The effect of hexagonal grid topology on wireless communication networks based on network coding

INTERNATIONAL JOURNAL OF COMMUNICATION SYSTEMSInt. J. Commun. Syst. 2014; 27:1319–1337Published online 25 March 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/dac.2782

The effect of hexagonal grid topology on wireless communicationnetworks based on network coding

Tao Shang*,†, Fu-Hua Huang, Ke-Fei Mao and Jian-Wei Liu

School of Electronic and Information Engineering, Beihang University, Beijing, China

SUMMARY

Network coding has emerged as a promising technology that can provide significant improvements in theperformance of wireless communication networks. Inspired by the recent advances in complex networks, weherein study the effect of topology structure on the performance benefit of network coding based on a highlystructured wireless communication network with hexagonal grid topology. We propose a new concept called‘network intensity’, namely � to characterize the property of hexagonal grid topology. We first derive a valueof 12=7 for the upper bound on performance benefit in a single regular hexagon network. This value holdsonly if the network intensity is in the case of

p3 6 � < 2. Based on the results in a single regular hexagon

network, we then derive a value of 16=7 for the upper bound on performance benefit in a general hexagonalgrid network only if the network intensity is in the case of 16 � <

p3. Furthermore, a comparative analysis

demonstrates that the network intensity affects the performance benefit of network coding in terms of theinterference and the coding number. These findings will contribute to the design of network topology and theanalysis of the bound on the performance benefit of network coding in wireless communication networks.Copyright © 2014 John Wiley & Sons, Ltd.

Received 8 September 2013; Revised 18 February 2014; Accepted 25 February 2014

KEY WORDS: network coding; complex networks; hexagonal grid; performance benefit; networkintensity; upper bound

1. INTRODUCTION

Network coding has been proved to be a promising technology for communication networks, and itcan increase throughput, reduce delay, improve robustness and improve load balance [1–6]. Espe-cially, the flooding and broadcasting peculiarity makes it more suitable for wireless communicationnetworks. As a paradigm, the network coding scheme Complete OPportunity Encoding (COPE)proposed in [7], which is the first practical network coding protocol in multihop wireless networks,makes use of broadcast channel by overhearing native packets and broadcasting encoded packets toexecute the network coding. Network coding has superb prospects in wireless communication net-works, and thus its essential performance benefit to wireless communication networks is a stronglydesirable to clarify.

Research on complex networks helps achieve better understanding of communication networks.Complex networks can be characterized by large-scale topologies, distributed resource manage-ment, extreme heterogeneity of the constituent elements, high clustering coefficients, and so on,which are highly correlated to wireless communication networks such as mobile ad-hoc network,wireless sensor network, wireless mesh network or WiMAX network. Further analysis of wirelesscommunication network paradigms in the context of complex networks will be very useful to net-work design. The key challenge is how to impose it on wireless communication networks. In the

*Correspondence to: Tao Shang, School of Electronic and Information Engineering, Beihang University, Beijing, China.†E-mail: [email protected]

Copyright © 2014 John Wiley & Sons, Ltd.

T. SHANG ET AL.

past few years, many researchers [8] studied the structure and function of complex networks, andthey have increasingly recognized that the characterization and the modeling of the structure of anetwork would lead to a better knowledge of its dynamical and functional behavior. Furthermore,the structural complexity of a network can be influenced from both node and connection diversity.Meanwhile, the network topology plays a crucial role in determining the emergence of collectivedynamical behavior, such as synchronization, or in governing the main features of relevant processesthat take place in complex networks, such as the spreading of information. Apparently, it remains achallenge to answer the fundamental question, that is, ‘How does topology structure affect wirelesscommunication networks based on network coding? ’ or ‘Exactly how many performance benefitscan be expected from network coding in the context of different topology structures of wirelesscommunication networks?’.

The studies in [9, 10] show that the performance benefit of network coding depends highly onthe topology structure of multihop wireless networks. For instance, in Figure 1(a), a maximum offour packets can be encoded by an intermediate node within the ‘cross-style’ topology structure.By comparison, in Figure 1(b), a maximum of six packets can be encoded by an intermediate nodewithin the ‘hexagon-style’ topology structure, which is three-halves times compared with the formercase, and the details are described in Section 2.1. This phenomenon clearly shows that the topol-ogy, especially the basic unit of topology, significantly affects the performance benefit of networkcoding. By taking the topology constraints into consideration, Le et al. [11] proved that for a gen-eral wireless network structure, the performance benefit using the XOR coding scheme (COPE orCOPE-like scheme) is upper bounded by 2n=.nC 1/ , where n denotes the maximum number ofpackets, which can be encoded within one single coding structure. This result illustrates that thetopology can affect the performance benefit of network coding by changing the value of n.

To seamlessly cover a full plane with regular polygons, there are three possible choices for thestructural property of network topology, including regular triangle, square and regular hexagon. Anetwork with hexagonal grid topology the basic unit of which is a regular hexagon is an optimaltwo-dimensional (2D) wireless communication network. This configuration can cover the geometri-cal range with the least number of nodes. In other words, if circles are used to cover a 2D plane withidentical radius, their centers should coincide with the center of each hexagon in the hexagonal grid[12] in order to reach the largest geometrical range. A hexagonal grid network can be extensivelyapplied in wireless communication networks. By means of network deployment and topology con-trol, the hexagonal grid network facilitates the implementation of geometrical coverage of a certainarea with the least number of nodes and provides a QoS approach for multiclass traffic wirelesscommunication networks [13,14]. Especially, the intrinsic characteristic of hexagonal grid topologymakes it easy for the clustering management of nodes and hierarchical processing of a network. Ifthe central node of each hexagon is treated to be a cluster header, intracluster session can be imple-mented in a hexagonal structure unit. If a hexagonal grid structure unit is located at the center ofsix surrounding structure units, the central node of the hexagonal grid structure unit can be treated

Figure 1. Difference of topology structures.

