A Two-Level Game Theory Approach for Joint Relay Selection ...chuhan/wp-content/uploads/chuhan_TMC_NCD2D.pdfmulti-antenna relay employing space-time analog network coding. However,

1536-1233 (c) 2016 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.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMC.2016.2642190, IEEETransactions on Mobile Computing

1

A Two-Level Game Theory Approach for JointRelay Selection and Resource Allocation in

Network Coding Assisted D2D CommunicationsChuhan Gao, Yong Li, Senior Member, IEEE , Yulei Zhao, and Sheng Chen, Fellow, IEEE

F

Abstract—Device-to-device (D2D) communication, which enables di-rect transmissions between mobile devices to improve spectrum ef-ficiency, is one of the preferable candidate technologies for the nextgeneration cellular network. Network coding, on the other hand, is widelyused to improve throughput in ad hoc networks. Thus, the performanceof D2D communications in cellular networks can potentially benefit fromnetwork coding. Aiming to improve the achievable capacity of D2D com-munications, we propose a system with inter-session network codingenabled to assist D2D transmissions. We formulate the joint problem ofrelay selection and resource allocation in network coding assisted D2Dcommunications, and obtain the overall capacity of the network undercomplex interference conditions as a function of the relay selection andresource allocation. To solve the formulated problem, we propose a two-level de-centralized approach termed NC-D2D, which solves the relayselection and resource allocation problems alternatively to obtain sta-ble solutions for these two problems. Specifically, a coalition formationgame associates relays with D2D pairs to enable network coding aidedtransmissions, and a greedy algorithm based game allocates limitedcellular resources to D2D pairs and relays in NC-D2D, respectively. Theperformances of the proposed scheme is evaluated through extensivesimulations to prove its superiority.

Index Terms—Device-to-device communication, network coding, relayselection, resource allocation, game theory

1 INTRODUCTION

Demand for mobile Internet access is growing at a tremen-dous rate. To satisfy this explosive traffic demand, device-to-device (D2D) communication has been proposed for Long-Term Evaluation-Advanced [1]. In D2D communications, userequipment (UE) in close proximity set up direct links for datatransmissions with licensed cellular spectrum resources, insteadof through base stations (BSs). The benefits of such proximitycommunication is manifold [2]. It has the potential to provide

C. Gao, Y. Li and Y. Zhao([email protected],[email protected], [email protected]) are with Departmentof Electronic Engineering, Tsinghua University, Beijing 100084, China.S. Chen ([email protected]) is with Electronics and Computer Science,University of Southampton, Southampton SO17 1BJ, UK, and also with KingAbdulaziz University, Jeddah 21589, Saudi Arabia.This work was partially supported by the National Natural Science Foundationof China (NSFC) under grants No. 61201189 and No. 61132002, the CreativeResearch Groups of NNSFC under grant No. 61021001, the National S&TMajor Project under grant No. 2011ZX03004-001-01, the National HighTech (863) Projects under Grant No. 2011AA010202, the Research Fund ofTsinghua University under grants No. 2011Z05117 and No. 20121087985,and Tsinghua University Initiative Scientific Research Program.

extremely high bit rate as well as low end-to-end delay and powerconsumption due to short-range transmissions. Since the cellularresources can be simultaneously shared and utilized by D2D UEs,the spectrum efficiency and reuse gain are improved. In addition,D2D communications enable mobile traffic offloading by usercooperations for content downloading and sharing, which alsobenefits cellular (non-D2D) users. Therefore, D2D communicationis expected to be a key feature supported by the next-generationcellular network [3].

