Receiver Cooperation in Topology Control for Wireless Ad-Hoc Networks

Abstract—We propose employing receiver cooperation in centralizedtopology control to improve energy efficiency as well asnetwork connectivity. The idea of transmitter cooperation hasbeen widely considered in topology control to improve networkconnectivity or energy efficiency. However, receiver cooperationhas not previously been considered in topology control. In particular,we show that we can improve both connectivity and energy efficiencyif we employ receiver cooperation in addition to transmittercooperation. Consequently, we conclude that a system based bothon transmitter and receiver cooperation is generally superior toone based only on transmitter cooperation. We also show that theincrease in network connectivity caused by employing transmittercooperation in addition to receiver cooperation is at the expense ofsignificantly increased energy consumption. Consequently, systemdesigners may opt for receiver-only cooperation in cases for whichenergy efficiency is of the highest priority or when connectivityincrease is no longer a serious concern.Index Terms—Ad-hoc network, energy efficiency, multi-hopcommunications, network connectivity, receiver cooperation,topology control, transmitter cooperation.I. INTRODUCTIONTHE wireless ad-hoc network has been receiving growingattention during the last decade for its various advantagessuch as instant deployment and reconfiguration capability. Ingeneral, a node in a wireless ad-hoc network suffers fromconnectivity instability because of channel quality variation andlimited battery lifespan. Therefore, an efficient algorithm forcontrolling the communication links among nodes is essentialfor the construction of a wireless ad-hoc network. In a topologycontrol scheme, communication links among nodes are definedto achieve certain desired properties for connectivity, energyconsumption, mobility, network capacity, security, and so on.In this paper, we propose topology control schemes that aimManuscript received February 23, 2014; revised July 24, 2014 and November12, 2014; accepted November 12, 2014. Date of publication December 4, 2014;date of current version April 7, 2015. Part of this work was presented at IEEEWCNC, Shanghai, China, April 2013. This work was supported in part byBasic Science Research Program through the National Research Foundationof Korea (NRF) funded by the Ministry of Education (NRF-2010-0025062 andNRF-2013R1A1A2011098). The associate editor coordinating the review ofthis paper and approving it for publication was M. Elkashlan. (Correspondingauthors: Do-Sik Yoo and Seong-Jun Oh).K. Moon, W. Lee, and S.-J. Oh are with Korea University, Seoul 136-701,Korea (e-mail: keith@korea.ac.kr; wlee@korea.ac.kr; seongjun@korea.ac.kr).D.-S. Yoo is with the Department of Electronic and Electrical Engineering,Hongik University, Seoul 121-791, Korea (e-mail: yoodosik@hongik.ac.kr).Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TWC.2014.2374617to increase the energy efficiency and the network connectivitysimultaneously.In a wireless ad-hoc network, two nodes that are not directlyconnected may possibly communicate with each other throughso-called multi-hop communications [1], [2]. By employingmulti-hop communication, a node in a wireless ad-hoc networkcan extend its communication range through cascaded multihoplinks and eliminate some dispensable links to reduce the totalrequired power. Various efforts have been made to study howthe links must be maintained and how much power must be associatedwith each of those links for optimal network operationsdepending on the situation at hand. For example, Kirousis et al.[3] and Clementi et al. [4] studied the problem of minimizingthe sum power consumption of the nodes in an ad-hoc networkand showed that this problem is nondeterministic polynomialtime(NP) hard. Because the sum power minimization problemis NP hard, the authors in [4] proposed a heuristic solution forpractical ad-hoc networks. Ramanathan and Rosales-Hain, in[5], proposed two topology control schemes that minimize themaximum transmission power of each node with bi-directionaland directional strong connectivities, respectively. When thenumber of participating nodes is very large, it is crucial toreduce the transmission delay due to multi-hop transmissions.To maintain the total transmission delay within a tolerable limit,Zhang et al. studied delay-constrained ad-hoc networks in [6]and Huang et al. proposed a novel topology control scheme in[7] by predicting node movement.In [3]–[7], it was assumed that there exists a centralizedsystem controlling nodes so that global information such asnode positions and synchronization timing is known by eachnode in advance. However, such an assumption can be toostrong, especially in the case of ad-hoc networks. For thisreason, a distributed approach has been widely considered [8]–[11], where each node has to make its decision based on the informationit has collected from nearby neighbor nodes. Li et al.proposed a distributed topology control scheme in [8] andproved that the distributed topology control scheme preservesthe network connectivity compared with a centralized one.Because the topology control schemes in [3]–[8] guarantee onlyone connected neighbor for each node, the network connectivitycan be broken even when only a single link is disconnected.Accordingly, a reliable distributed topology control scheme thatguarantees at least k-neighbors was proposed in [9]. The resultin [9] was extended to a low computational complexity schemein [10], to a mobility guaranteeing scheme in [11], and to anenergy saving scheme in [12].1536-1276 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.MOON et al.: RECEIVER COOPERATION IN TOPOLOGY CONTROL FOR WIRELESS AD-HOC NETWORKS 1859In [13], the concept of cooperative communications was firstemployed in centralized topology control, where it was shownthat cooperative communications can dramatically reduce thesum power consumption in broadcast network. Cardei et al.applied the idea of [13] to wireless ad-hoc networks in [14].Yu et al. further showed that cooperative communications canextend the communication range of each node with only amarginal increment in power consumption so that networkconnectivity is increased in an energy efficient manner [15],[16]. Because of these various advantages, the idea of cooperativecommunications has been widely considered in recentstudies on topology control to maximize capacity [17], improverouting efficiency [18], and mitigate interference from nearbynodes [19], [20]. The idea of cooperative communications inthese previous works [13]–[20] is realized in the followingway. First, a transmitting node sends a message to its neighbornodes (called helper nodes). After the helper nodes decode themessage, they (as well as the transmitting node in some cases)retransmit the message to a receiving node, and the receivingnode decodes the message by combining the signals frommultiple nodes. Therefore, strictly speaking, only