Copyright © 2014 John Wiley & Sons, Ltd. Int. J. Commun. Syst. 2014; 27:1319–1337DOI: 10.1002/dac

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THE EFFECT OF HEXAGONAL GRID TOPOLOGY ON NETWORK CODING

to be a higher level cluster header, and the total nodes of the hexagonal grid structure unit canbe treated to be relay nodes so as to implement intercluster communications. Furthermore, for thecurrent construction of wireless metropolitan, based on current wireless infrastructure, newly-builtwired and wireless base stations can be arranged according to hexagonal grid topology so that itcan be deployed to optimize wireless coverage and provide the maximum-throughput transmissionperformance with the economical minimum-node infrastructure.

Until now, there is still no explicit conclusion about the performance benefit of network codingin a network with hexagonal grid topology. We herein attempt to derive the upper bound on theperformance benefit of network coding in hexagonal grid topology and to generalize the resultsto the other types of topology. In addition, the highly structured geometrical shape of hexagonalgrid is also helpful to analyze the upper bound and provide the ground for the quantification of therelationship between the topology structure and the performance benefit of network coding.

In order to analyze the effect of topology structure on the performance benefit of network coding,we herein first propose a quantitative network parameter called ‘network intensity’, namely � tocharacterize the connectivity between nodes or the density of a network, we will discuss ‘networkintensity’ in details in Section 3. On one hand, we quantify the exact relationship between the net-work intensity � and the performance benefit of network coding, that is, we study how exactly thetopology structure affects the performance benefit of network coding. On the other hand, we try toobtain a tightest upper bound on the performance benefit of network coding in the multihop wirelessnetwork of hexagonal grid topology on the basis of Le’s work.

As the main contributions of our work,

(1) we introduced the network intensity � to characterize the property of topology structure andto quantify the exact relationship between the network intensity � and the upper bound on theperformance benefit of network coding.

(2) We derived the tightest upper bound on the performance benefit of network coding in a singleregular hexagon network for now, and we found that the exact upper bound is 12=7 only ifp36 � < 2 is satisfied.

(3) Furthermore, we derived the tightest upper bound on the performance benefit of network cod-ing in a general hexagonal grid network, and we found that the exact upper bound is 16=7 onlyif 16 � <

p3 is satisfied.

(4) We verified that the topology structure described in this way can certainly affect the per-formance benefit of network coding, and we concluded that the compensative effect of theinterference and the coding number results in the improvement of performance benefit inhexagonal grid topology.

This paper is structured as follows. Section 2 presents the main works related to this paper andhighlights the problems encountered in these works. Section 3 describes the network model in detailsand the definition of network intensity. Section 4 discusses the relationship between network inten-sity and performance benefit and derives the upper bounds on performance benefit in a single regularhexagon network under different cases. Section 5 extends the upper bounds from a single regularhexagon network to a general hexagonal grid network. Finally, Section 6 concludes the paper.

2. RELATED WORKS

2.1. Complete opportunity encoding scheme

Here, we describe the working mechanism of COPE and take the typical scenario in Figure 2 as anexample. COPE incorporates three main techniques as follows [7]:(a) Opportunistic listening: Wireless is a broadcast medium, creating many opportunities for nodesto overhear packets when they are equipped with omnidirectional antenna. COPE sets the nodes inpromiscuous mode, makes them snoop on all communications over the wireless medium and storethe overheard packets for a limited period.


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Figure 2. Typical topology of wireless network coding.

In addition, each node broadcasts reception reports to tell its neighbors which packets it hasstored. Reception reports are sent by annotating the packets the node transmits. A node that has nopackets to transmit periodically sends the reception reports in special control packets.

In Figure 2, let us suppose that the node S1 has a packet a to transmit to the node D1 via theintermediate node C and that the node S2 has a packet b to transmit to the node D2 via the node C .First, the node S1.S2/ transmits the packet a.b/ to C . Using the opportunistic listening, D2.D1/overhears the packet a.b/. And thenD2.D1/ broadcasts reception reports to tell C that it has storedthe packet a.b/.(b) Opportunistic coding: Packets from multiple unicast flows may have encoded together at someintermediate hop. The nodes that perform encoding should aim to maximize the number of nativepackets delivered in a single transmission, while ensuring that each intended nexthop has enoughinformation to decode its native packet. This can be achieved using the following simple rule: totransmit n packets, p1, : : : ,pn, to n nexthops, r1, : : : , rn, a node can XOR the n packets togetheronly if each nexthop ri has all n� 1 packets pj for j ¤ i .

In Figure 2, C knows that D2.D1/ has stored the packet a.b/ via D2.D1/’s reception reports;therefore, the rule is satisfied. Then C XORs a and b, and broadcasts the encoded packet a˚b toD1 and D2. After receiving a˚b, D2.D1/ makes use of the overheard packet a.b/ to decode thepacket a˚b. The decoding processes in D1 and D2 are executed as aD a˚b˚b and b D a˚b˚a,then D1.D2/ receive the packet a.b/.(c) Learning neighbor state: Besides the reception report, COPE provides another method for nodesto know what packets its neighbors have. This method is that COPE estimates the probability that aparticular neighbor has a packet as the delivery probability of the link between the packet’s previoushop and the neighbor.

In Figure 2, C may guess that D2.D1/ has stored the packet a˚b for the delivery probability ofthe link between S1.S2/ and D2.D1/ is high enough.

Now, let us explain the situation in Figure 1. In Figure 1(a), by using COPE, the intermediatenode can encode a maximum of four packets when there are four data flows crossing it at the sametime. Similarly, in Figure 1(b), by means of COPE, the intermediate node can encode a maximumof six packets when there are six data flows crossing it at the same time.

In conclusion, COPE is executed in a multihop wireless network by overhearing and broad-casting packets. The characteristic of overhearing generates coding opportunities that can benefitboth encoding and decoding, and reduce the times of packets transmission, thus improving networkthroughput. In the example, S1 and S2 transmitting the packets a and b to D1 and D2 need threetransmissions with network coding compared with four transmissions without network coding (forC should send a and b separately). The times of packet transmissions decrease when using networkcoding, thus improving the network throughput.