Although D2D communication enhances the system perfor-mance in many aspects, it also causes severe interference, whichmay degrade the transmission rates of both cellular and D2Dusers. To solve this problem, current works focus on resourceallocation [1], [4]–[6] and power control [7], [8]. Gu et al. [1]utilized two stable matching algorithms to optimize the overallsystem throughput while simultaneously meeting the quality ofservice (QoS) requirements for cellular and D2D users. In themodel of [1], D2D users seek channel reuse partners from cellularusers to share their spectrum resources for data transmissions. Liet al. [4] considered a similar system model to solve the resourceallocation problem by a coalition formation game based scheme.In the approach of [4], different transmission modes, mutualinterferences and resource sharing policy are combined in a utilityfunction, which is used by D2D users to determine the spectrumresource reuse partners in the coalition formation game. Yu etal. [5] optimized the throughput over the shared resources forD2D communications to improve local services, while fulfillingprioritized cellular service constraints. Xu et al. [6] introduced areverse iterative combinatorial auction as the resource allocationmechanism to optimize the system sum rate. Lee et al. [7]proposed a random network model for D2D communications usingstochastic geometry and developed centralized and distributedpower control algorithms, while Liu et al. [8] analyzed the benefitsof power control in enhancing the transmission capacity region.

Network coding has the potential to improve throughput effi-ciency [9], [10]. Current studies on network coding can be dividedinto two categories: intra-session coding [11], [12], and inter-session coding [13], [14]. Intra-session coding usually relies onrandom linear network coding to organize packets in groups tobe linearly combined using randomly chosen coefficients from theelements of a finite field. Ho et al. [11] introduced dynamic al-gorithms for multicast routing, network coding, power allocation,session scheduling and rate allocation across correlated sourcesfor intra-session network coding. Cai et al. [12] investigated the



2

problem of selecting the candidate forwarder set and allocatedtraffic among candidate forwarders to achieve optimal routingin opportunistic routing with intra-session network coding. Sinceonly packets from the same flow are combined in intra-sessioncoding, it is not a good fit for D2D communications underlayingcellular networks, where each D2D pair is viewed as a flow, andthe cooperation among different D2D pairs should be considered.Inter-session coding combines the packets from different networkflows. Upon identifying sets of nodes that can form a codingregion, the packets can be mixed (XORed) in order to attain higherspectrum efficiency. Katti et al. [13] proposed an architecture forwireless mesh networks, where routers mix packets from differentsources to increase the information content of each transmissionin addition to forwarding packets. Liu et al. [14] addressed thedistributed control problem in heterogeneous-service networkswith multi-rate multicast and unicast services, and proposed adecentralized rate control algorithm for inter-session networkcoding. Some recent works combined both intra-session codingand inter-session coding to enhance the system performance [15],[16].

Pahlevani et al. [10] discussed the potential of enabling net-work coding in D2D communications to enhance communicationefficiency and security. In the scenario where a D2D communityis composed of multiple devices connected in a multi-hop fashion,network coding’s ability to recode coded packets on the flyprovides the means of improving D2D communication’s through-put, delay, and energy efficiency. A few schemes that enablenetwork coding in D2D communications have been proposed sofar, and they can be categorized into two types: either studying therelay selection problem [17], [18], or investigating the resourceallocation problem [19], [20]. Specifically, Bhorkar et al. [17]investigated the relay selection and scheduling problems, whileMaher et al. [18] utilized idle devices in D2D communicationsas relay nodes to enable network coding. Wu et al. [19] devel-oped a radio resource management mechanism to optimize powercontrol and subchannel allocation with network coding, while Weiet al. [20] investigated multi-pair D2D communications with amulti-antenna relay employing space-time analog network coding.However, in D2D communications underlaying cellular networks,both relay selection and resource allocation have major impacton the achievable performance of network coding and, moreover,these two problems are actually coupled and cannot be solvedseparately without compromising the overall system performance.Detailed discussions on the coupling of these two problems willbe given later.