the conceptof transmitter cooperation has been employed, and receivercooperation has not been considered.In this paper, we propose to employ the idea of receivercooperation in centralized topology control schemes, possiblyin combination with transmitter cooperation, to increase thenetwork connectivity in an energy efficient way. Consequently,we propose two centralized topology control schemes, onebased solely on receiver cooperation, and the other basedboth on transmitter and receiver cooperation. For comparisonwith proposed schemes, we consider a cooperative topologycontrol scheme in [16] that is based solely on transmittercooperation. We show, through extensive simulations, that wecan improve both network connectivity and energy efficiencyif we employ receiver cooperation in addition to transmittercooperation. We conclude that the system based both ontransmitter and receiver cooperation is generally superior tothat based only on transmitter cooperation. We also showthat the system based solely on receiver cooperation is asenergy efficient as one based both on transmitter and receivercooperation despite a slight decrease in network connectivity.Although the system based both on transmitter and receivercooperation achieves higher network connectivity than onebased only on receiver cooperation, we show that the additionalconnectivity increase requires significantly increased energyconsumption. For this reason, system designers may opt forreceiver-only cooperation, if energy efficiency is of the highestpriority or connectivity increase is no longer of seriousconcern.The remainder of this paper is organized as follows.In Section II, we describe the channel model consideredthroughout this paper. In Section III, we explain the topologycontrol scheme without cooperation that underlies the twocooperative topology control schemes considered in this paper.The two cooperative topology control schemes are then describedin Section IV. Furthermore, the performance of the twocooperative topology control schemes are numerically analyzedin Section V. Finally, we draw conclusions in Section VI.II. SYSTEM MODELIn this section, we describe the system model consideredthroughout this paper. We consider a network V ≡{v1, v2, . . . , vn} consisting of n nodes that are assumed to beuniformly distributed over a certain region in R2. The nodesare assumed to communicate with one another by transmittingsignals over a wireless channel with given bandwidth W. Weassume that the physical location of each node does not changewith time.To model a practical wireless channel, we assume that thepath loss PL(di j) between nodes vi and vj is given byPL(di j)[dB] = PLd0 +10k log_di jd0_+2loghi j +Xó+c. (1)Here, PLd0 is the reference path loss at unit distance d0 obtainedfrom the free space path loss model [21], and k denotes the pathloss exponent that represents how quickly the transmit powerattenuates as a function of the distance. The variables di j andhi j respectively denote the distance and the randomly varyingfast fading coefficient between vi and vj . In addition, Xó is arandom variable introduced to account for the shadowing effect.We assume that hi j and Xó vary independently from packetto packet, but remain constant during each packet duration.We assume further that h2i j follows a ÷2-distribution with twodegrees of freedom and Xó follows a normal distribution withzero mean and standard deviation ó. Finally, the variable c is theoffset correction factor between the mathematical model andfield measurement. We note that the values of PLd0 , d0, k, ó,and c vary depending on channel scenario, urban or suburban[22]. For given PLd0 , d0, k, ó, and c, when node vi transmitsa signal to node vj with power Pi, the received signal to noiseratio (SNR) ãi j(Pi) is given asãi j(Pi) =PiN0, jW×100.1×PL(di j), (2)where N0, j denotes the one-sided noise power spectral densityat vj . Throughout this paper, we assume that the maximumtransmit power of each node is given by Pmax.As the final issue in the system model, we briefly discuss networksynchronization. Communication in a completely asynchronousmanner is impossible, or at least be very difficult toachieve. In fact, synchronization can be a particularly importantissue in ad-hoc networks [23]–[25]. In this paper, we assumethat symbol level synchronization is maintained among participatingnodes. Although detailed synchronization techniquesare not the main focus of this paper, we briefly describe howthe issue of synchronization can be resolved with existingmethods. Synchronization techniques have been reported thatit can achieve time errors around 3 ∼ 7 μs. At such a level ofsynchronization, it will become desirable to maintain symbolduration longer than 50 μs, which corresponds to symbol rateof up to 20 kilo-symbols per second. A symbol rate of 20 kilosymbolswith rudimentary binary phase shift keying (BPSK)modulation results in a data-rate of only 20 kbps, which is notvery high. However, we can employ multi-carrier techniquessuch as orthogonal frequency division multiplexing (OFDM)to increase the data rate while maintaining or reducing thesymbol rate. For example, if we employ an OFDM system1860 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 4, APRIL 2015Fig. 1. A pictorial representation of G = (V,E) with V = {v1, . . . , v8} and E = {(v1;v2)NN, (v1;v3)NN, (v4;v5)NN, (v4;v6)NN, (v4;v7)NN}.with 512 subcarriers, the data rate can be increased to about10 Mbps using a simple BPSK sub-carrier modulation scheme.Consequently, even with existing techniques such as the OFDMscheme and synchronization algorithms proposed in [25], it ispossible to maintain the symbol-level synchronization requiredto implement the algorithms proposed in this paper.III. NODE-TO-NODE TOPOLOGY CONTROLIn this section, we explain a topology control scheme, whichwe refer to as the node-to-node topology control (NNTC)scheme, that is based solely on node-to-node communicationlinks. To describe the NNTC scheme, we first consider theconcept of a wireless communication link between two nodesand its related definitions. In this paper, a wireless link betweentwo nodes is said to exist if the received SNR exceeds a certainthreshold, meaning that the packet error probability is below acertain level (corresponding to the threshold). More formally,we say that there exists a node-to-node (N-N) link from node vito node vj if and only iff (ãi j(Pi)) ≤ áô, (3)for a certain transmit power Pi ≤ Pmax from vi. Here, f : R + →[0,1] denotes the packet error probability function associatedwith the given coding and modulation scheme and áô is thegiven threshold on the packet error probability, which we callthe error threshold hereafter. We assume that f is a monotonicallydecreasing continuous function and that all the nodesshare the same packet error probability function f .1When there exists a uni-directional N-N link from vi to vj ,the power Pi that satisfies (3) with equality, which we denoteby PNN(vi → vj), is called the minimum N-N routable powerof N-N link from vi to vj . We note