2.2. Performance benefit

In general, performance benefit is defined as the ratio of performance with and without networkcoding. It evaluates the performance improvement for the employment of network coding.

Liu et al. [15] were first to study the upper bound on the performance benefit of network coding ina multihop wireless network. They found that for arbitrary network coding in 2D random topology,the upper bound on performance benefit is 2c

p�.1 C �/=� for large n, where � > 0 is a


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parameter of the wireless medium that characterizes the intensity of interference between nodes,n is the number of nodes in the network, and c Dmax¹2,

p�2C 2�º. This conclusion contributes

to the analysis of relationship between performance benefit and network intensity. However, thederived upper bound is not tight enough to characterize a practical performance benefit. In addition,the effect of topology structure on performance benefit is not considered as well. If the topologystructure was to be considered, a tighter bound would have been achieved.

Goseling et al. [9] studied the bound on the performance benefit of network coding from theviewpoint of energy consumption. They found that if coding and noncoding solutions use the sametransmission range, the benefit in d -dimensional network is at least 2d=b

pdc. Moreover, if the

transmission range can be optimized for coding and noncoding individually, the benefit in 2D net-works is at least 3. Meanwhile, a constructive coding scheme is established in a hexagonal gridnetwork and quadrilateral grid network, respectively, but the coding scheme is too theoretical torepresent general practical cases.

Koutsonikolas’s empirical study [10] shows that the performance benefit of network coding in amoderate-size multihop wireless network for general traffic patterns is extremely limited. The per-formance of a noncoding scheme even outperforms the performance of a coding scheme. Empiricalexperiments were conducted in a sparse network with quadrilateral grid topology and one-hop trans-mission range, as shown in Figure 3. In fact, such configuration restricted the performance benefit ofnetwork coding. Considering the scenario in Figure 3, the coding opportunities emerge only whenthere are two opposite flows traversing the same node simultaneously, and even under this condition,the number of encoded packets is just 2, which is the least in all feasible coding schemes. This isdifferent from the original results reported in [7], where in a random network and with a randomtraffic pattern, the original COPE protocol offers a three to four times throughput improvement overa noncoding scheme. The significant gain in [7] is due to the small (20 nodes) and dense network itused. Such a configuration offers high overhearing probabilities, giving nodes chances to XOR morethan two packets together. In contrast, the evaluation in [10] was conducted in a larger and sparsernetwork where the overhearing cannot bring any benefit. Meanwhile, the parameter � in [15] alsoimplies the influence of the network intensity on performance benefit to a certain extent. The formerresults show that the network intensity greatly affects the performance benefit of network coding.Thus the question of how the network intensity exactly affects the performance benefit of networkcoding is still open.

Le et al. [16] proposed the concept of ‘coding number’, which is equal to the number of packetsthat can be encoded in a single ‘coding structure’. Le pointed out that all the coding opportunitiesare within one single coding structure and that the performance benefit in one coding structure isassociated with the coding number. The upper bound on the coding number in any possible codingstructure is O..r=ı/2/ in a 2D random wireless network, where r is the transmission range, andı is the transmission gap. This result is contradictive with the result in [7], which indicates thatthe coding number can be infinite. In fact, as described in [10], there exist geometrical constraintswithin a multihop wireless network, that is, the network intensity can affect the performance benefit

Figure 3. Quadrilateral grid topology.


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of network coding. By taking the geometrical constraints into consideration, Le et al. [11] provedthat for a general wireless network, the upper bound on the performance benefit by using the XOR-style coding scheme is 2n=.nC1/ , where n denotes the maximum coding number within one singlecoding structure. This upper bound is tighter compared with the one in [15], but the drawback is thatthis upper bound is the same with the one in a general network with the interference of uncorrelatednodes, which implies that an actual upper bound could be tighter.

3. NETWORK MODEL

3.1. Description of network model

As shown in Figure 4, the topology with a hexagonal grid is optimal in a 2D plane. It can cover thegeometrical range with the least number of nodes. Our study on the performance benefit of networkcoding in a multihop wireless network is based on hexagonal grid topology. The basic structure unitof hexagonal grid topology shown in Figure 5 is a single regular hexagon network, which consistsof six vertices and one central node.

As shown in Figure 5, we studied the multiple unicast traffic pattern as follows: each sourcenode, denoted as Si .i D 1, : : : , 6/, has packets to deliver to its opposite destination node, which isDi .i D 1, : : : , 6/. C denotes the intermediate node. We schedule the source packets transmissionin order to maximize the throughput with or without network coding. We assume that the nodes Siand Di cannot communicate with each other directly, and they have to use the intermediate nodeC to relay the communication. Provided that j � j denotes the Euclid distance and jSDj denotes thedistance between the node S and the node D, we assume that the node D successfully receives

Figure 4. Hexagonal grid topology.

Figure 5. Single regular hexagon network.


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the packets originating from the node S with the probability of 100% if r > jSDj and with theprobability of 0% if r < jSDj.

We consider a stationary wireless network, because dynamic node will destroy the validity ofopportunistic coding of COPE. The static hexagonal grid can offer good characteristics to make fulluse of the performance of COPE. We also assume all nodes in the network have identical trans-mission range. This configuration can cover the geometrical range with the least number of nodes.Practically, there are many types of multihop wireless network the nodes of which are stationaryand have identical transmission range, such as the mesh router of wireless mesh networks and thesensor node of wireless sensor networks.

The essential limit to network throughput is the shared physical channel within a single reg-ular hexagon network. All seven participant nodes are within one shared wireless channel. Weassume that the capacity of the shared channel is 1, and the bandwidth resource is time-sharedsuch as distributed coordination function in 802.11. Only one node is allowed to send data in atime slot. Consequently, there exists no transmission collision in a neighbor scope, that is, thereare no lost packets in a neighbor scope. Thus such assumption is helpful to the analysis of theupper bound on performance benefit. Meanwhile, because the half-duplex mode is easier to cal-culate performance benefit than the full-duplex mode, we assume that nodes operate at half-duplex mode.

