) f (k) j (q,p. Thus, we have V = ū + j N;j j N;j M j f (k) j (p,p ) f (k) j (q,p )}. 4 For a treatse on potental games, we refer to [15] and [16]. (6) Snce f (k) (p) s monotonc and M p,max, t follows from (5) that 0 f p >q and f (k) (p) >f (k) (q,p ); 0 f p <q and f (k) (p) <f (k) (q,p ); ū = < 0 f p >q and f (k) (p) =f (k) (7) (q,p ); > 0 f p <q and f (k) (p) =f (k) (q,p ) The sgn of the second term n (6) s the same as the sgn of ū for the frst two cases of (7). For the last two cases of (7), the second term n (6) s 0 and 0, respectvely. In general, sgn ( V ) = sgn ( ū ) V s an OPF and Γ an OPG. A potental game has two mportant propertes: stablty and convergence. In other words, exstence of a pure strategy NE s assured for a potental game, and any selfsh strategy s bound to converge to a NE. In a selfsh algorthm such as the RATC, selectng reduced power levels that preserves Ñ (k) s the ratonal decson, as shown n the followng proposton. Lemma 2: In every teraton of the RATC game, every node must preserve ts orgnal k-neghborhood Ñ (k). Proof: We prove ths by contradcton. Suppose node reduces ts power level from p to p (k) to reduce Ñ (to say, N (k) ) and ncrease ts utlty. Ths mples that ū (p,p ) = M N (k) p >M Ñ (k) p, where N (k) ( ) < Ñ (k). Ths mples, M Ñ (k) N (k) <p p, an mpossble nequalty, because the term on the LHS s larger than p max and the term on the RHS s smaller than p max. Note that RATC s a local algorthm where nodes only preserve connectvty wth ther 1-hop neghbors from the prevous round. Techncally, nodes do not know whether or not they are even connected to all ther k-neghbors from the prevous round. We, however, clam that t s suffcent for nodes to only check connectvty wth ther prevous 1-hop neghbors; ths automatcally ensures that nodes are connected to all ther prevous k-neghbors as well, thus establshng the correctness of the fath-based RATC algorthm. Theorem 3: If every node makes power selectons accordng to RATC, then Ñ (k) s preserved for all nodes, k >0. Proof: There are two cases to examne: 1) Every node must preserve ts own Ñ (k) ; and 2) No node should reduce Ñ m (k) of any other node m. In the frst case, we show that node makng power selectons accordng to RATC preserves ts Ñ (k) n any gven round. Wthout loss of generalty consder a path S j =(, p,..., j) and let p be the 1-hop neghbor of. Suppose there exsts a k- hop path (k >1) S p that selects after power reducton. Snce the route S pj remans unaffected by s power reducton, the route S j s preserved as well. Note that, n any gven round, loses a drect connecton wth at most one 1-hop neghbor (accordng to C.1). In the second case, we show that the power reducton by does not reduce the for any other node m. If m Ñ (k), then mantans connectvty wth m accordng to Ñ (k) m

5 the precedng case, and hence Ñ m (k) the only way for to lose connectvty wth m, or wth some s preserved. If m/ Ñ (k), member of Ñ m (k), s to dsconnect wth an exstng member of Ñ (k), whch s dsallowed by precedng argument. It follows that topologes preserve Ñ (k), k n every round of the RATC algorthm. Determnng f (k) and whether or not a route exsts to all orgnal k-neghbors may requre global knowledge n the worst case. The prevous theorem, however, establshes that t s not necessary to check for global connectvty to evaluate the ndvdual utltes. The RATC algorthm that we propose actually works wth less than global knowledge. We expect that ths reducton wll lead to sgnfcant savngs n the overhead cost, especally for small k values. Theorem 4: If every node makes power selectons accordng to RATC, then wthn a fnte number of adaptatons, the nodes wll reach a stable state n whch they cease to make further adaptatons. Proof: There are only a fnte number of power levels that can be stepped through (correspondng to the power thresholds ω(j)). From Theorem 3, we see that Ñ (k) s preserved for every node as they decrease ther power levels accordng to (4). Ths also means that nodes wll never have to ncrease ther power as t wll not lead to an ncrease n ther objectve functon. Snce transmsson powers wll only be reduced and there are a fnte number of power levels, the algorthm must eventually stablze. Because there can be at most n(n 1)/2 connectons (n the case of complete networks), the convergence rate of the RATC algorthm s O(n 2 ). Gven the connectvty preservng propertes of the RATC algorthm and the above convergence result, the followng corollary s a straghtforward deducton. Corollary 5: If G max s connected, then