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Censorng for Bayesan Cooperatve Postonng n Dense Wreless Networks Kallol Das, Student Member, IEEE, Henk Wymeersch, Member, IEEE Abstract Cooperatve postonng s a promsng soluton for locaton-enabled technologes n GPS-challenged envronments. However, t suffers from hgh computatonal complexty and ncreased network traffc, compared to tradtonal postonng approaches. The computatonal complexty s related to the number of lnks consdered durng nformaton fuson. The network traffc s dependent on how often devces share postonal nformaton wth neghbors. For practcal mplementaton of cooperatve postonng, a low-complexty algorthm wth reduced packet broadcasts s thus necessary. Our work s bult on the nsght that for precse postonng, not all the ncomng nformaton from neghborng devces s requred, or even useful. We show that blockng selected broadcasts transmt censorng and dscardng selected ncomng nformaton receve censorng based on a Cramér-Rao bound crteron, leads to an algorthm wth reduced complexty and traffc, wthout sgnfcantly affectng accuracy and latency. Index Terms Indoor postonng, lnk selecton, cooperatve postonng, dstrbuted wreless localzaton, Cramér Rao bound, censorng. I. INTRODUCTION POSITIONAL nformaton s consdered to be of great mportance n many applcatons, such as navgaton [], search-and-rescue operatons [2], dsaster management [3], sensor networks [4], supply chan montorng [5], and traffc control [6]. Focusng on range-based systems, dfferent technques are currently avalable for postonng, whch can be classfed nto two maor categores: non-cooperatve and cooperatve [7]. In a non-cooperatve settng, devces rely on dstance estmates wth reference nodes, whereas n a cooperatve settng, devces addtonally use dstance estmates between each other. These addtonal measurements can enable postonng n GPS-challenged envronments, such as ndoors or n urban canyons. Dependng on the use of pror nformaton, cooperatve postonng algorthms can be further dvded nto two categores: non-bayesan and Bayesan. In non-bayesan methods, devces exchange poston estmates [7], whereas n Bayesan methods, devces exchange full statstcal nformaton [8]. Whle cooperaton leads to mproved performance, t also results n a hgh computatonal complexty per devce, due to the addtonal nformaton from neghborng devces that Kallol Das s wth the Pervasve Systems Group, Unversty of Twente, Enschede, The Netherlands Emal: k.das@utwente.nl. Henk Wymeersch s wth the Department of Sgnals and Systems, Chalmers Unversty of Technology, Gothenburg, Sweden Emal: henkw@chalmers.se. Ths research was supported, n part, by the European Research Councl, under Grant No. 25848 COOPNET, and the Swedsh Research Councl, under Grant No. 200-5889. needs to be processed and fused. Moreover, cooperatng devces broadcast ther postonal nformaton pont estmates or dstrbutons, leadng to ncreased network traffc and packet loss. The mpact of packet loss on postonng performance was consdered n [9], [0], showng severe degradatons. These drawbacks make cooperatve postonng algorthms challengng to mplement n practce. When more than the mnmum number of reference nodes for postonng s avalable to a gven devce, some form of lnk selecton can be appled [] [6]. Such lnk selecton can be seen as nformaton censorng, prevously appled n decentralzed detecton for sensor networks [7], [8]. For postonng, the use of the closest reference nodes as a censorng crteron was proposed n []. The closest reference nodes may not be the most nformatve for postonng as the geometrc confguraton also affects the postonng performance. Ths problem has been partally addressed n [2], [3], where the Cramér-Rao bound CRB was used to choose the best reference nodes. In [5], [6] geometrc dluton of precson GDOP was appled to select the best four satelltes for a GPS recever. None of the methods above are desgned for cooperatve postonng. Recently, [9] consdered non- Bayesan cooperatve postonng and proposed to use the neghbors wth the hghest receved sgnal strength to reduce complexty and energy consumpton n sensor nodes. In [20], we have shown that n non-bayesan cooperatve postonng, complexty and traffc can be reduced smultaneously, wthout degradng postonng performance, by usng a CRB-based crteron. Ths s acheved by blockng the broadcasts of the nodes that do not have relable estmates transmt censorng and selectng the most nformatve lnks after recevng sgnals from neghbors receve censorng. In ths paper, we extend transmt and receve censorng to Bayesan cooperatve postonng, where the nodes share full statstcal postonal nformaton nstead of pont estmates. Our man contrbutons are as follows: We propose a smple, yet effectve censorng crteron based on the modfed Bayesan CRB n conuncton wth a smple message approxmaton; We show that the complexty of Bayesan cooperatve postonng can be reduced sgnfcantly, by applyng receve censorng; We show that network traffc can be reduced to some extent when devces block the broadcast of unrelable nformaton, by applyng neghbor-agnostc transmt censorng; We show that the network traffc can be reduced sgnfcantly when devces block the broadcast of nformaton