We use COPE as a representative protocol and expect to generalize the results of COPE to othersimilar protocols [17–23]. We show how COPE or COPE-like coding scheme works under the traf-fic pattern described previously in a hexagonal grid network. When each source node Si sends apacket ai , all other nodes except for opposite nodeDi should be able to overhear this packet. WhenC receives all six packets from Si .i D 1, : : : , 6/, it then XORs these packets and broadcasts themas a single packet, denoted as the following: a1 ˚ a2 ˚ a3 ˚ a4 ˚ a5 ˚ a6. All the six nodesDi .i D 1, : : : , 6/ receive the broadcast packet and decode it to obtain the target packet by means ofthe overheard packets, that is, ai D .a1˚a2˚a3˚a4˚a5˚a6/˚.a1˚: : :˚ai�1˚aiC1˚: : :˚a6/.So far, the multiple unicast traffic has been fulfilled by the coding scheme.

Thus, in our network model, we give the following assumptions:

(1) The hexagonal grid is static.(2) All nodes in the network have identical transmission range.(3) The communication mode between nodes is half-duplex.(4) The bandwidth resource is time-shared.(5) The COPE or COPE-like scheme is used as typical coding scheme.

On the other hand, we define the coding number nc as the number of packets, which can beencoded by the central node C at one time. With COPE or COPE-like coding scheme, the codingnumber in a single regular hexagon network achieves a maximum value 6 when all six source nodessend packets to its opposite destination nodes.

Let Ti .i D 1, : : : , 6/ denote the end-to-end throughput of the data flow from the source nodeSi .i D 1, : : : , 6/ to the destination node Di .i D 1, : : : , 6/, which is equal to the incoming data ofDi .i D 1, : : : , 6/. Then we define the throughput of a network as the sum of Ti .i D 1, : : : , 6/, that

is, T D�P6

iD1 Ti

�.

Let Tc and Tnc denote the throughput with coding scheme and noncoding scheme under sametopology structure and network intensity, respectively. Let T �c and T �nc denote the maximum of Tcand Tnc . Then we define the performance benefit of network coding as Tb D Tc=T �nc to evaluate thethroughput improved by network coding. Thus the upper bound on performance benefit is definedas T �

bD T �c =T

�nc .

3.2. Definition of network intensity

As we know, network topology can affect the performance benefit of network coding. Here, we pro-pose a new concept called ‘network intensity’ to quantify the effect of hexagonal grid topology onthe performance benefit of network coding.


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Definition 1‘Network intensity’ is defined by � D r=R , where R denotes the distance between two neighbornodes in a hexagonal grid, and r denotes the transmission range of a node, which considers theeffect of both the transmission range and the density of nodes, as shown in Figure 6.

Network density generally denotes the number of nodes in a certain transmission range. Becauseit does not consider the topology structure and opportunistic coding, the concept of network den-sity is insufficient to characterize the hexagonal grid topology. Instead of network density, networkintensity combines the factor of distance between two neighbor nodes with the factor of transmis-sion range. Furthermore, the factor of the distance between two neighbor nodes can be related toopportunistic coding. Thus network intensity can describe the relationship of acquiring encodingopportunity and evaluate how network topology affects the performance benefit of network coding.Specifically, the performance benefit depends on the coding number nc , and the coding numberfurther depends on network intensity.

For hexagonal grid topology, R is isometric. When r is different, the coding number is alsodifferent due to the opportunistic listening ability. The smaller the value of � , the smaller thecoding number, just like the strength of wireless signal. Such phenomenon leads to the concept ofnetwork intensity.

Note that the network intensity is a general concept, and it can be expanded to general wire-less networks. For a wireless network with nodes randomly distributed, if the network intensity islarger, the coverage is relatively larger. That mean that there are more neighbor nodes receiving thetransmitted packets, and the coding number is also larger.

According to this definition, the network intensity actually describe the relative relationshipbetween R and r . Virtually, it is a precise measurement of the coding number, and it refines theessential characteristics of network topology. The network intensity in different ranges will leadto different coding number, which is an important property of the network intensity. Theorem 1describes this property in details.

Theorem 1The coding number in a single regular hexagon network is nc D 6 and nc D 2 when

p3 6 � < 2

and 16 � <p3 are satisfied, respectively.

ProofIn a single regular hexagon network with COPE or COPE-like coding scheme, there exist geomet-rical constraints for the coding scheme to achieve a maximum throughput. After the central nodeC broadcasts the encoded packet, all six vertices Di .i D 1, : : : , 6/ should be able to decode theencoded packet, which means that each transmission of the source node Si .i D 1, : : : , 6/ shouldbe overheard by the vertices Dj .j D 1, : : : , 6, j ¤ i/, but cannot be overheard by the vertex Di ,

Figure 6. Network intensity � D r=R.


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Figure 7. The effect of � on nc .

Figure 8. Two types of topology.

just as shown in Figure 7. Each transmission of the source node Si .i D 1, : : : , 6/ will be over-heard by the destination node Dj .j D 1, : : : , 6, j ¤ i/ if r >

p3R. Moreover, each transmission

of Si .i D 1, : : : , 6/ will not be overheard by the opposite node Di if r < 2R. In summary, ifp36 � < 2, then the central node C in COPE or COPE-like coding scheme will be able to encode

all six packets from Si .i D 1, : : : , 6/ and broadcast the encoded packet to Dj .j D 1, : : : , 6/, thatis, nc D 6. On the other hand, under the condition of 1 6 � <

p3, because of lack of over-

hearing opportunities, there only exist two kinds of feasible topology structure for coding in thesingle regular hexagon network: (1) ‘X-style’ topology, and (2) topology with two opposite flows. InFigure 8(a), both the node Si and the node Di send packets to the node C and store them in theirbuffers, the node C XORs the received packets and broadcasts the encoded ones; after receiving theencoded packet, Si and Di use the packet they store in the buffer to decode and obtain the targetpackets. Obviously, the number of the packets that C can encode is 2. The other case in Figure 8(b)has the same coding number as the scenario in Figure 8(a), which is also 2. So the coding numberis nc D 2 when 16 � <

p3.