RATC converges to a network that s also connected for all k>0. Proof: The proof s an mmedate consequence of Theorems 3 and 4. If all nodes preserve connectvty wth all ther orgnal k-neghbors and the topology s connected to begn wth, then the NE topology s also connected. IV. SIMULATION RESULTS To determne the effcacy of the RATC algorthm, we develop a smulaton consstng of N nodes placed accordng to a unform random dstrbuton wthn a unt square. The power thresholds ω(j) requred to close a lnk between nodes and j were assumed to be equal to d 2 (, j) (we choose a path loss exponent of 2, although our basc conclusons reman the same for other channel attenuaton factors as well), where d s the eucldean dstance metrc. The ntal node transmt power level was chosen usng the formula from [17] (adjustng the value for fnte networks), such that the nduced network was 1-connected wth 85% probablty. We consder only the connected nstances of G n our smulatons (meanng that there exsts a path from every node to every other node n the network). All smulatons are carred out for 1000 dfferent topology scenaros wth nodes randomly placed at dfferent locatons n each case; the performance results are then averaged over all these scenaros. Each node s a selfsh player n the RATC game, selectng power levels accordng to (4) that mproves ts utlty (3). All nodes ntalze ther powers to p max before adaptng ther power selectons. Each node n the network s assgned a random backoff wthn a fxed wndow. The backoff perods nduce an orderng that represents a random permutaton. When the backoff ends, nodes randomly select an acton from ts set (4). When no mprovng acton exsts, nodes revert to ther prevously chosen power level. We frst study the performance benefts of ncreasng the amount of k-hop knowledge avalable to the nodes n the network. As a proxy measurement for the network performance, we measure the data packet energy. The data packet energy s calculated as the amount of energy requred to transmt, va uncast, a data packet from every node to every other node, usng the least-power route between every par of nodes n the network. The power used by the node at each hop along the route n the topology s summed and ths value s totaled for every node par. To convert from power to energy, we multply ths total power by a constant equal to the length of tme for a packet transmsson assumng that all packets are of equal transmsson length. Ths gves us the end-to-end energy effcency performance of the topology under a gven level of k-hop knowledge that nodes possess. Average consumed by a NE topology k hop knowledge Fg. 2. Average data packet energy requred for 50 node network Fgure 2 shows, n an expected sense, the performance varaton of NE topologes wth k-hop knowledge. Wth ncreasng k, nodes possess greater knowledge about connectvty wth nodes that are further away. Ths enables the nodes to reduce ther power levels more aggressvely wthout losng connectvty. Thus, as expected, ncreasng knowledge decreases the packet energy requred for data, meanng that the NE topologes that emerge are more energy effcent. Note that the network energy consumpton decreases, albet slowly, as the amount of knowledge avalable to nodes becomes large. Notce also that the network performance doesn t mprove any

6 further beyond a certan k value because here k approaches the dameter of the network and all nodes possess complete knowledge about the entre network connectvty state. The performance benefts of ncreasng the amount of k- hop knowledge avalable to the nodes n the network are clear. However, knowledge comes at a cost, requrng more overhead transactons to obtan connectvty nformaton as the amount of knowledge ncreases. It thus stands to reason that n many topologes havng less than full knowledge may, n fact, be optmal when one takes nto account the cost-performance tradeoff. To understand the mpact of partal network state knowledge and acqurng such nformaton, we use the same energy metrc used to evaluate the network performance; ths allows for an unbased comparson of the network performance and the cost of acqurng network connectvty nformaton. To measure the cost of buldng a network topology under a gven level of k-hop knowledge, we measure the total packet energy requred for obtanng connectvty nformaton. Ths s determned by calculatng the amount of energy needed to transmt control messages to check f a 1-hop neghbor s wthn k hops after each power decrement. 