that PNN(vi → vj) directlyfollows from the definition thatPNN(vi →vj) =N0, jW f−1(áô)100.1×PL(di j). (4)If both the uni-directional N-N links from vi to vj and from vjto vi exist, we say that there exists an N-N bi-directional link,or simply an N-N link between the two nodes vi and vj that1In many previous works on topology controls [14]–[17], (3) is equivalentlywritten as ãi j(Pi) ≥ SNRô ≡ f−1(áô). However, to consider the receivercooperation scheme in a unified framework, we directly consider the packeterror probability function f .we denote by (vi; vj)NN. The minimum N-N round-trip powerPNN(vi, vj) of the bi-directional N-N link (vi; vj)NN is definedas the sum of the two uni-directional minimum N-N routablepowers, namely, asPNN(vi, vj) = PNN(vi →vj)+PNN(vj →vi). (5)We note that there are some situations in which two nodes viand vj can communicate with each other even if there is no NNlink between vi and vj . For example, we consider the case inwhich there are two N-N links (v1; v2)NN and (v1; v3)NN. In thiscase, v2 and v3 can exchange a message through v1 even if thereis no N-N link between v2 and v3. To route a message throughmultiple N-N links, all available N-N links should be knownto the nodes. To reduce the routing complexity, only some ofthe existing N-N links are used for communications in practice.By eliminating redundant links, we can simplify the messagerouting protocol and save power consumed for exchangingreference signals such as pilot and channel information [26],[27].We denote the set of N-N links to be used for routing by E.Consequently, (vi; vj)NN ∈ E means that there exists N-N link(vi; vj)NN and this N-N link is to be used for routing. Here, wenote that (vi; vj)NN /∈E does not necessarily mean that there isno N-N link between vi and vj . In graph theory, the combinationG = (V,E) of V and E is called a graph with vertex set V andedge set E. In the remainder of this paper, nodes and links shallalso be referred to as vertexes and edges, respectively.For a given E, if (vi; vj)NN ∈ E, vi is said to be a neighborof vj and vice versa. We denote by N(vi|E) the setof neighbors of vi. For illustration, we consider the graphG = (V,E) with V = {v1, v2, . . . , v8} and E = {(v1; v2)NN,(v1; v3)NN, (v4; v5)NN, (v4; v6)NN, (v4; v7)NN}, which compactlydescribes the situation in Fig. 1. In this example, v5, v6 andv7 are neighbors of v4, therefore, N(v4|E) = {v5, v6, v7}. Here,we note that v5 is not a neighbor of v7, however, it is possiblefor v5 to send a message to v7 if (v4; v5)NN and (v4; v7)NNare cascaded. Likewise, if vi and vj can send a message bidirectionallyusing a single or cascaded multiple N-N edges, wesay that vi and vj are connected by N-N edges. The maximal setof nodes connected by N-N edges in E is referred to as a cluster.For notational convenience, a given cluster {vi1 , vi2 , . . . , vim} isdenoted by Ùmax{i1,i2,…,im}. For instance, in Fig. 1, there arethree clusters {v1, v2, v3}, {v4, v5, v6, v7}, and {v8}, which aredenoted by Ù3, Ù7, and Ù8, respectively. As shown in thisMOON et al.: RECEIVER COOPERATION IN TOPOLOGY CONTROL FOR WIRELESS AD-HOC NETWORKS 1861Fig. 2. Steps to construct the edge set E for a given node distribution V. (a) Identification of all N-N links. (b) A typical example of a spanning forest ofGL = (V,L).example, several clusters can exist for a given graph.We denotethe set of all clusters by V . We note that V = {Ù3,Ù7,Ù8} inthe above example.We now describe precisely how the set E of N-N edges tobe used for routing in the NNTC scheme is constructed. For agiven node set V, the set L of all existing N-N links and the setV of clusters defined by the graph GL = (V,L) are identified.Next, the edge set E is defined as a subset of L such that thegraph G = (V,E) also leads to the same cluster set V as graphGL = (V,L). Several candidate algorithms exist that can buildE such as breath-first search (BFS) [28] and depth-first search(DFS) [29]. In this paper, we use the minimum-weight spanningforest (MSF) algorithm that aims to build a sparse edge setusing the optimal average power required for network structureconstruction [1], [8], [15], [16]. In the MSF algorithm, first a setTÙ called a minimum spanning tree (MST), is defined for eachcluster Ù ∈ V . After obtaining all the MSTs, the set FV , calledthe minimum spanning forest of V, is defined as the union of allthe MSTs, namely, asFV = _Ù∈VTÙ, (6)which is defined to be edge set E in the NNTC scheme.It now remains to describe how the MST TÙ is obtained foreach cluster Ù ∈ V . If Ù is a singleton, then TÙ is defined to bethe empty set /0. If Ù contains more than one node, to obtain TÙ,it is necessary to consider the set L|Ù of all edges that connectnodes in Ù. For instance, we consider the example depictedin Fig. 2(a) in which the network consists of three singletonclusters and nine non-singleton clusters. For a non-singletoncluster Ù encircled by a red colored line, the edge set L|Ù isdefined as the set of all edges inside the red circle. We call asubset T of L|Ù a spanning tree of Ù if and only if there are nocycles (loops) in T and if any two nodes in Ù are connected byedges in T. For example, the edge set of each cluster depictedin Fig. 2(b) is a spanning tree of that cluster. Among all theexisting spanning trees of Ù, the one that leads to the minimumedge-weight sum is referred to as the MST TÙ of Ù. Here, theminimum N-N round-trip power PNN(vi, vj) of the N-N link isused for the weight of each edge (vi, vj)NN ∈ L.We note that transmission through the link in FV is not completelyerror-free, but has a packet error probability of áô. However,in the following, we assume that the communication linkin FV is error-free, possibly with the help of an automatic repeatand request (ARQ) scheme. Clearly, the repeated transmissionwill consume additional energy. However, even with the simplestARQ scheme, the average required energy to complete asuccessful transmission is increased from a single transmission(with packet error rate áô) by a factor of 1/(1−áô) [30]. Wenote that the factor 1/(1−áô) is reasonably close to 1 if áô ischosen to be small, say, less than 0.1. Therefore, if áô is sufficientlysmall, the additional cost for error-free communicationis only a small fraction of the total cost and hence is negligible.IV. COOPERATIVE TOPOLOGY CONTROLWe note that inter-cluster communication, namely, communicationbetween nodes belonging to different clusters is notpossible solely through cascaded N-N links. To make interclustercommunications possible, [16] employed the idea oftransmitter cooperation in which multiple nodes in one clustersimultaneously transmit the same message to a single node inanother cluster. In [16], to keep the additional complexity due tothe employment of cooperative transmission manageable, it wasassumed that a pair of nodes belonging to two communicatingclusters were pre-assigned so that communications between thetwo clusters could only happen between these two nodes withthe help of nodes in their neighborhoods. We note that notonly the