Besides the constraints stated in Theorem 1, it is evident that there will be no communicationwhen � < 1 and communication will be excessive in the single regular hexagon network where it isnot necessary for the node C to be an intermediate node when � > 2.

In conclusion, the property of the network intensity is summarized as follows: the coding num-ber increases from nc D 2 to nc D 6 as the network intensity increases from 1 6 � <

p3 top

36 � < 2,nc D 0 when � < 1, and nc is undefined when � > 2. �

4. PERFORMANCE BENEFIT IN A SINGLE REGULAR HEXAGON NETWORK

As the basic unit of a hexagonal grid network, the single regular hexagon network is the placewhere the coding process of COPE or COPE-like protocols happens. In this section, we discuss theperformance benefit of network coding within a single regular hexagon network.

The main mathematical symbols used in this section are defined in Table I.The work in [11] illustrates that the coding number will affect the throughput of network coding

and provides two lemmas to calculate the values of T �c and T �nc in a circle topology with a central


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Table I. The definition of mathematical symbols.

Symbol Definition

� Network intensitync Coding numberTb The performance benefit of network codingT �b

The upper bound on the performance benefit˛i .i D 1, : : : , 6/ The draining rate of the source node Si .i D 1, : : : , 6/� The draining rate of the central node C

node and n nodes homogeneously distributed on the circumference. We extend the related conclu-sions to deduce the value of T �c and T �nc . As proved in Theorem 1, in a single regular hexagonnetwork, the coding number strongly depends on the network intensity, so we discuss the perfor-mance benefit of network coding in a single regular hexagon network under two different conditions:p36 � < 2 and 16 � <

p3.

4.1. Upper bound on performance benefit under the condition ofp36 � < 2

Lemma 1In a noncoding scheme, when

p36 � < 2, the maximum throughput T �nc is equal to 1=2 under the

condition that the sum of bandwidth allocated to six vertices Si .i D 1, : : : , 6/ should be the same asthe bandwidth allocated to the central node C .

ProofLet ˛i .i D 1, : : : , 6/ denote the draining rate of vertex Si .i D 1, : : : , 6/, and let � denote the drain-ing rate of the central node C . Because C is the bottleneck of the network, the total throughputis equal to �. Obviously we get � 6

P6iD1 ˛i . Because all seven nodes share the same channel,

we getP6iD1 ˛i C � 6 1. Therefore, the throughput reaches the maximum T �nc D 1=2 when

� DP6iD1 ˛i . �

Lemma 2In a coding scheme, when

p3 6 � < 2, the maximum throughput T �c is equal to 6=7 under the

condition that the transmission schedule follows some cyclic pattern such as S1,S2, : : : ,S6,C , andequal bandwidth is allocated to each node.

ProofThe proof is almost the same as that in Lemma 1, except that the total throughput is now equalto the node C ’s draining rate nc�. Similar to the proof of Lemma 1, we have nc� 6

P6iD1 ˛i

andP6iD1 ˛i C � 6 1. According to these two formulae, we obtain nc� 6 nc=.1 C nc/. Whenp

3 6 � < 2, the coding number is nc D 6. So the throughput reaches the maximum T �c D 6=7 on

the condition that � D�P6

iD1 ˛i

�=6. �

Theorem 2In a single regular hexagon network, when

p3 6 � < 2.nc D 6/, the upper bound on the

performance benefit of COPE or COPE-like coding scheme is equal to T �bD 12=7.

ProofFrom Lemma 1, Lemma 2 and the definition of performance benefit, we derive that when

p36 � <

2,T �bD T �c =T

�nc D 12=7. �

Example 1In Figure 5, we assume that Si .i D 1, : : : , 6/ wants to send packet pi .i D 1, : : : , 6/ to its oppo-site vertex Di .i D 1, : : : , 6/. The size of packet is one unit. Figure 9 shows how packets are


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Figure 9. An example without coding.

Figure 10. An example with coding.

transmitted to achieve the maximum throughput without coding in one cycle. In this scenario, sixpackets are transmitted in 12 time slots, thus the maximum throughput is 1=2. On the other hand,Figure 10 shows how packets are transmitted to achieve the maximum throughput with coding inone cycle. In this scenario, under the condition of

p3 6 � < 2, each node can overhear every

other node. Si .i D 1, : : : , 6/ first transmits pi .i D 1, : : : , 6/ to C , then C encoded six packets aspal l D p1 ˚ p2 ˚ p3 ˚ p4 ˚ p5 ˚ p6 and broadcasts pal l to all six nodes Sal l , which includesSi .i D 1, : : : , 6/. In this scenario, six packets are transmitted in seven time slots; thus the maximumthroughput is 6=7.

4.2. Upper bound on performance benefit under the condition of 16 � <p3

Lemma 3In a noncoding scheme, when 16 � <

p3, the maximum throughput T �nc is equal to 1=2 under the

condition that the sum of bandwidth allocated to six vertices Si .i D 1, : : : , 6/ should be the same asthe bandwidth allocated to the central node C .

ProofThe proof is the same as that in Lemma 1. �

Remark. In fact, Tnc can reach the maximum throughput as long as 16 � < 2.

Lemma 4In a coding scheme, when 16 � <

p3, the maximum throughput T �c is equal to 2=3 under the con-

dition that the transmission schedule follows some patterns such as S1,S4,C ,S2,S5,C ,S3,S6,C ,and the bandwidth allocated to the node C is half as much as the sum of bandwidth allocated toother nodes.