In this paper, we aim to assist the D2D communicationsunderlaying cellular network with inter-session network coding.Specifically, we consider the scenario where relays assist nearbyD2D pairs, forming coding regions, and perform network coding.In order to achieve high sum capacity, the problems of relayselection and resource allocation need to be solved. Firstly, thecapacity gain highly depends on the locations of the relays andthe assisted users. Therefore, D2D pairs need to select properrelays to assist their transmissions for maximizing the benefits ofnetwork coding. Secondly, the spectrum resources should also beallocated wisely to the D2D pairs and relays to mitigate the severeinterference. We formulate a joint problem of relay selectionand resource allocation, where these two problems are optimizedjointly. Theoretically, the joint optimal solution can be obtainedby exhaustive search. However, it is very difficult to apply anyexisting optimization approaches to solve this joint optimization

efficiently owing to its extremely high computation complexity.Moreover, global network information, such as network topology,are required, which either impose high synchronizing overhead orare not available. Thus, we address these two problems from agame theory point of view, using a coalition formation game anda greedy algorithm based game. Utilizing game theory enablesus to obtain the solutions for the relay selection and resourceallocation problems efficiently, where the nodes only require localnetwork information. More specifically, we propose a two-leveloptimization approach, termed NC-D2D, to obtain near-optimalsolutions. In our NC-D2D, relay selection game and resourceallocation game operate alternately, each using the results obtainedby the other game as inputs, and the alternating optimizationprocedure continues until the system reaches a stable state in termsof both relay selection and resource allocation. Our contribution isthree-fold, as summarized in the following.

• We introduce inter-session coding to assist D2D com-munications underlaying cellular networks with realisticmultiple D2D pairs, relays, and cellular users, where D2Dpairs select relays and form coding regions to improvethe achievable capacity. Relays XOR the received packetsbefore multicasting them to the corresponding D2D users.

• We formulate the joint problem of relay selection andresource allocation to maximize the sum transmissioncapacity of all D2D pairs and cellular users. The sumcapacity of the network is derived as a function of theresults of relay selection and resource allocation, whiletaking into account the interference among users.

• Since the joint optimal solution of relay selection andresource allocation imposes extremely high complexity,rendering it impractical for large-scale systems, we pro-pose a two-level optimization approach, termed NC-D2D,to obtain stable solution for the joint optimization, whererelay selection and resource allocation are performedalternately, utilizing a coalition formation game and agreedy algorithm based game, respectively. In each round,the input of one problem is provided by the solution ofthe other problem. In addition, each node only requireslocal information to solve the two problems. Through ourextensive evaluation, we prove that our NC-D2D is ableto obtain a stable near-optimal solution with very lowcomputation complexity in a few rounds.

The rest of the paper is organized as follows. After presentingthe system model of network coding assisted D2D communica-tions in Section 2, we derive the system capacity and formulatethe joint problem of relay selection and resource allocation inSection 3. Then we give an overview of the proposed two-levelNC-D2D optimization approach in Section 4, and detail the relayselection coalition formation game and resource allocation gamein Sections 5 and 6, respectively. The performance of our proposedscheme is evaluated through extensive simulations in Section 7.Finally, we conclude this paper in Section 8.

2 SYSTEM OVERVIEW AND MODEL

2.1 System OverviewWe focus on the scenario of a single cell involving multiple D2Dpairs, relays and cellular users. In D2D communications, a pair ofUEs in close proximity are able to enjoy extremely high data rateby setting up a direct link between them. However, it is well known



3

Interference

Fig. 1. Illustration of the network coding aided D2D communi-cations underlaying cellular network, where there are 3 cellularusers, 4 D2D pairs and 1 cooperative relay.