neighboring nodes around the transmitting node butalso the nodes around the receiving node can help to establishinter-cluster communications. Consequently, in this paper, wepropose to employ receiver cooperation in which the interclustercommunication is regarded as successful if the receivingnode or any of the neighboring nodes succeeds in receiving the1862 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 4, APRIL 2015message correctly. If the neighboring nodes only around thereceiving node participate in the cooperation, the establishedlink between two clusters is referred to as the node-to-cluster(N-C) link. Furthermore, if neighboring nodes around boththe transmitting and receiving nodes participate in the linkestablishment, the inter-cluster communication link is calledcluster-to-cluster (C-C) link.In this section, we describe two centralized cooperativetopology control schemes based on N-C and C-C links thatare referred to as node-to-cluster topology control (NCTC) andcluster-to-cluster topology control (CCTC) schemes, respectively.In each of these cooperative topology control schemes,cooperative links are employed to connect the clusters obtainedfrom the graph G = (V,E) described in Section III. Consequently,the network configuration defined in a cooperativetopology control scheme is described by four sets, namely, theset V of nodes, the set E of edges used for routing in the NNTCscheme, the set V of clusters defined by the graph G = (V,E),and the set E of cooperative edges. For this reason, the networkconfigurations defined in the NCTC and CCTC schemes areidentified by GNC = (V,E,V ,ENC) and GCC = (V,E,V ,ECC),respectively. Here, ENC and ECC consist only of N-C and C-Cedges, respectively.A. NCTCIn this subsection, we describe how the network configurationGNC = (V,E,V ,ENC) corresponding to the NCTC schemeis defined. Given graph G = (V,E) and corresponding clusterset V , the edge set ENC is obtained in three steps. First, theset LNC of all N-C links connecting clusters in V is identified.Next, for each N-C link in LNC, the weight of the link isdefined as the minimum power required to establish it. Finally,the desired edge set ENC is defined as the MSF of the graphGLNC = (V ,LNC).To describe the NCTC scheme, we first define the nodeto-cluster (N-C) link. For more concrete understanding ofN-C link, we consider a simple example of receiver cooperationbetween two clusters Ù3 ={v1, v2, v3} and Ù7 ={v4, v5, v6, v7}.For illustration, we assume that the inter-cluster communicationlink between two clusters is established if the error probabilityis less than or equal to 0.1. We assume that the decoding errorprobabilities at nodes v4, v5, v6 and v7 are, respectively, given as0.3, 0.4, 0.8, and 0.9 when v1 sends a message with maximumpower. Consequently, node v1 and a node in Ù7 cannot establishinter-cluster communications between Ù3 and Ù7 throughN-N links. However, if any of the nodes in Ù7 succeed incorrectly decoding the message, the message can be routed toany of the desired nodes in Ù7. If such receiver cooperation isemployed, communication fails only when all four nodes v4, v5,v6, and v7 fail to decode the message at the same time. We notethat such a probability is 0.3×0.4×0.8×0.9 = 0.0864 < 0.1.For this reason, we say that cooperative communication linkbetween Ù3 and Ù7 is established.In the above example, all nodes in the receiving cluster tryto decode the transmitted message. However, if the size of thereceiving cluster is large, the routing protocol and maintenancecost can become very burdensome. For this reason, we assumethat a certain receiving node and its one hop neighbors participatein the receiver cooperation. To be more precise, for a givenpair of clusters, a certain node is selected from each clusterand the signal is assumed to be transmitted from either of thesetwo nodes and then received by the other node and its one-hopneighbors.We note that there exists a more aggressive method of receivercooperation than the one described above. For example,the bridge node can achieve a huge combining gain if thehelper nodes transmit observed soft information rather thandecoded bits. However, the transmission of the observed datagenerally consumes large amount of energy and bandwidth.Consequently, a sufficiently fine quantization must be consideredto employ soft combining. Because this problem ishighly complex, we assume in this paper that the helper nodesdecode the message and deliver it to the bridge node. However,considering the importance of this problem, serious researchemploying soft combining schemes should be pursued.For a more formal description, we consider two non-emptyclusters Ùl and Ùm from the given graph G = (V,E) defined inthe NNTC scheme. We formally define the concept of an N-Clink as follows.Definition 1: Let vbl∈ Ùl and vbm∈ Ùm. Then, we saythat there exists a bi-directional N-C link, or simply, a N-Clink denoted by (vbl ,N(vbl|L); vbm,N(vbm|L))NC between Ùland Ùm, if and only ifÐvr∈{vbm}∪N(vbm|L)f_ãbl r(Pbl )_≤ áô (7)andÐvr∈{vbl}∪N(vbl|L)f (ãbmr(Pbm)) ≤ áô (8)for some Pbl≤ Pmax and Pbm≤ Pmax.Here, L denotes the set of all N-N links described inSection III. In other words, all one-hop neighbors of the receivingnode are assumed to participate in receiver cooperationregardless whether they belong to E. We note that the errorprobability between helper and bridge node is assumed tobe zero, as mentioned in Section III. For a given N-C link(vbl ,N(vbl|L); vbm,N(vbm|L))NC, nodes vbl and vbm and setsN(vbl|L) and N(vbm|L) are called the bridge nodes and helpersets, respectively.In Definition 1, we note that the sum of the Pbl and Pbm valuesthat satisfy (7) and (8) with equality is the minimum total transmissionpower required to make round-trip communicationbetween Ùl and Ùm through (vbl ,N(vbl|L); vbm,N(vbm|L))NC.Because the sum Pbl +Pbm depends on the choice of the N-Clink, it is natural to choose the N-C link that minimizes the sumpower Pbl +Pbm. The minimized sum power shall be referred toas the minimum N-C round-trip power and the correspondingN-C link as the minimum power N-C link between Ùl and Ùm.We denote by PNC(Ùl ,Ùm) the minimum N-C round-trip powerbetween Ùl and Ùm.We now describe how we establish communications betweenÙl and Ùm. First, let vbl∈ Ùl and vvm∈ Ùm be the bridgenodes of the minimum power N-C link between Ùl and Ùmand let Hl and Hm be the helper sets of the link. We nowMOON et al.: RECEIVER COOPERATION IN TOPOLOGY CONTROL FOR WIRELESS AD-HOC NETWORKS 1863Fig. 3. Steps to construct the edge set ENC for the given graph G = (V,E). (a) Identification of all N-C links. (b) A typical example of a spanning forest ofGLNC = (V ,LNC).assume that a source node vs in Ùl −{vbl} attempts to senda message to destination node vd in Ùm −{vbm}. In this case,vs sends the message to bridge node vbl through cascaded N-Nedges, and then bridge node vbl transmits the message to Ùm.The message sent from vbl is then decoded at bridge nodevbm and all the nodes in the helper set Hm. Because of thedefinition of the N-C link, the