ProofThe proof is almost the same as that in Lemma 2, except that when 1 6 � <

p3; the coding

number is nc D 2. So the throughput reaches the maximum T �c D 2=3 under the condition of

� D�P6

iD1 ˛i

�=2. �

Theorem 3In a single regular hexagon network, when 1 6 � <

p3.nc D 2/, the upper bound on the

performance benefit of COPE or COPE-like coding scheme is equal to T �bD 4=3.


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ProofFrom Lemma 3, Lemma 4 and the definition of performance benefit, we derive that when 1 6 � <p3, T �

bD T �c =T

�nc D 4=3. �

Example 2In this example, the setting is the same as that in Example 1. The example of noncoding schemeunder the condition of 1 6 � <

p3 is the same as that in Example 1, which is shown in Figure 9.

Figure 11 shows how packets are transmitted to achieve the maximum throughput with coding inone cycle. In Figure 9, S14 denotes the set of S1,S4IS25 denotes the set of S2,S5IS36 denotes theset of S3,S6I andp14 D p1 ˚ p4,p25 D p2 ˚ p5,p36 D p3 ˚ p6. The pair of opposite nodestransmit packet to C , then C encodes two packets together and broadcast back to this pair of nodes.In this scenario, six packets are transmitted in nine time slots; thus the maximum throughput is 2=3.

In conclusion, as we can see in the proof of Lemma 1–4, it is the capacity of the shared wire-less channel that restricts the throughput in a single regular hexagon network. Network coding canimprove the efficiency of channel utilization by decreasing the number of packet transmission timesof the central node and finally improving the throughput. How many packet transmission times canbe reduced depends on the coding number, which further depends on the network intensity. In otherwords, we illustrate that the network intensity affects the performance benefit of network coding ina single regular hexagon network.

5. PERFORMANCE BENEFIT IN A GENERAL HEXAGONAL GRID NETWORK

As shown in Figure 12, we extend the single regular hexagon network to a general hexagonal gridnetwork, which consists of seven single regular hexagons. In this two-layer topology structure, eachvertex in the seven hexagons tries to send packets to its opposite node and the number of packetsoriginating from the vertices in every regular hexagon is same. We assume that all nodes withina transmission range share the same physical channel. In other words, when there is a node send-ing packets, all other nodes within its transmission range can neither send nor receive packets. Weassume that the capacity of one shared physical channel is equal to 1, then the whole capacity ofthis topology structure is equal to 7.

The definitions of main mathematical symbols used in this section are shown in Table II.Let �i .i D 0, : : : , 6/ denote the seven hexagons in the topology structure, respectively; let

Ci .i D 0, : : : , 6/ denote the central node in the hexagon �i ; let �i .i D 0, : : : , 6/ denote the drain-ing rate of Ci ; and let ˛ij .i D 0, : : : , 6I j D 1, : : : , 6/ denote the draining rate of vertex j in thehexagon �i .

In Section 4, we have discussed the performance benefit of network coding under two differentconditions:

p36 � < 2 and 16 � <

p3. Here, we also deduce the upper bound on the performance

benefit of network coding in a general hexagonal grid network under two different configurationsof network intensity and analyze the effect of network intensity on the performance benefit ofnetwork coding.


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Figure 12. Two-layer extension of a single regular hexagon network.

Table II. The definition of mathematical symbols.

Symbol Definition

�i The seven hexagons in a general hexagonal grid networkCi The central node in the hexagon �i�i The Draining rate of Ci˛ij The draining rate of vertex j in the hexagon �i�i The draining rate of nodes in the group III�i The draining of rate nodes in the group IV

5.1. Upper bound on performance benefit under the condition ofp36 � < 2

Lemma 5The maximum throughput in a general hexagonal grid network with a noncoding scheme is equal toT �nc D 1=14 when

p36 � < 2 is satisfied.

ProofConsider the central node C0, when C0 is sending packets, it cannot receive packets at the sametime. Furthermore, when the network intensity satisfies

p3 6 � < 2,Ci .i D 1, : : : , 6/ is under

the transmission range of C0, so these nodes cannot send or receive any packets. Then we havethe following:

6XiD0

�i C

6XiD0

6XjD1

˛ij 6 1 (1)

For each node Ci .i D 0, : : : , 6/, the draining rate should be less than or equal to its incoming rate,so we have the following:

�i 66XjD1

˛ij .i D 0, : : : , 6/ (2)


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The total throughput of the general hexagonal grid network is the sum of �i .i D 0, : : : , 6/, so wehave the following:

Tnc D

P6iD0 �i

7(3)

Clearly, the optimal solution to max .T �nc/ is a linear programming problem. We can searchthe optimal solution to the maximum throughput by means of MATLAB. The result is shown inTable III. �

Remark. As we can see the result shown in Table III, the throughput reaches the maximum valueunder the conditions that ˛ij .i D 0, : : : , 6I j D 1, : : : , 6/ is identical and �i D

P6jD1 ˛ij .i D

0, : : : , 6/. This result is in accordance with Lemma 1, and within one regular hexagon, the sum ofbandwidth allocated to six vertices should be as much as the bandwidth allocated to the central node.Note that here the maximum throughput .T �nc D 1=14/ in a general hexagonal grid network is farless than the maximum throughput .T �nc D 1=2/ in a single regular hexagon network. This situationarises because there exists more interference between the hexagons of the general hexagonal gridnetwork when

p36 � < 2.

Lemma 6The maximum throughput in a general hexagonal grid network with a coding scheme is equal toT �c D 6=49 when


ProofThe proof is almost the same as that in Lemma 5, except that now we use the coding number for thedraining rate in Lemma 6.