5 As wth the data measurement, the cost s calculated by utlzng the least power route between two nodes that lose each other from ther 1-hop neghborhood because of a power reducton by one of these two nodes. The power used by all nodes on such a route s summed up. We use the same tme constant as wth the data packets to convert from power to energy. Ths gves us the energy cost assocated wth constructng a topology wth local nformaton. Fgure 3 shows the energy consumed n obtanng the k hop connectvty knowledge. Note that, as expected, the energy consumed ncreases monotoncally wth k. When nodes possess more knowledge, more control packets are generated n the network because of longer routes, and the network consumes more energy. Consder k =1, n whch case the RATC algorthm settles n an equlbrum state mmedately n one teraton. The control cost s the energy spent n broadcastng beacon messages to ensure that all 1- hop neghbors are stll reachable. For k =2, RATC typcally takes more than one teraton to converge, gven condton C.1. In the frst teraton the energy spent s the same as n the case of k =1. Addtonal energy s spent n the subsequent teratons. Smlar argument holds for k 3 cases as well. Ths establshes the monotonc (non-decreasng) trend n the 5 The cost of obtanng the reachablty knowledge that s accounted for n our analyss may be mplemented accordng to some on-demand routng algorthm: Suppose a source node s reduces ts power and loses a drect connecton wth ts currect 1-hop neghbor (x). Then x sends out a broadcast message to all ts 1-hop neghbors wth a TTL value set to k hops. Ths broadcast message eventually reaches s f t s wthn k hops of x, otherwse the tmer expres and the message s dscarded. In ths way s can determne whether x s wthn k hops or not. Because the actual cost of obtanng connectvty knowledge depends on the underlyng routng algorthm, we assume there exsts some optmzed routng algorthm wth an oracle knowledge about mnmum energy paths. The mnmum cost of obtanng the knowledge about whether or a not 1-hop neghbors are wthn k hops s the sum of of energy requred to transmt, va uncast, a control packet va least power routes over all such s and x. Note that ths smplfcaton, n some sense, provdes a lower bound on the control energy, and wll not change our man concluson as the true knowledge cost wll also follow a smlar trend. knowledge cost observed n Fgure 3. Note that n our cost computaton we do not consder the overhead cost nvolved n knowng that a node s stll connected to some prevous k- neghbor. Gven the localzed nature of the RATC algorthm, the source assumes that t s connected to those k-neghbors (recall that ths s shown to be true analytcally n Theorem 3). We only count the transactons nvolved n knowng that 1-hop neghbors are no more than k hops away, for each node, and sum up all such transactons over the entre network. Ths gves us the cost assocated wth the network (n buldng t). Average energy consumed to obtan knowledge k hop knowledge Fg. 3. Average control packet energy requred for 50 node network Fgure 4 shows the total packet energy requred for data and knowledge packets, f data packets are sent at the same frequency as knowledge requests. There s a sweet spot for energy at 3-hop knowledge, where the sum total packet energy s lower than at full knowledge. Ths s the pont where the total energy cost s mnmzed. The performance mprovement beyond ths pont s not sgnfcant, whereas the assocated energy cost contnues to rse. Takng cost-performance tradeoff nto account, t s network-optmal for the nodes to operate wth only a 3-hop knowledge as opposed to havng entre network connectvty nformaton. Smlar conclusons can be derved for other topologes as well. We also noted that ths optmal k value remans same across varous network szes, wth the densty kept fxed. Ths ndcates that our algorthm scales well wth network sze. The above example s for a low rato of data to knowledge requests (specfcally, for a 1:1 rato). Assumng the amount of data stays constant, as a network becomes more moble, the number of requests requred to ensure connectvty of 1-hop neghbors (wth k hop knowledge) ncreases proportonally to the data. Fgure 5 shows the relatve dfference between the mnmum total packet energy (at 3-hop partal knowledge) and the global knowledge total packet energy for dfferent ratos of data transmts to knowledge requests. As expected, when the network s relatvely stable, and the rato of data to control s hgh, havng global knowledge gves the best performance because of the domnatng data energy term.