message must be decoded, withnegligible failure rate, at least at one node in {vbm} ∪ Hm.Because Hm consists only of the one hop neighbors of vbm, thenodes that successfully decode the message can be determinedby vbm with little overhead. After determining the nodes thatsuccessfully decoded the message, vbm delivers the message totarget destination node vd through the cascaded N-N edges.Finally, we describe how the edge set ENC is constructedin the NCTC scheme. First, the minimum power N-C link isidentified for each pair of clusters between which N-C linksexist. Let LNC denote the set of the minimum power N-C linksobtained as the result. For each (vbl ,Hl ; vbm,Hm)NC ∈ LNC,the weight is then defined as the corresponding minimum N-Cround-trip power. After computing all the weights of LNC, thesparse edge set ENC is defined as theMSF of GLNC =(V ,LNC).Note that the MSF construction procedure described inSection III can be directly applied here by substituting V andL with V and LNC, respectively. In Fig. 3, the procedure isillustrated. For instance, Fig. 3(a) indicates all the minimumpower N-C links between clusters by solid red lines andFig. 3(b) illustrates the shape of a typical spanning forest thatdoes not include any loops. Likewise, after finding all thespanning forests of GLNC = (V ,LNC), the one that minimizesthe sum weight is defined as the MSF ENC. After obtainingthe ENC, the desired final graph GNC = (V,E,V ,ENC) for theNCTC scheme is constructed.B. CCTCIn this subsection, we describe the CCTC scheme and explainhow the network configuration GCC = (V,E,V ,ECC) correspondingto the CCTC scheme is defined. We first explainthe concept of a cluster-to-cluster (C-C) link and the relatedrouting protocol with a simple example. We assume that sourcenode vs ∈ Ùl attempts to send a message to destination nodevd ∈ Ùm. In this case, vs sends a message through cascadedN-N edges to a pre-defined bridge node vbl . After receivingthe message, vbl disseminates the message to the nodes in apre-defined helper set Hl . After decoding the message, vbl andvhl∈ Hl simultaneously transmit the message to Ùm in thenext time frame. In Ùm, a pre-defined bridge node vbm andthe nodes in a pre-defined helper set Hm attempt to decode themessage with the multiple signal replicas from the transmitters.If the maximum ratio combiner (MRC) [31] is employed at thereceiving node vr ∈ {vbm}∪Hm, the combined average receivedSNR ¯ãr at vr can be written as¯ãr = ãbl r(Pbl)+ Óvhl∈Hlãhl r(Phl ), (9)and the decoding error probability at vr is given as f (¯ãr). Toestablish the symbol combining in (9), the same signals fromthe multiple transmitters should be received at the same timeas assumed in [13]. We note that problems related to timesynchronization were discussed in Section II. Similarly to thecase for N-C links, we say that the message is decodable, withnegligible failure rate, at least at one node in {vbm}∪Hm ifÐvr∈{vbm}∪Hmf (¯ãr) ≤ áô (10)with small enough áô, where f (·) denotes the common packeterror probability function for given received SNR, as defined inSection II. If the inequality (10) holds, we say that there exists aC-C link from Ùl to Ùm. Once the message is decoded at nodesin {vbm}∪Hm, the message is delivered to destination node vdthrough cascaded N-N edges to complete the routing procedure.To maintain the C-C link power efficiently, it is necessaryto choose appropriately the node pair (vbl , vbm), the helper set1864 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 4, APRIL 2015(Hl ,Hm), and the transmission power from each transmittingnode to minimize the power consumption. However, the computationalcomplexity makes such an optimization algorithmhardly feasible not only in practical systems but also in simulationenvironments [14]. For this reason, it is widely assumedthat nodes participating in transmitter cooperation use the samepower [15], [16]. Consequently, we adopt the same assumptionwhen designing the CCTC scheme.For a more formal description, we consider two non-emptyclusters Ùl and Ùm from a given graph G = (E,V). We definethe concept of a C-C link in the following definition.Definition 2: Let vbl∈ Ùl , vbm∈ Ùm, Hl ⊂ N(vbl|L), andHm ⊂ N(vbm|L). Then, we say that there exists a bi-directionalC-C link, or simply, a C-C link denoted by (vbl ,Hl ; vbm,Hm)CCbetween Ùl and Ùm if and only ifÐvr∈{vbm}∪Hmf⎛⎝ãbl r(Pcl)+ Óvhl∈Hlãhl r(Pcl )⎞⎠ ≤ áô, (11)andÐvr∈{vbl}∪Hlfãbmr(Pcm)+ Óvhm∈Hmãhmr(Pcm)_≤ áô (12)for some Pcl≤ Pmax and Pcm≤ Pmax.Here, Pcl and Pcm denote the common transmission powersof transmitting nodes in Ùl and Ùm, respectively. For a givenC-C link (vbl ,Hl ; vbm,Hm)CC, the nodes vbl and vbm are calledthe bridge nodes and the sets Hl and Hm are called the helpersets between Ùl and Ùm. Such terminology is the same for ofN-C links. However, in the case of C-C links, the nodes in thehelper set participate not only in receiver cooperation but alsoin transmitter cooperation.In Definition 2, we note that the total transmission powerminimally required to make round-trip communication betweenÙl and Ùm is given by (|Hl |+1)Pcl +(|Hm|+1)Pcm using thevalues for Pcl and Pcm that satisfy (11) and (12) with equality.Here, |X| denotes the cardinality of set X. We also note that therequired total transmission power (|Hl |+1)Pcl +(|Hm|+1)Pcmvaries depending on the choice of the C-C link. Consequently, itis natural to choose the C-C link that leads to the smallest totalrequired transmission power. The smallest total required transmissionpower and the corresponding C-C link are referred to asthe minimum C-C round-trip power and minimum power C-Clink between Ùl and Ùm, respectively. We denote the minimumC-C round-trip power between Ùl and Ùm by PCC(Ùl ,Ùm).We now describe how the edge set ECC is constructed inthe CCTC scheme. We note that the procedure for obtainingECC is essentially the same as that for obtaining ENC. Therefore,we describe it with brevity. First, the set LCC of all theminimum power C-C links between clusters is identified. Foreach (vbl ,Hl ; vbm,Hm)CC ∈ LCC, the weight is defined as thecorresponding minimum C-C round-trip power. After computingall the weights of LCC, the sparse edge set ECC is definedas the MSF of GLCC = (V ,LCC). After obtaining ECC, thedesired final graph GCC = (V,E,V ,ECC) for CCTC scheme isconstructed.Next, we briefly remark on the additional receiver processingcosts required for the NCTC and CCTC schemes. Comparedto the transmitter cooperative topology control scheme in [16],additional decoding power is required in the NCTC and CCTCschemes because of multiple-node decoding. This additionaldecoding increases not only the power consumption, but alsothe overall system complexity. Furthermore, each receivinghelper node should report the received message decodabilityto the bridge node, which increases system overhead. Thereare some analytical studies on receiving power consumption[32], [33] and overhead [34] because it could be a critical issuein the case of ad-hoc networks. However, we note that thedecoding power consumption and