2 that wll be gnored by neghbors, by applyng neghboraware transmt censorng; We propose a combned censorng scheme that leads to reduced complexty and reduced traffc, wthout sgnfcantly affectng the postonng performance or the latency. The remander of ths paper s arranged as follows. In Secton II, we descrbe our model and assumptons. In Secton III the censorng crteron s ntroduced, and then appled to develop three censorng schemes. Results from smulatons are dscussed n Secton IV. Fnally, we present our conclusons n Secton V. A. System Model II. PROBLEM FORMULATION We consder a wreless network comprsng two classes of nodes: agents and anchors. Agents have unknown postons, whle anchor have a pror known postons. The goal of the agents s to determne ther postons, based on the postons of the anchors and dstances estmates between nodes. We denote by x the poston of node and by S the ndces of nodes from whch node can receve sgnals. Through a rangng protocol e.g., tme of arrval TOA or receved sgnal strength RSS wth node S, node can estmate the dstance ˆd = x x + n, where n s the rangng nose. For smplcty, as n [8], we assume n N 0, σ 2. Our model assumes all nodes are statc, but our fndngs can be extended to a moble scenaro where nodes move n dscrete tme slots. B. Drawbacks of Cooperatve Postonng Dfferent algorthms for cooperatve postonng have been proposed see [7], [8] and references theren. In ths paper, we wll consder the sum-product algorthm over a wreless network SPAWN from [8], as t offers excellent performance wth low latency. In SPAWN, every agent has an assocated a pror dstrbuton, b 0 x. Statstcal nformaton s exchanged and computed teratvely through messages, correspondng to dstrbutons of two- or three-dmensonal contnuous random varables. At every teraton k, every agent updates ts dstrbuton, wrtten as b k x, named the belef. SPAWN s summarzed n Algorthm, for a agent at teraton k. Ths algorthm s executed n parallel by every agent n the network untl the belefs have converged. Intally, the belefs b 0 x are set to unform dstrbutons whch are not broadcast for the agents and delta Drac dstrbutons for the anchors. In Algorthm, lnes 2 and 5 are not part of standard SPAWN, but form the focus of ths paper. The messages and belefs n SPAWN are dstrbutons of multdmensonal random varables. Exact representaton of these dstrbutons s generally mpossble, so one must resort to non-parametrc [2] or parametrc [22] representatons. Whle the representaton has a drect mpact on the complexty of SPAWN, we wll not make any assumptons on the message representaton. As a performance example, we have smulated a 00 meter 00 meter envronment wth 00 agents havng 20 meter Algorthm SPAWN teraton k, agent. : receve b k X from neghbors S 2: select the set S k of most nformatve lnks through receve censorng 3: convert b k X to a dstrbuton over CCDFerror m X x 4: compute new belef b k x b 0 x p k ˆd x,x b X x dx S k 5: decde f transmt censored 6: broadcast b k f not censored 0 0 0 0 2 Fg.. m X x 0 0.5.5 2 2.5 3 3.5 4 4.5 5 error [m] teraton teraton 2 teraton 3 Postonng performance of SPAWN at dfferent teratons. communcaton range and 3 systematcally placed anchors [8, see Fgure 3], wth a rangng nose varance of σ 2 = 0 cm 2. The postonng performance of SPAWN n terms of the complementary cumulatve dstrbuton functon CCDF of the postonng error at dfferent teratons from top to bottom s shown n Fgure. Observe that after 5 teratons, 99% of the agents have less than meter postonng error. The remanng % agents could not satsfactorly converge due to ther bad geometrcal placement or lmted connectvty. Despte the fast convergence and excellent postonng performance, SPAWN suffers from two mportant drawbacks. Frst of all, the complexty of SPAWN per agent to be detaled n Secton IV-B grows lnearly wth the number of neghbors. In our example, the average number of lnks per agent s roughly 3.7, whereas n a non-cooperatve envronment wth the same communcaton range, ths number n only.5. Hence, the complexty s almost ten tmes larger. Secondly, at every teraton of SPAWN, every agent broadcasts a packet, contanng ts locaton nformaton. Ths results n a large amount of network traffc.