We assume that the number of encoded packets in each node Ci .i D 0, : : : , 6/ is the same, anduse nc.26 nc 6 6/ to denote this number. For each node Ci .i D 0, : : : , 6/, the draining rate shouldbe less than or equal to its incoming rate, so we have the following:

nc�i 66XjD1

˛ij .i D 0, : : : , 6/ (4)

The total throughput of the general hexagonal grid network is the sum of �i .i D 0, : : : , 6/, so wehave the following:

Tc D

6XiD0

nc�i

7(5)

The optimal solution to max .T �c / is shown in Table III. �

Remark. As we can see in Table III, the throughput reaches the maximum value under the con-ditions that ˛ij .i D 0, : : : , 6I j D 1, : : : , 6/ and �i .i D 0, : : : , 6/ are identical. This result is inaccordance with Lemma 2, and within one regular hexagon, the bandwidth allocated to each vertexshould be as much as the bandwidth allocated to the central node. Again, as in Lemma 5, here the

Table III. The optimal solution forp36 � < 2.

Scheme Noncoding scheme Coding scheme

�i .i D 0, : : : , 6/ 1/14 1/49˛ij .i D 0, : : : , 6I j D 1, : : : , 6/ 1/84 1/49nc / 6T � T �nc D 1=14 T �c D 6=49T �bD T �c =T

�nc 12/7


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maximum throughput .T �c D 6=49/ is far less than the maximum throughput .T �c D 6=7/ in thesingle regular hexagon network. This is due to the interference between the hexagons of the generalhexagonal grid network as well.

Theorem 4The upper bound on the performance benefit of network coding in a general hexagonal grid networkis equal to T �

bD 12=7 when


ProofFrom Lemmas 5 and 6 previously and from the definition of the upper bound on performance benefit,we derive the upper bound as T �

bD T �c =T

�nc D 12=7. �

Example 3In Figure 12, we assume that Sij .i D 0, : : : , 6I j D 1, : : : , 6/ wants to send the packet pij .i D0, : : : , 6I j D 1, : : : , 6/ to its opposite vertex Dij .i D 0, : : : , 6I j D 1, : : : , 6/. The size of the packetis one unit. Figure 13 shows how packets are transmitted to achieve the maximum throughput with-out coding in one cycle. In this scenario, the packets in each regular hexagon are sent in the sameway as Figure 9, 42 packets are transmitted in 84 time slots, by dividing the number of regularhexagons (7), thus the average maximum throughput is 1=14. On the other hand, Figure 14 showshow these packets are transmitted to achieve the maximum throughput with coding in one cycle. Inthis scenario, the packets in each regular hexagon are sent in the same way as Figure 10, 42 pack-ets are transmitted in 49 time slots, by dividing the number of regular hexagons, thus the averagemaximum throughput is 6=49.

Remark. This result shows that the upper bound on the performance benefit in a general hexag-onal grid network is the same as that in a single regular hexagon network, that is, both 12=7. Thereason is that when

p3 6 � < 2 is satisfied, the node C0 becomes the bottleneck of the network.

In fact,p3 6 � < 2 is required only by the coding scheme to reach the maximum throughput,

whereas it is not necessary for the noncoding scheme. There exists much interference in the non-coding scheme when

p3 6 � < 2 is satisfied. When � becomes smaller, such as 1 6 � <

p3, the

maximum throughput in the noncoding scheme will become larger. Afterwards, in Lemmas 7 and 8,we discuss the maximum throughput when 16 � <

p3 is satisfied.

5.2. Upper bound on performance benefit under the condition of 16 � <p3

As shown in Figure 12, for convenient discussion, we classify all nodes in the topology structureinto five groups:

Figure 13. An example without coding.



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T. SHANG ET AL.

(1) I : the central node C0.(2) II : the central nodes Ci .i D 1, : : : , 6/.(3) III : the vertices belong to three hexagons of all seven hexagons.(4) IV : the vertices belong to two hexagons of all seven hexagons.(5) V : other nodes (the vertices only belong to one hexagon of all seven hexagons).

Let �i .i D 1, : : : , 6/ denote the draining rate of six nodes belonging to the group III . Then wehave the following:

�i D ˛0i C ˛i.iC2/C ˛.iC1/.i�2/, .i D 1, : : : , 6/ (6)

�i 6 1, .i D 1, : : : , 6/ (7)

All the operations ‘C’ and ‘�’ in the subscripts in our formulae are based on the module of 6.Specially, we define 0mod.6/� 6mod.6/� 6 rather than 0mod.6/� 6mod.6/� 0.

Let �i .i D 1, : : : , 6/ denote the draining rate of six nodes belonging to the group IV . Then wehave the following:

�i D ˛i.iC1/C ˛.iC1/.i�1/, .i D 1, : : : , 6/ (8)

�i 6 1, .i D 1, : : : , 6/ (9)

Lemma 7The maximum throughput in a general hexagonal grid network with a noncoding scheme is equal toT �nc D 3=28 when 16 � <

p3 is satisfied.

ProofConsider the nodes in group I, we have the following:

�0C

6XiD1

�i C

6XiD1

�i C

6XjD1

˛0j 6 1 (10)

Consider the nodes in group II, we have the following:

�i C �.iC1/C �.i�1/C �0C ˛i i C ˛i.i�1/

C�i C�.i�1/C �i C �.i�1/ 6 1, .i D 0, : : : , 6/(11)

Consider the nodes in group III, we have the following:

�i C �.iC1/C �0C

6XjD1

.˛ij C ˛.iC1/j C ˛0j /

C˛.iC2/.i�1/C ˛.i�1/.iC1/ 6 1, .i D 0, : : : , 6/

(12)

Consider the nodes in the group IV, we have the following:

�i C �.iC1/C

6XjD1

.˛ij C ˛.iC1/j /C ˛0i 6 1, .i D 0, : : : , 6/ (13)

We assume that all draining rates in each regular hexagon are the same, we have the following:

˛ij D ˛ik , .i D 0, : : : , 6I j D 1, : : : , 6I k D 1, : : : , 6I j ¤ k/ (14)

For each node Ci .i D 1, : : : , 6/, the output rate should be less than or equal to its input rate, wehave the following:

�i 66XjD1

˛ij , .i D 0, : : : , 6/ (15)


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The total throughput of the general hexagonal grid network is the sum of �i .i D 0, : : : , 6/, wehave the following:

Tnc D

P6iD0 �i

7(16)

The optimal solution to max .T �nc/ is a linear programming problem, same as the problem inLemmas 5 and 6. We show the optimal solution in Table IV. �

Remark. The maximum throughput .T �nc D 3=28/ here is larger than the maximum throughput.T �nc D 1=14/ when

p3 6 � < 2. It is because the interference between the hexagons here is not

as large as the interference whenp3 6 � < 2. Note that the values of �0 and ˛0j .j D 1, : : : , 6/

are all zeroes. This implies that in our network model, the central hexagon should be sacrificed tomaximize the throughput.