related overhead are heavilydependent on the receiving strategy. For example, one can chosea receiving strategy in which the receiving helper nodes decodethe message in the order of channel conditions until a successfuldecoding node appears. In this case, the average decodingpower consumption and system complexity can be reduced.In addition, the serach for the optimal receiving strategy ishighly non-trivial and requires serious and independent study.However, despite its importance, in this primary effort ontopology control, we do not consider such issues any furtherto keep the problem tractable.Finally, we briefly consider the impact of mobility on the proposedtopology control schemes. Unfortunately, the proposedschemes are basically inapplicable except when the mobilityis very low. When a node moves, three situations can happen.First, in some situations in which only minor movement isinvolved, there may be no changes in the network topologyexcept for the configurations inside the cluster to which themoved node belongs. Second, in other situations, the clusterto which the moved node originally belonged, must be dividedinto more than one cluster. Finally, in still other situations,some clusters could be unified into one cluster by the N-Nlinks newly defined by the node movement. In the first case,the mobility problem is relatively simple. If the moved node isnot a bridge or helper node, the moved node could be simplyattached to the nearby cluster. On the other hand, if the movednode is a bridge or helper node, the bridge and/or helper nodesof the corresponding cooperative link are changed to one of thealternatives among the pre-stored alternative bridge and helpernodes. However, if there is no alternative bridge and/or helpernode or if the second or the third situation occurs, clusters andcooperative edges should be redefined. In addition, if severalnodes move at the same time, the second and third situationsmay happen more frequently and this is why the proposedschemes are applicable only when the mobility is very low.V. PERFORMANCE EVALUATIONAND NUMERICAL RESULTSIn this section, we analyze through simulations the performanceof the two proposed centralized topology controlschemes, namely, the NCTC and CNTC schemes, and comparethem to the NNTC scheme and cooperative topology controlscheme in [16] that is based solely on transmitter cooperation.For convenience, we call the topology control scheme in[16] the cluster-to-node topology control scheme (CNTC). ToMOON et al.: RECEIVER COOPERATION IN TOPOLOGY CONTROL FOR WIRELESS AD-HOC NETWORKS 1865TABLE ISIMULATION CONFIGURATION PARAMETERSour best knowledge, the CNTC scheme achieves the highestconnectivity with a power requirement that is onl marginallygreater than other existing topology control schemes. In thissection, we show that the proposed NCTC scheme providesbetter energy efficiency with marginal connectivity loss and theCCTC scheme allows both better energy efficiency and higherconnectivity than the CNTC scheme.A. Simulation ConfigurationThe system performance is evaluated through simulations inthis paper. Although analytic evaluation is generally more desirable,the performance of topology control schemes is very hardto analyze. To the best of our knowledge, only some analyticalresults have been obtained for the case of non-cooperativecommunications among an infinite number of nodes [35], [36]and previous studies [13]–[16] on cooperative topology controlschemes have only been evaluated through numerical simulations.For this reason, we study the performance throughsimulations. However, we provide partial analytical reasoningwhenever possible. Furthermore, to improve the value of theresults, we reflect practical situations as much as possiblein simulation configuration by employing channel parametersbased on actual field measurement [22] and the design parametersin the 3GPP standard [37].To describe the system configuration used for performanceevaluation, we need to specify the values of various parameters,which we divide into two categories: channel parameters andsystem design parameters. The channel parameters includethe reference path loss PLd0 , path loss exponent k, shadowingrandom variable Xó, offset correction factor c, and noisepower spectral density N0,i. First, we assumed that N0,i, i =1, . . . ,n, were identically given as −174 dBm/Hz, the noisepower spectral density at the room temperature. For the otherchannel parameters PLd0 , k, Xó, and c, we consider two setsof values, given in Table I, that represent suburban and urbanscenarios [22].The system design parameters considered in this section arethe number of nodes n, simulation area A, error threshold áô,packet error function f , and maximum transmit power Pmax.Parameters n and A are closely related to the node density,which determines the number of nodes participating in thecooperation. Therefore, we varied n and A to observe how theperformance is influenced by the node density. The choice oferror function f depends on the error correction coding schemeemployed. In this study, we assume that a convolutional codewith a constraint length of two is used as the error correctioncoding scheme with a packet length of 1,024 [38]. Hence, weused the actual packet error rate obtained through extensivesimulations with the aforementioned convolutional code for thepacket error function f . For the choice of áô, we used 10−2,a value often adopted as the target packet error rate in manysituations. Finally, we assumed that the node power Pi is limitedby Pmax = 250 mW, and Pi is uniformly distributed over a10 MHz bandwidth. Detailed values of the above channel andsystem parameters are summarized in Table I.B. ConnectivityTo compare the level of performance achievable with theproposed topology control schemes, we first consider a metriccalled connectivity to measure the average proportion of nodesconnected to a node. Before proceeding with the formal definitionof metric connectivity, we observe that the performance ofa given topology control scheme depends not only on the valuesof n and A but also on the distribution of these n nodes over areaA. For this reason, we assume that n(≥ 2) nodes are randomlyand uniformly distributed over a given area A in the followingdiscussion.To formally define connectivity, we first denote the set ofall nodes connected to node vi by R(vi). We note that the setR(vi) depends on the choice of topology control schemes. Forinstance, in the NNTC scheme, R(vi) is the set of all nodesconnected to vi by an N-N edge. On the other hand, in acooperative topology control scheme, R(vi) consists of all thenodes that are connected through cascaded N-N and cascadedcooperative edges. Therefore, the connectivity à (of a giventopology control scheme) is defined asà =1nE_nÓi=1|R(vi)|n−1, (13)where |R(vi)| denotes the cardinality of R(vi). Here, the expectationE[·] has been taken because the cardinality |R(vi)|depends on how the nodes are distributed over a given area.We note that R(vi)/(n − 1) is the proportion of nodes thatare connected