3 A B Rx Censorng 2 C 3 Tx Censorng Fg. 2. Transmt and receve censorng schemes n a cooperatve network, wth 3 agents, 2, and 3 and 3 anchors A, B, and C. A. Concept III. CENSORING We wll descrbe the ntended censorng schemes by consderng a small example wth three agents and three anchors, depcted n Fgure 2. Agent 3 s connected to only one anchor, so ntally t has lmted knowledge about ts poston. Hence, ths agent can only provde lmted nformaton to ts neghbors. In turn, ths mples that f ths agent blocks the broadcast of ts postonal nformaton, the overall performance wll not be greatly affected. We defne ths blockng as transmt censorng TxC. Agent 2 s connected to two anchors, whch gves t a poston ambguty. Its nformaton may be useful for other agents. So agent 2 should broadcast ts postonal nformaton. Agent can get nformaton from three anchors and also from agent 2. By gnorng the nformaton from agent 2, ts postonng accuracy may be relatvely unaffected. We defne ths gnorng as receve censorng RxC. B. Censorng Crteron The Modfed Bayesan Cramér-Rao Bound: Transmt and receve censorng as ntuted n the prevous secton, requre a rgorous crteron based on whch agents decde whether or not to censor. Ths crteron should reflect the qualty of the rangng; the local geometry of the agents and ts neghbors; the uncertanty of the agent s poston, and the uncertanty of the neghbors postons. In addton, the crteron should allow fast computaton. One crteron that satsfes these condtons s the modfed Bayesan Cramér- Rao bound MBCRB [23], defned as follows. Assume that both x, the poston of the agent n queston, and {x } S, the postons of the neghbors of agent, are random varables wth correspondng dstrbutons p X x, S {}, then the so-called modfed Bayesan Fsher nformaton matrx As we wll see later, these dstrbutons wll be smple approxmatons to the belefs b k X x, computed n SPAWN. MBFIM s defned as F = 2 log p ˆd x,x E n,x,x x 2 S }{{} =F M, { 2 } log p X x E x x 2. }{{} =F P, The expectaton n occurs over the rangng nose and the nodes postons. The MBFIM can be broken up nto a term related to measurements F M, and a term related to a pror nformaton F P,. When p X x s a Gaussan dstrbuton wth covarance matrx Σ, then F P, = Σ. Assumng Gaussan rangng nose, the expectaton over the rangng nose can be carred out analytcally [24], leadng to F M, = S σ 2 { x x E x,x x x x x T x x }. 2 The expectaton over x and x s generally dffcult to perform analytcally, so we resort to Monte Carlo ntegraton. Assumng we can draw N samples {x n } N n= from p X, S {}, we fnd that F M, = S σ 2 N S σ 2 p X x p X x x x x x N x n x n x n n=0 x n x n Fnally, the MBCRB can be calculated as x n x x T x x dx dx x n x n T. 3 MBCRB = trace F. 4 Ths MBCRB s also defned when S =,.e., even when there are no neghbors for the update, or when there are no measurements. In ths case, MBCRB = traceσ. Incdentally, we note that when p X x s unform, and p X x, S are delta Drac dstrbutons, 4 reverts to the censorng crteron consdered n [20]. 2 Message Approxmaton for Censorng: In order to be able to compute the MBCRB effcently, the dstrbutons p X x should not be too complex. On the other hand, the true belefs b k x can have many dfferent shapes. For our purpose, the detals of the shape of b k x are not so mportant, but rather we wsh to capture how concentrated the dstrbuton s, and the poston of the centers of mass. A smple Gaussan approxmaton s not suffcent as t cannot capture the common case when an agent can communcate wth two anchors, leadng to a bmodal belef wth two hghly concentrated components. Hence, we propose to approxmate the belefs wth a mxture of two Gaussans: we frst determne the number of components N k {, 2} of the belef b k x of agent x at teraton k. For every component, we then determne the mean µ k, and µk 2, and the covarance