7 Average total energy consumed for data and knowledge k hop knowledge Fg. 4. Average total packet energy requred for 50 node network When the network s dynamc, and the rato of data to control s low, havng partal knowledge gves a lower total packet energy because the cost of constantly obtanng connectvty knowledge domnates. In some sense, ths means that dynamc network condtons (or moblty) actually helps n nformatonconstraned networks, where nodes possess only local knowledge (low k hop connectvty nformaton). Percentage energy saved by usng 3-hop knowledge 35% 30% 25% 20% 15% 10% 5% 0-5% Rato of data to knowledge packets Fg. 5. Percent addtonal total packet energy requred under global knowledge as compared to mnmum total packet energy under partal knowledge for 50 node network V. CONCLUSIONS The performance of a mult-hop network s dependent on, among other factors, nodes awareness about the current network operatng condton when determnng ther transmsson parameters accordng to some dstrbuted optmzaton algorthm. Dependng on the problem at hand, sometmes havng more knowledge llumnates better solutons, whle other tmes t may just add redundancy and overhead to the system. Regardless of the network beneft these partalknowledge solutons provde, there s always a network cost to acqurng, communcatng, and mantanng knowledge. Both of these factors must be taken nto account n determnng how much nformaton the nodes actually need. To study the mpact of partal nformaton on network optmalty, we have studed the TC problem of mnmzng energy consumpton whle mantanng network connectvty. We cast the problem as a dstrbuted non-cooperatve game and developed a local RATC algorthm to cope wth lmted network connectvty nformaton that nodes possess. We have shown the exstence of a stable state for the game and evaluated the energy effcency performance of the topologes that emerged under dfferent levels of k hop knowledge. Havng full network state knowledge naturally leads to energy-effcent topologes when takng performance alone nto consderaton. When takng the cost of acqurng and mantanng knowledge about connectvty nto account, we have shown that havng partal knowledge (3 hop knowledge n a 50 node network) as opposed to full knowledge leads to optmal topologes. These results on the mpact of partal knowledge begs a thorough analytcal understandng of nformaton-constraned networks that mght eventually lead to development of robust dstrbuted protocols. REFERENCES [1] Z. J. Haas and M. R. Pearlman, The performance of query control schemes for the zone routng protocol, IEEE/ACM Trans. Netw., vol. 9, no. 4, pp , [2] P. Samar, M. R. Pearlman, and Z. J. Haas, Independent zone routng: an adaptve hybrd routng framework for ad hoc wreless networks, IEEE/ACM Transactons on Networkng, vol. 12, pp , Aug [3] R. W. Thomas, L. A. DaSlva, and A. B. Mackenze, Cogntve networks, n Proc. of IEEE DySPAN 2005, pp , Nov [4] R. W. Thomas, D. H. Frend, L. A. DaSlva, and A. B. MacKenze, Cogntve Networks: Adaptaton and Learnng to Acheve End-to-end Performance Objectves, IEEE Communcatons Magazne, vol. 44, pp , December [5] R. W. Thomas, L. A. DaSlva, M. V. Marathe, and K. N. Wood, Crtcal desgn decsons for cogntve networks, n Proc. of IEEE ICC 2007, June [6] M. Petrova and P. Mähönen, Evoluton of rado resource management: A case for cogntve resource manager, n Proceedngs of the IEEE Internatonal Conference on Communcatons (ICC), pp , June [7] P. Mähönen and M. Petrova, Mnorty game for cogntve rados: Cooperatng wthout cooperaton, Physcal Communcaton, vol. 1, pp , June [8] P. Sant, S. Edenbenz, and G. Resta, A framework for ncentve compatble topology control n non-cooperatve wreless mult-hop networks, n DIWANS 06: Proceedngs of the 2006 workshop on dependablty ssues n wreless ad hoc networks and sensor networks, pp. 9 18, [9] S. Edenbenz, V. Kumar, and S. Zust, Equlbra n topology control games for ad hoc networks, ACM/Kluwer Moble Networks and Applcatons (MONET), vol. 11, no. 2, pp , [10] R. S. Komal, A. B. MacKenze, and R. P. Glles, Effect of selfsh node behavor on effcent topology desgn, IEEE Transactons on Moble Computng (TMC), vol. 7, pp , [11] P. Sant, Topology control n wreless ad hoc and sensor networks, ACM Computng Surveys (CSUR), vol. 37, pp , March 2005.

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