that the channel quality between two users degrades rapidly as thedistance between the transmitter and receiver increases. When thedistance between the D2D transmitter and receiver is too longto support a direct link, these D2D users will have to switch tocellular mode [21], [22]. Therefore, we enable cooperative relaysto assist D2D pairs for data transmissions in order to overcomethe long distance between D2D users. Each D2D pair can eithertransmit data via the direct link between the two users, or assistedby a cooperative relay, depending on their circumstance. Hencethere are two kinds of D2D pairs in the system, i.e., (i) ordinaryD2D pair: two D2D users transmit via the direct link betweenthem, and (ii) relay Assisted D2D pair: two D2D users are assistedby a relay which employs network coding aided transmission. Wedenote the sets of all D2D pairs, relays and cellular users as D,R and U , respectively. D2D pair i is denoted by di, while itstransmitter and receiver is denoted by dti and dri . Relay i andcellular user j is denoted as ri and uj , respectively. We allowmultiple D2D pairs and relays to share the spectrum resourceof one cellular user in order to increase the possible number ofconcurrent transmissions. Each cellular spectrum resource can beshared by multiple D2D users at the same time as well. To fullyexploit this spatial reuse gain, therefore, the interference causedby this spectrum sharing must be taken into consideration.

As illustrated in Fig. 1, D2D pairs d1 and d2 are relay assistedD2D pairs, aided by relay r1, while d3 and d4 are ordinary D2Dpairs. d1 and d2 occupy the uplink spectrum resource of cellularuser u1 and u2, respectively, and u3 shares its resource with r1, d3and d4. The interferences between d2 and u2 as well as betweenr and d3 are plotted as examples.

We introduce network coding to aid D2D communications,which utilizes the relay to improve throughput efficiency. Due

to the spectrum sharing, mutual interferences exist among D2Dpairs, regular cellular users and cooperative relays. In order toachieve high transmission rates for D2D users and cellular users,both the relay selection for network coding and the spectrumresource allocation need to be optimized. In our system, the D2Dpairs can choose different relays to assist their transmissions byapplying network coding. However, the throughput gain highlydepends on the distances and channel qualities of the links amongthe D2D users and relays that form a coding region, as will beanalyzed in detail later. Moreover, the relays that are available inthe network may not be enough to assist all the D2D pairs. Inthis case, they should assist the D2D pairs that have long linkdistance or poor channel quality to maximize the throughput gainof network coding. Therefore, the D2D pairs should select properworking modes, namely, whether to rely on relays and to selectwhich relays in order to achieve the maximum throughput gain. Inour system, decode-and-forward is adopted as the relay strategyby all relay nodes, instead of amplify-and-forward. The limitedspectrum resources shared by the cellular users should also beallocated properly to the D2D pairs and relays to ensure the QoSof all users and to achieve high sum rates of the network.

Although the selfishness of D2D users are addressed anddiscussed in previous works, we assume operator controlled D2Dcommunications in our system, where the behavior pattern of theterminals is defined and controlled by the service provider oroperator. Operator controlled D2D communication is also adoptedby the 3GPP standard.

2.2 System ModelWe define the binary variables xd,u, xr,u and yd,r to depict therelay selection and resource allocation policies for D2D pairsand relays. Specifically, xd,u = 1 indicates that D2D pair duses the uplink resource of cellular user u, otherwise xd,u = 0,while xr,u = 1 if relay r shares the spectrum resource of u,otherwise xr,u = 0. Similarly, yd,r = 1 indicates that relayr assists the D2D pair d’s transmission, otherwise, yd,r = 0.We denote the matrices of xd,u, xr,u and yd,r as XD , XRand Y , respectively. Each row in XD and XR represents thespectrum resource sharing of the corresponding D2D pair andrelay, respectively, while each row in Y represents the relayselection of the corresponding D2D pair. For example, in thescenario of Fig. 1, these three matrices can be written as:

XD =

1 0 00 1 00 0 10 0 1

, (1)

XR =[

0 0 1], (2)

Y =[

1 1 0 0]T. (3)

2.3 Network Coding Assisted D2D TransmissionsWe adopt an inter-session network coding to assist the transmis-sions of D2D pairs. A typical coding region is presented in Fig. 2,which consists of two relay assisted D2D pairs and one relay.This scheme is described in [10], [16]. We will first describe theoperation of this scheme in our system. Then we will derive theachievable rate of this coding region, and analyze the capacity gainof network coding in Section 3.