4 Algorthm 2 Receve censorng for SPAWN teraton k, agent. : receve b k X x from neghbors, S 2: f N k 3: f trace = then < γ RX then Σ k 4: set S k = 5: else 6: select L neghbors from S : see Algorthm 3 7: end f 8: else 9: remove ambguty n b k X x 0: goto lne 3 : end f 2: use S k for update matrx Σ k, and Σ k 2,. For smplcty and robustness, we further only consder the covarance matrx wth the largest trace: Σ k = argmax Σ nσ o trace Σ. Fnally, we k,,σk 2, approxmate all belefs at teraton k by a mxture of two Gaussans as p k x b k x, where p k x = 2 N µ k,,σk + 2 N µ k 2,,Σk. 5 When N k =, we have that µ k, = µ k 2,. We note that ths approxmaton s used only wthn the censorng methods, whle the messages computed and propagated n SPAWN reman unaffected. More sophstcated approxmatons to b k x can of course be consdered, but as we wll see, a mxture of two Gaussans s suffcent for our scenaro. C. Censorng Schemes Neghbor-Agnostc Transmt Censorng: In neghboragnostc transmt censorng, an agent wll decde to broadcast or censor ts postonal nformaton based on the uncertanty of ts own belef. After calculatng ts belef b k x at teraton k, agent can determne the covarance matrx Σ k assocated wth p k x, ndcatng how concentrated the belef s. An agent wll transmt-censor when trace Σ k γ TX. 6 The transmt censorng threshold γ TX, expressed n m 2, depends on the rangng model and the performance requrements. Durng the frst teraton non-cooperatve phase agents that can only communcate wth zero or one anchors wll have belefs that are not concentrated. Hence, these agents wll censor ther belefs. In later teratons, agents can obtan more nformaton from neghbors, leadng to more concentrated belefs, and thus less transmt censorng. We note that the censorng crteron does not drectly depend on the neghbors belefs. For that reason, we call ths censorng scheme neghbor-agnostc. Algorthm 3 Lnk selecton of L most nformatve lnks. : f S > L then 2: create S L = {allsubsetsof S of sze L} 3: for l = to S L do {subset ndex} 4: let S L [l] be the l-th subset n S L 5: determne where F [l] = + MBCRB [l] = trace F [l], σ S 2 L[l] [ ] Σ k 6: end for 7: select the best subset 8: set S k to S L[ˆl] 9: else 0: set S k to S : end f { x x E x,x x x ˆl = argmn MBCRB [l] l } x x T x x 2 Receve Censorng: In receve censorng, an agent wll decde to dscard unnformatve ncomng nformaton from neghborng agents. To allow pror Fsher nformaton of the form F P, = Σ, we perform a separate pre-processng step to remove ambgutes see Algorthm 2: based on ts belef b k x at the prevous teraton, agent can determne the covarance matrx Σ k and the number of components N k {, 2}. When N k = 2, the agent wll try to remove the ambguty n ts belef by consderng the nformaton from the neghbors. Ambguty removal can smply be performed by checkng the consstency between the components n b k x and the belefs of all the neghbors b k X x, S. After ambguty removal, 2 a lnk selecton algorthm see Algorthm 3 s executed to select the most nformatve subset of L 3 neghbors. However, when trace Σ k < γ RX, the agent dscards all ncomng nformaton 3 by settng S k =. The sze of the subset ndcated by L n Algorthm 3 should be at least 3 for twodmensonal postonng. 3 Neghbor-Aware Transmt Censorng: Whle transmt censorng as descrbed n Secton III-C can reduce the network traffc, t s clear that n combnaton wth receve censorng further reductons n network traffc are possble: when all neghbors of agent satsfy the receve censorng 2 When the ambguty n b k X x cannot be removed, lne 6 of Algorthm 2 s executed based on one arbtrarly chosen component of b k X x. Ths approxmaton turns out to have lttle mpact on the fnal performance, as ths case only occurs when agent or all of ts neghbors have belefs that are not concentrated. 3 Essentally, we consder the agent as well-localzed, so no further processng s requred.