to vi and hence à is the expected value of itsarithmetic mean. For notational convenience, the connectivitiesof CCTC, NCTC, NNTC, and CNTC schemes are denoted byÃCC, ÃNC, ÃNN, and ÃCN, respectively.In Fig. 4, the connectivity for various topology controlschemes is shown as a function of the number of nodes nfor three different areas and two different environments. Mostimportantly, we observe that ÃCC ≥ ÃCN ≥ ÃNC ≥ ÃNN forall values of n and A and for any environment considered.We clearly see that either transmitter or receiver cooperationimproves connectivity. The fact that the CCTC scheme achievesthe highest connectivity is hardly surprising, hence what weactually need to observe is how the NCTC and CNTC schemes1866 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 4, APRIL 2015Fig. 4. Connectivity as a function of the number of nodes for various topology control schemes in various communication environments. (a) Urban. (b) Suburban.perform in comparison to it. In particular, since ÃCN ≤ ÃNC,we conclude that transmitter cooperation is more effective thanreceiver cooperation at achieving connectivity.C. Power ConsumptionSo far, we have observed that the CCTC scheme achievesthe highest connectivity and that the connectivity gap betweenthe CNTC and CCTC schemes is not large. In fact, it is notmore than 8% in most cases. Consequently, it is possible tosay that the CNTC scheme is a good alternative to the CCTCscheme if we consider connectivity only. However, the CNTCscheme is not as efficient as the CCTC scheme in terms ofpower consumption. Before proceeding with the analysis ofpower consumption, we define ˆ ECC to be the set of cluster pairscorresponding to the edges in ECC. In other words, (Ùl ,Ùm) ∈ˆ ECC, if and only if the edge set ECC contains the C-C edgebetween Ùl and Ùm. In a similar way, we denote the sets of thecluster pairs corresponding to edges in ENC and ECN by ˆ ENCand ˆ ECN, respectively.To quantitatively compare the power consumption of theCCTC and CNTC schemes, we now consider the following twoquantities¯PCC =1nE⎡⎣ Óð∈ˆECC∩ˆECNPCC(ð)⎤⎦ (14)and¯PCN =1nE⎡⎣ Óð∈ˆECC∩ˆECNPCN(ð)⎤⎦, (15)where PCN(ð) denotes the minimum C-N round-trip powerbetween the pair ð of clusters, similarly to PCC(ð) and PNC(ð)as defined in Section IV.We note that these quantities representthe average power required per each node to establish cooperativeedges between clusters in ˆ ECC ∩ ˆ ECN. Consequently, bycomparing ¯PCC and ¯PCN, we intend to compare the power requiredfor the CCTC and CNTC schemes to establish commoncooperative edges.Before proceeding with the evaluation of ¯PCC and ¯PCN, wefirst note that the two sets ˆ ECC− ˆ ECN and ˆ ECN − ˆ ECC of clusterpairs are not necessarily empty. Because the CCTC schemeemploys receiver cooperation in addition to transmitter cooperation,it appears reasonable to expect ˆ ECC − ˆ ECN to containsome sizable number of cooperative edges and ˆ ENC − ˆ ECC tobe empty. In fact, the average number of elements in ˆ ECC −ˆ ECN reaches as much as 25% of that of ˆ ECC ∩ˆ ECN in manysituations. However, interestingly, ˆ ECN − ˆ ECC is not necessarilyempty. This is because of the employment of MSF algorithm,that removes some redundant links. In other words, in CCTCschemes, some links used in the CNTC scheme are eliminatedby applying the MSF algorithms in some rare situations. Fromour numerical analysis, we found that the average cardinalityof ˆ ECN − ˆ ECC sometimes reaches as much as 8% of that ofˆ ECC∩ ˆ ECN. However, in most cases, the set ˆ ECN− ˆ ECC is emptyand hence ˆ ECC ∩ ˆ ECN is the same as ˆ ECN.Fig. 5(a) illustrates how the values of ¯PCC and ¯PCN changeas a function of the number of nodes n. We note that ¯PCCfirst increases as n increases and then decreases after n reachesa certain value. A similar tendency can be found in ¯PCN. Toexplain this non-monotonic performance of ¯PCC and ¯PCN, wedefine two quantitiesFCC =E_Óð∈ˆECC∩ˆECNPCC(ð)_E_|ˆ ECC ∩ ˆ ECN|_ (16)andFCN =E_Óð∈ˆECC∩ˆECNPCN(ð)_E_|ˆ ECC ∩ ˆ ECN|_ , (17)MOON et al.: RECEIVER COOPERATION IN TOPOLOGY CONTROL FOR WIRELESS AD-HOC NETWORKS 1867Fig. 5. The average additional power required per each node to establish cooperative edges in CCTC and CNTC schemes. (a) ¯PCN and ¯PCC. (b) ¯PCN over ¯PCC.to describe the average power consumed to establish a C-C linkand a C-N link, respectively. As a result, ¯PCC and ¯PCN can berewritten as¯PCC =1n·FCC ·N (18)and¯PCN =1n·FCN ·N , (19)where N = E[|ˆ ECC ∩ ˆ ECN|].While we cannot provide fully analytical behaviors of thequantities ¯PCC and ¯PCN, which is very difficult, it will be meaningfulto consider their qualitative behaviors. First, we note thatthe quantities FCC and FCN are mainly affected by the distancebetween clusters. It is natural to expect that the average clusterto-cluster distance will decrease with an increased number ofnodes n. However, the average cluster-to-cluster distance decreasesas a very slowly varying function of n, particularly aftern reaches a certain critical value. This is because two clustersare merged into one if the distance between them becomes tooclose. As a consequence, FCC and FCN decrease very slowly asn increases. For example, the minimum observed value of FCCwas only about 25% lower than the maximum observed value inthe simulation performed for an urban 2 × 2 km situation wheren ranged from 10 to 100. Because the quantities FCC and FCNare relatively unaffected by the variation of n, the behaviors of¯PCC and ¯PCN can possibly be accounted for by the behaviors ofthe average number of elementsN in ˆ ECC∩ ˆ ECN, which, in fact,varies very significantly as n varies. Let us observe, when thenode density is sufficiently low, that N increases as n increases,since increased n results in an increased number of clustersand then in an increased number of edges. However, when thenode density is high enough, adding nodes no longer makes thenumber of clusters larger because the addition of nodes nowresults in cluster unification. For this reason, N first increasesup to a certain critical value of n and then decreases againas n grows further. However, it is very difficult to predict thebehavior of N in a fully analytical manner, since N depends ontoo many factors such as node distribution, channel and fadingmodels, error probability function, and so on. As far as weknow, only a few analytical results [35], [36] have been derivedfor non-cooperative communications with an infinite number ofnodes and none for general cases or cooperative environments.We now discuss the simulation results of comparing ¯PCC and¯PCN. Because FCC and FCN vary slowly as functions of n, thevariations of ¯PCC and ¯PCN are dominantly determined by 1/nand N . When n = 10, N is almost zero since a very smallnumber of clusters exist and they are located too far away.As n increases up to a certain value, the number of clustersincreases so that the chance of cooperative communication alsoincreases. In this region, N grows faster than n, therefore, ¯PCCbecomes larger. On the other hand, if n exceeds a certain value,the number