For an ordinary D2D pair, the data is transmitted via the directlink between the two devices. For the relay assisted D2D pairs



4

dt1

r

dt2

dr1dr2

Flow 1 Flow 2

Fig. 2. Illustration of a coding region in network coding assistedD2D communications.

as shown in Fig. 2, sender dt1 multicasts the packets to relay rand dr2, while dt2 multicasts the packets to r and dr1. Relay rXORs the packets received from dt1 and dt2 bit by bit beforemulticasting them to dr1 and dr2. Then dr1 and dr2 are able todecode the packets after receiving the XORed packets from r andthe multicasted packets from dt2 and dt1, respectively. The assistof relay by applying network coding not only helps to overcomethe low transmission rate of the direct link between users, causedby long transmitting distance or poor channel quality, but alsoimproves the spectrum efficiency.

It should be noted that the optimal coding region may consistarbitrary number of relays and D2D pairs. We adopt this simplemodel to achieve efficient optimization. In such a large scalenetwork as we investigated in this paper, jointly optimizing thecoding region formation of all the relays and D2D pairs could beextremely difficult and time consuming. Since optimizing relayselection and resource allocation would require solving these twoproblems iteratively, the low efficiency might make the overallscheme impractical in cellular systems.

3 PROBLEM FORMULATION

We first derive the achievable capacity of network coding assistedD2D communications and compare it with that of ordinary D2Dcommunications without the assist of network coding and cooper-ative relays, based on which we then obtain the sum capacity of thenetwork as a function of XD , XR and Y . Finally, we formulatethe optimization problem of relay selection and resource allocationwith system constrains.

3.1 Network Coding Assisted D2D Capacity

We first consider one coding region, which consists of one relayand two relay assisted D2D pairs as shown in Fig. 2, and deriveits capacity. Assume that dt1 wishes to send packet A to dr1 anddt2 wishes to send packet B to dr2. If d1 and d2 work underordinary mode, i.e., there is no relay to assist them, dt1 and dt2 willtransmit A and B via links (dt1, d

r1) and (dt2, d

r2), respectively.

Therefore, the achievable capacity of these two D2D pairs isc(dt1, d

r1

)+ c

(dt2, d

r2

), where c(i, j) represents the capacity of

link (i, j). With the aid of network coding, dt1 multicasts A to dr2and r, and dt2 multicasts B to dr1 and r. Then r combines A andB, and multicasts AXORB to dr1 and dr2, who will be able todecode A and B, respectively.

Since relay r needs to combine the packets from both dt1and dt2 and then multicasts the result, the rate that r sendsAXORB to dr1 and dr2 is actually limited by the transmitting

rates of the four links: (dt1, r), (dt2, r), (r, dr1) and (r, dr2).Firstly, the maximum rate that r is able to send AXORB islimited by the lower-rate link of the two, (dt1, r) and (dt2, r),which has the rate min

{c(dt1, r), c(d

t2, r)

}. The computing time

of r to combine A and B is neglected compared to packets’transmission time. Then AXORB is transmitted through onemore hop, link (r, dr1) to dr1. For the purpose of deriving theachievable rate, the process of dr1 receiving AXORB can bemodeled as a two-hop transmission, where r receives AXORBfrom a virtual source node and then transmits it to dr1. Thetransmitting rate of the second hop is obviously c(r, dr1), andthe transmitting rate of the first virtual hop is equivalent to therate of r receiving A and B, which is min

{c(dt1, r), c(d

t2, r)

}.

For this two-hop transmission, therefore, the achievable rate isequal to min

{c(r, dr1),min

{c(dt1, r), c(d

t2, r)

}}. In addition

to AXORB, dr1 also needs to receive packet B from dt2 inorder to decode A from AXORB. The rate of dr1 receiv-ing B is c(dt2, d

r1). Thus, the rate of dr1 receiving (decoding)

A is min{c(dt2, d

r1),min

{c(r, dr1),min

{c(dt1, r), c(d

t2, r)

}}}.