5 Algorthm 4 Neghbor-aware transmt censorng for SPAWN teraton k, agent. : f trace Σ k > γ TX then 2: block the broadcast of b k X x 3: else 4: broadcast = FALSE 5: for = to S do {neghbor s ndex} 6: f N k = 2 OR trace 7: broadcast = TRUE 8: break 9: end f 0: end for : f broadcast then 2: broadcast b k X x 3: end f 4: end f Σ k > γ RX then threshold, 4 the broadcasts of agent wll be gnored by all neghbors. Hence, those broadcasts are unnecessary. We can thus develop a neghbor-aware transmt censorng scheme, as outlned n Algorthm 4, whch blocks broadcasts that wll be gnored by all the neghbors [25]. Observe that neghboragnostc transmt censorng corresponds to lnes 3 n Algorthm 4. It s mportant to note that ths scheme suffers from a hdden node problem: when an agent s not aware a neghbor s present due to packet loss, transmt censorng, or asymmetrc lnks, t may decde to transmt censor too early. A. Smulaton Setup IV. NUMERICAL RESULTS We consder random networks smlar to those descrbed n Secton II-B, wth 00 randomly placed agents, 3 anchors, a 00 m by 00 m map, 20 m communcaton radus, and 0 cm rangng nose standard devaton. Our focus s on a lneof-sght LOS scenaro, though the censorng methods can be appled unaltered n non-los NLOS condtons when NLOS detecton s employed [26]. We wll frst fx the receve censorng threshold γ RX and transmt censorng threshold γ TX, both expressed n m 2. The value of γ Rs drectly related to desred postonng accuracy, wth more aggressve censorng.e., larger values of γ RX leadng to faster convergence, lower complexty, but a reducton n accuracy. Receve censorng s swtched off when γ RX = 0. The value of γ TX reflects when an agent s deemed nformatve for neghbors. More aggressve censorng.e., smaller value of γ TX leads to fewer broadcasts, as only hghly nformatve nformaton s shared, but also to less nformaton n the network. Transmt censorng s swtched off when γ TX = +. For combned transmt and receve censorng, we requre that γ TX γ RX : when an agent s belef has met the receve censorng threshold, t should not block ts broadcasts. We have chosen γ RX = 0.28 m 2 and γ TX = 0.45 m 2, whch are both on the order of the rangng 4 I.e., traceσ k < γ RX AND N k =, S. nose varance. As we wll see n Secton IV-B5, the system s not very senstve to the value of ether threshold. We set L = 3 n Algorthm 3. We wll denote by NoC the SPAWN algorthm wth no censorng, by TxC when only neghbor-agnostc transmt censorng s used, by RxC when only receve censorng s used, and by TxRxC when receve censorng wth neghbor-aware transmt censorng s used. B. Smulaton Dscusson Reducton n Complexty: The complexty of SPAWN s manly related to the number of messages used durng message multplcaton lne 4 n Algorthm. In partcular, n a sample-based message representaton, the complexty of the message multplcaton scales as OQ 2 S k, where Q s the number of samples per message typcally 000 0000 and S k denotes the cardnalty of the set S k. In a parametrc message representaton, the complexty scales as OC S k, where C s a generally large constant related to the computaton of the message parameters, whch typcally nvolves solvng a non-convex optmzaton problem [22]. As we wll see later, wthout lnk selecton, S k 0, whle wth lnk selecton S k 3. The complexty of the lnk selecton algorthm Algorthm 3 scales as O N S k, where N from 3 s relatvely small say, 200. For small L, the lnk selecton process s much less complex than the message multplcaton, whch drectly motvates the need to reduce the number of multpled messages. The complexty can be further reduced by performng a greedy, rather than exhaustve search of the L most nformatve lnks n Algorthm 3. As a ndcaton, Table I shows the normalzed smulaton tmes for a parametrc message representaton [27]. We observe that wth the TxRxC strategy, SPAWN can be executed roughly 9 tmes faster than wthout censorng. For a nonparametrc representaton, results not shown ndcated smlar complexty reductons. Fgure 3 shows, as a functon of the teraton ndex, the average number of multpled messages per agent for the dfferent censorng strateges. When no censorng s employed, over 0 lnks are consdered per agent at every teraton except the frst, non-cooperatve teraton. The TxC strategy results n a margnal reducton, as some broadcasts are blocked. In contrast, the RxC strategy leads to a sgnfcant reducton n the number of lnks used. After two teratons, most agents meet the receve censorng threshold, so the number of lnks wll be close to zero. The combnaton TxRxC leads to addtonal TABLE I NORMALIZED SIMULATION TIME FOR SPAWN WITH DIFFERENT CENSORING SCHEMES, FOR 0 ITERATIONS. smulaton tme [normalzed] SPAWN NoC L TxC RxC TxRxC 8.9 8.2..0