Lemma 8The maximum throughput in a general hexagonal grid network with a coding scheme is equal to.T �c D 12=49/ when 16 � <

p3 is satisfied.

ProofThe proof is almost the same as that in Lemma 7, except that here we use the coding number for thedraining rate.

As proved in Theorem 3, the number of packets that can be encoded by the central node is only 2when 16 � <

p3, and nc D 2, so we have the following:

nc�i 66XjD1

˛ij , .i D 0, : : : , 6/ (17)

Tc D

P6iD0 nc�i

7(18)

The optimal solution to max .T �c / is shown in Table IV. �

Remark. It is surprising that the maximum throughput .Tc D 12=49/ here is larger than the max-imum throughput.Tc D 6=49/ when

p3 6 � < 2. Why does it happen like this? As we know, the

coding number .nc D 6/ whenp3 6 � < 2 is far more than the coding number .nc D 2/ when

1 6 � <p3. However, as � increases, the interference between the hexagons increases as well.

Thus, when � varies fromp3 6 � < 2 to 1 6 � <

p3, the coding number decreases and the

interference also decreases. The compensative effect of these two factors results in the improvementof the maximum throughput. Compared with the situation in the single regular hexagon network,where there is no interference from other hexagons, the maximum throughput simply increases from2=3 to 6=7 as the network intensity varies from 16 � <

p3 top36 � < 2.

Table IV. The optimal solution for 16 � <p3.

Scheme Noncoding scheme Coding scheme

�i .i D 1, : : : , 6/ 1/8 1/7˛ij .i D 1, : : : , 6I j D 1, : : : , 6/ 1/48 1/84�0 0 0˛0j .j D 1, : : : , 6/ 0 0nc / 2T � T �nc D 3=28 T �c D 12=49T �bD T �c =T

�nc 16/7


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Theorem 5The upper bound on the performance benefit of network coding in a general hexagonal grid networkis equal to T �

bD 16=7 when 16 � <

p3 is satisfied.

ProofFrom Lemmas 7 and 8 previously and the definition of the upper bound on performance benefit, wederive the upper bound as T �

bD T �c =T

�nc D 16=7. �

Example 4Note that the transmission of packets in this section is dynamic and complicated. Although it isconstrained by mathematical formulae, we provide an example of a nonoptimal solution instead ofan optimal solution. Figure 13 is the example without coding; here, when 1 6 � <

p3 is satisfied,

the throughput is 1=14 < 3=28. Figure 15 shows the coding scheme in one cycle; in this scenario, 42packets are sent in 63 time slots, by dividing the number of regular hexagons, thus the throughput is2=21 < 12=49.

Remark. The upper bound on the performance benefit .T �bD 16=7/ here is larger than the upper

bound on the performance benefit .T �bD 12=7/ when

p3 6 � < 2. The reason is that when

1 6 � <p3, the decrement of the interference increases the maximum throughput in the noncod-

ing scheme. And the decrement of the interference increases the maximum throughput higher inthe coding scheme compared with the noncoding scheme, even if the coding number decreases inthis situation.

In conclusion, in the case of noncoding scheme, the network intensity affects the throughputby changing the interference of a network. Specifically, the throughput increases as the networkintensity decreases. In the case of coding scheme, the network intensity affects the throughput bychanging both the interference and the coding number. When the network intensity is larger, theinterference and the coding number are also larger. On the contrary, when the network intensity issmaller, the interference and the coding number are also smaller. How the network intensity affectsthe performance benefit of network coding is determined by the compensative effect of both theinterference and the coding number. In our general hexagonal grid network, the upper bound onperformance benefit varies from 12=7 to 16=7 as the network intensity varies from

p3 6 � < 2 to

16 � <p3.

6. CONCLUSION

Inspired by the recent advances in complex networks, we focused our study on the effect of generalhexagonal grid topology on the performance benefit of network coding. In this paper, we proposeda new concept called ‘network intensity’, which describes the characteristics of hexagonal gridtopology. We analyzed the effect of the network intensity on the performance benefit of networkcoding. In a single regular hexagon network, the upper bound on performance benefit varies from3=4 to 12=7 as the network intensity � varies from 1 6 � <

p3 top3 6 � < 2. In a general

hexagonal grid network, the upper bound on the performance benefit varies from 16=7 to 12=7 asthe network intensity varies from 1 6 � <

p3 top3 6 � < 2. Our analysis demonstrates that

the network intensity affects the performance benefit by changing the interference and the coding


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number of a network. Compared with previous studies in [15] and [11], our results first derived thequantitative relationship between the performance benefit of network coding and topology structure,and we also derive the upper bound on the performance benefit of network coding that is tighter thanthe result in the work [15].

Furthermore, we still need to explore more complex network topology to analyze the performancebenefit of network coding in wireless communication networks. Our conclusion is very helpful tothe design of the network topology and the bound analysis of the performance benefit of networkcoding in wireless communication networks.

ACKNOWLEDGEMENTS

The authors would like to thank the National Natural Science Foundation of China (no.61272501),the National Basic Research Program of China (no.2012CB315905), and the Beijing Natural ScienceFoundation (no.4132056) for their valuable help.

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The effect of hexagonal grid topology on wireless communication networks based on network coding