Similarly, the rate dr2 receiving (decoding) B can be shown to bemin

{c(dt1, d

r2),min

{c(r, dr2),min

{c(dt1, r), c(d

t2, r)

}}}. The

total achievable capacity of the coding region formed by relayr and D2D pairs d1 and d2 is therefore given by

cCR(r, d1, d2) =

min{c(dt2, d

r1),min

{c(r, dr1),min

{c(dt1, r), c(d

t2, r)

}}}+

min{c(dt1, d

r2),min

{c(r, dr2),min

{c(dt1, r), c(d

t2, r)

}}}, (4)

which can be simplified to

cCR(r, d1, d2) = min{c(dt2, d

r1), c(r, dr1), c(dt1, r), c(d

t2, r)

}+ min

{c(dt1, d

r2), c(r, dr2), c(dt1, r), c(d

t2, r)

}. (5)

With the achievable capacity of network coding assisted D2Dcommunications in (5), we also need the capacity of each link inorder to derive the system capacity as a function of XD , XR andY . The transmission capacity of a link is

c = log2

( PRI +N

+ 1), (6)

where PR is the receiving power at the receiver of the link, while Iand N denote the interference and noise powers, respectively. Weassume Rayleigh fading and adopt the Friis transmission equationto calculate the path loss of the transmitted signal [23]. Therefore,PR can be written as

PR = PT +GT +GR + 20 lg( λ

4πl

), (7)

where PT is the transmitting power, GT and GR are the antennagains of the transmitter and receiver, respectively, while λ is thesignal wavelength and l is the link distance. The transmittingpowers of relay, D2D user and cellular user are labeled as PRT , PDTand PUT , respectively. Similarly, we have GRT , GDT , GUT , GRR , GDRand GUR for these three kinds of users, and GBSR for the BS. Forsimplicity, we denote PT = PT +GT +GR in the sequel. Underthe Rayleigh fading channel model, although the instantaneouschannel taps are the function of time and spatial locations, thestatistical characteristic of the channels, is constant within the BSscoverage area.

For the purpose of relay selection and resource allocation aswell as deriving the system capacity, we need to know which nodesform coding regions with a given matrix Y . In other words, given



5

a relay rj , we want to represent the two D2D pairs it assisted. Welabel the D2D pairs assisted by rj as α(j) and β(j). According tothe definition of Y , ydα(j),rj = ydβ(j),rj = 1, while ydi,rj = 0∀i 6= α(j) , β(j). Therefore, α(j) and β(j) can be obtained by,{

α(j) = min{i | yi,rj = 1},β(j) = max{i | yi,rj = 1}.

(8)

Representing the index of the relay assisted D2D pairs with Ywould make it possible to optimize Y to maximize sum capacity.In other words, we apply this step to make the math in the problemformulation and optimization more clearly.

To obtain the capacity of each link, we need considering theinterference. There are three types of links involving D2D usersin each coding region: the links from D2D transmitters to D2Dreceivers (links (dt1, d

r2) and (dt2, d

r1) in Fig. 2), the links from

D2D transmitters to the relay (links (dt1, r) and (dt2, r)), andthe links from relay to D2D receivers (links (r, dr1) and (r, dr2)).Thus, the interference can be categorized into three kinds: theinterference from other D2D users, interference from relays, andthe interference from cellular users.

For the coding region of relay rj , first consider (dtα(j), drβ(j))

and (dtβ(j), drα(j)). For (dtα(j), d

rβ(j)), the interference from other

D2D transmitters, denoted by ID(dtα(j)

,drβ(j)

), is

ID(dtα(j)

,drβ(j)

) =∑m

∑k 6=α(j)

xdα(j),um

· xdk,um(PT + 20 lg

( λ

4πl(dtk,drβ(j))

)), (9)

while the interference from relays, denoted by IR(dtα(j)

,drβ(j)

), canbe expressed as

IR(dtα(j)

,drβ(j)

) =∑m

∑k

xdα(j),um

· xrk,um(PT + 20 lg

( λ

4πl(rk,drβ(j))

)). (10)

Similarly, the interference from cellular users, denoted byIU(dt

α(j),drβ(j)

), can be expressed as,

IU(dtα(j)

,drβ(j)

) =∑m

xdα(j),um

(PT + 20 lg

( λ

4πl(um,drβ(j))

)).