6 number of multpled messages per agent 0 0 0 0 NoC TxC RxC TxRxC 0 2 2 3 4 5 6 7 8 9 0 teraton Fg. 3. Complexty: comparson of the average number of used lnks or messages multpled for dfferent censorng schemes. CCDFerror 0 0 NoC TxC RxC TxRxC 0 0 2 0 0.5.5 2 2.5 3 3.5 4 4.5 5 error [m] Fg. 5. Postonng performance comparson after 0 teratons wth and wthout censorng. average number of broadcasts per agent 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. NoC & RxC TxC TxRxC CCDFerror = meter 0 0 NoC TxC RxC TxRxC 0 0 2 0 2 3 4 5 6 7 8 9 0 teraton 2 3 4 5 6 7 8 9 0 teraton Fg. 4. Network traffc: comparson of average number of broadcasts per agent, for dfferent censorng schemes. Fg. 6. Convergence speed of dfferent censorng schemes. gans, as agents are more lkely to receve useful nformaton, and hence more quckly meet the receve censorng threshold. The quanttatve reducton n complexty due to the reducton n the number of multpled messages depends on the partcular message representaton. 2 Reducton n Network Traffc : The hgh network traffc n SPAWN s due to every agent broadcastng ts belef at every teraton. Fgure 4 shows, as a functon of the teraton ndex, the average number of broadcasts per agent for the dfferent censorng strateges. Wthout censorng, almost every agent wll broadcast ts belef at every teraton, except the frst one. Applyng the TxC strategy results n a reducton of the number of broadcasts, especally n the frst few teratons, when many agents are not yet well-localzed. In later teratons, when most agents are well-localzed, no censorng takes place, resultng n almost the same number of broadcasts compared to conventonal SPAWN. The TxRxC strategy follows the same trend as TxC for the frst few teratons. Then, neghbor-aware transmt censorng can harness the fact that an agent s neghbors have met the receve censorng threshold and block that agent s broadcast. Hence, the average number of broadcasts drop close to zero wth further teratons. 3 Postonng Performance : We now nvestgate the postonng performance of the dfferent censorng schemes. Fgure 5 shows complementary cumulatve dstrbuton functon CCDF of the postonng error,.e., the probablty that the postonng error exceeds a certan value, after 0 teratons. We can observe that the CCDF of TxC follows the CCDF of conventonal SPAWN because most of the avalable lnks are used see also Fgure 3. On the other hand, RxC results n a performance degradaton compared to conventonal SPAWN, as agents only use a subset of L = 3 lnks from the avalable lnks durng message multplcaton. Interestngly, the TxRxC strategy outperforms RxC. The reason for ths s that unnformatve belefs are transmt-censored, so that durng
7 receve censorng, the lnks to choose from all correspond to concentrated belefs. 4 Convergence Speed: The convergence speed of the algorthm s drectly related to the latency and the refresh-rate. In Fgure 6 we compare the postonng performance as a functon of the teraton ndex. The postonng performance s measured n terms of the CCDF at a fxed value of the postonng error m. For nstance, a curve wth label TxC shows, under the TxC strategy, the probablty that an agent wll have a postonng error greater than m, at every teraton. We observe that RxC converges the slowest, whle TxC and TxRxC requre 5 6 teraton to converge, rrespectve of the error value. 5 Senstvty to Parameters γ TX, γ RX, and L: We vared γ TX, γ RX, and L. Changng γ TX around 0.45 m 2 dd not lead to a sgnfcant change n performance or traffc, but too conservatve transmt censorng causes ncreased network traffc. Any change n γ RX affects the complexty of the algorthm through the average number of used lnks, as well as the requred number of teratons for convergence. By reducng γ RX to 0.4 m 2, the gap n postonng performance between NoC and RxC can be reduced sgnfcantly, at a small cost n complexty, as fewer agents meet the receve censorng threshold. Fnally, changng L from 3 to 4 dd not yeld any sgnfcant performance mprovement, but results n addtonal complexty n Algorthm 3. V. CONCLUSIONS AND FUTURE WORK Motvated by the need to reduce complexty and network traffc n cooperatve postonng schemes, we have proposed and evaluated dfferent censorng schemes. All censorng decsons are dstrbuted and based on a modfed Bayesan Cramér-Rao bound crteron. By applyng the proposed censorng schemes to Bayesan cooperatve postonng, we have found that: receve censorng gnorng unnformatve nformaton can dramatcally reduce the complexty of nformaton fuson, but at the cost n postonng performance; transmt censorng blockng broadcasts of unrelable nformaton can reduce the network traffc durng frst few teratons wthout postonng performance degradaton; receve censorng wth neghbor-aware transmt censorng blockng broadcasts of nformaton that wll be gnored can further sgnfcantly reduce the network traffc. Overall, ths latter scheme mantans the excellent performance and low latency of Bayesan cooperatve postonng wthout censorng, but does so at a fracton of the computatonal cost, and at a fracton of the network traffc. These advantages of censorng schemes, along wth ther dstrbuted nature make them promsng for large-scale dense networks. 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