of clusters decreases, and eventually, it goes to one.Therefore, N quickly converges to zero with growing n, andthis is why ¯PCC decreases. In Fig. 5(a), we next observe that ¯PCCis always smaller than ¯PCN. To quantify the difference betweenthe two values, we illustrate the values of ¯PCN/¯PCC in Fig. 5(b),where we clearly see that ¯PCN is about 10–100% larger than¯PCC. From this figure, we clearly see that the CCTC schemerequires significantly less power than the CNTC scheme toestablish the same cooperative edges.Here, the question arises as to how the NCTC schemecompares to the CCTC scheme in terms of power consumption.First, we can compare the amount of power required for theCCTC and NCTC schemes to establish common cooperativeedges. In a similar comparison in Fig. 5, we noted that ¯PCCis significantly smaller than ¯PCN. However, in the case of theCCTC and NCTC schemes, there is virtually no differencebetween the powers required to establish common cooperativeedges. This is related to the assumption that the nodesparticipating in the cooperative transmission use the same1868 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 4, APRIL 2015Fig. 6. The relative amount of power required to establish one more additional cooperative edge with the CCTC scheme in comparison with the NCTC andCNTC schemes. (a) Urban. (b) Suburban.transmission power as in CCTC scheme. Because of this constrainton the transmission power, only one node is selected,even in the CCTC scheme, to transmit signals almost alwayswhenever the cooperative edge is contained in both ˆ ECC andˆ ENC. Therefore, it can be said that the NCTC scheme is almostas efficient as the CCTC scheme in terms of power consumption.Consequently, if the connectivity is of less priority thanthe power consumption or if the situation is such that theconnectivities of CCTC and NCTC are almost the same valuesbecause of a very high node density, the NCTC scheme can beconsidered to be a good alternative to the CCTC scheme. This isparticularly so because the average power required to establisha cooperative edge in ˆ ECC− ˆ ENC is significantly larger, in manycases, than the power required to establish cooperative edgein ˆ ENC.To illustrate this, we consider the metric ñCCNC defined asñCCNC =DCCNCKNC(20)in whichDCCNC =E_Óð∈ˆECC−ˆENCPCC(ð)_E_|ˆ ECC − ˆ ENC|_ (21)andKNC =E_Óð∈ˆENCPNC(ð)_E_|ˆ ENC|_ . (22)We note that DCCNC denotes the power required to establish oneC-C link that can not be established in NCTC scheme andthat KCCNC is the power consumption required for one N-C link.Consequently, the metric ñCCNC measures the relative amount ofpower required to establish one more additional cooperativeedge using the CCTC scheme in comparison to the NCTCscheme. In a similar manner, we define the metric ñCCCN byñCCCN =E_Óð∈ˆECC−ˆECNPCC(ð)_E_|ˆ ECC − ˆ ECN|_ ÷E_Óð∈ˆECNPCN(ð)_E_|ˆ ECN|_ (23)=DCCCNKCN(24)to quantify the relative amount of power required to establishone more additional cooperative edge using the CCTC schemein comparison to the CNTC scheme.In Fig. 6, we plot ñCCNC and ñCCCN as functions of n. Here,we first observe that the numerical values of ñCCNC and ñCCCNare around 3 and 1.2, respectively, for all cases considered.We note that, as mentioned in the explanation of Fig. 5, thepower consumed to establish a single cooperative link decreaseswith growing n so that DCCNC, DCCCN, KNC, and KCN are alldecreasing functions of n. In addition, we note that the powerrequired to establish a cooperative link is mainly affected bythe number of transmitting nodes and the transmitting powerof each node. We also note that the cooperative link betweentwo clusters is established by only a small number of nodeslocated near the boundary of each cluster, even when the clustersize is very large. This means that the number of transmittingnodes is almost constant, regardless of n. Therefore, the rateof decreasing power consumption is primarily affected by thetransmitting power of each node, which is closely related to thedistance between clusters. Because the configuration of clustersis identically given by the NNTC scheme, as n increases, thedecreasing rate of the power required to establish cooperativelinks is relatively similar for all three cooperative schemes,namely, the NCTC, CNTC, and CCTC schemes. For thisMOON et al.: RECEIVER COOPERATION IN TOPOLOGY CONTROL FOR WIRELESS AD-HOC NETWORKS 1869reason, the ratios DCCNC/KNC and DCCCN/KCN remain roughly thesame regardless of the value of n.We next observe that the values of ñCCNC, plotted by solid purplelines, are always around three. This means that to establishan edge that cannot be established in the NCTC scheme, theCCTC scheme requires about three times the power requiredto establish an edge in the NCTC scheme, regardless of thescenario and node density considered. Combining this resultwith the connectivity result in Fig. 4, we gain an importantinsight into the system design. When n = 50, the connectivityof the CCTC scheme is almost twice that of the NCTC scheme.Therefore, a three-fold increase in power consumption could bea reasonable choice if connectivity is of the highest priority.However, when n = 100, by employing the CCTC scheme,one would achieve 0.13% increase in connectivity, but threetimes more power would still be required. Therefore, somesystem designers may prefer the NCTC scheme to the CCTCscheme, for instance, where power efficiency is of the highestpriority or connectivity increase is not an issue. In contrast,ñCCCN, plotted by dotted by the green line, is about 1.2 in allcases. This means that only 20% more power is required to adda new cooperative edge using the CCTC scheme that cannotbe established in the CNTC scheme. Consequently, one canreplace the CNTC scheme with the CCTC scheme without aserious power consumption burden, regardless of node density.VI. CONCLUSIONIn this paper, we proposed to employ receiver cooperationin topology control to improve energy efficiency as well asnetwork connectivity. In particular, we proposed two centralizedtopology control schemes, one based solely on receivercooperation, and the other based both on transmitter and receivercooperations. For comparison, we also considered atopology control scheme that is based solely on transmittercooperation. By extensive simulation, we showed that we canimprove both connectivity and energy efficiency if we employreceiver cooperation in addition to transmitter cooperation.Consequently, it is generally more desirable to employ bothreceiver and transmitter cooperation than to employ transmittercooperation only. We also showed that the increase in networkconnectivity by employing transmitter cooperation in additionto receiver cooperation is at the expense of significantly increasedenergy consumption. For this reason, we conclude thatthe system based only on receiver cooperation could prove to bea good alternative to one based both on receiver and transmittercooperation, if energy efficiency is of the highest priority or theincrease in connectivity is no longer of serious concern.