(11)Therefore, we have

c(dtα(j), drβ(j)) =

log2

PT + 20 lg(

λ4πl(dt

α(j),drβ(j)

)

)ID(dtα(j)

,drβ(j)

)+ IR

(dtα(j)

,drβ(j)

)+ IU

(dtα(j)

,drβ(j)

)+N

+ 1

.(12)

In the same way, the capacity of link (dtβ(j), drα(j)) is

c(dtβ(j), drα(j)) =

log2

PT + 20 lg(

λ4πl(dt

β(j),drα(j)

)

)Id(dtβ(j)

,drα(j)

)+ Ir

(dtβ(j)

,drα(j)

)+ Iu

(dtβ(j)

,drα(j)

)+N

+ 1

.(13)

Next consider (dtα(j), rj) and (dtβ(j), rj). For (dtα(j), rj), theinterferences from other D2D users, relays and cellular users canbe expressed respectively as

ID(dtα(j)

,rj)=∑m

∑k 6=i

xdα(j),um


( λ

4πl(dtk,rj)

)), (14)

IR(dtα(j)

,rj)=∑m

∑k 6=j

xdα(j),um


( λ

4πl(rk,rj)

)), (15)

IU(dtα(j)

,rj)=∑m

xdα(j),um

(PT + 20 lg

( λ

4πl(um,rj)

)). (16)

Thus we havec(dtα(j), rj) =

log2

PT + 20 lg(

λ4πl(dt

α(j),rj)

)ID(dtα(j)

,rj)+ IR

(dtα(j)

,rj)+ IU

(dtα(j)

,rj)+N

+ 1

, (17)

c(dtβ(j), rj) =

log2

PT + 20 lg(

λ4πl(dt

β(j),rj)

)ID(dtβ(j)

,rj)+ IR

(dtβ(j)

,rj)+ IU

(dtβ(j)

,rj)+N

+ 1

. (18)

Finally, consider (rj , drα(j)) and (rj , d

rβ(j)). For (rj , d

rα(j)),

the three types of interference can be written respectively asID(rj ,drα(j)

) =∑m

∑k

xrj ,um


( λ

4πl(dtk,drα(j))

)), (19)

IR(rj ,drα(j)) =

∑m

∑k 6=j

xrj ,um


( λ

4πl(rk,drα(j))

)), (20)

IU(rj ,drα(j)) =

∑m

xdα(j),um

(PT + 20 lg

( λ

4πl(um,drα(j))

)).

(21)

Thus the link capacities can be expressed asc(rj , d

rα(j)) =

log2

PT + 20 lg(

λ4πl(rj,drα(j)

)

)ID(rj ,drα(j)

) + IR(rj ,drα(j)) + IU(rj ,drα(j)

) +N+ 1

, (22)

c(rj , drβ(j)) =

log2

PT + 20 lg(

λ4πl(rj,drβ(j))

)ID(rj ,drβ(j))

+ IR(rj ,drβ(j))+ IU(rj ,drβ(j))

+N+ 1

. (23)

From (5), the total capacity of the coding region of relay rjcan be rewritten ascCR(rj , dα(j), dβ(j)) = min

{c(dtβ(j), d

rα(j)), c(rj , d

rα(j)),

c(dtα(j), rj), c(dtβ(j), rj)

}+ min

{c(dtα(j), d

rβ(j)),

c(rj , drβ(j)), c(d

tα(j), rj), c(d

tβ(j), rj)

}, (24)



6

in which all the link capacities can be calculated according to (12),(13), (17), (18), (22) and (23). It is possible that not all the relaysform coding regions with D2D pairs and assist their transmissions.For any relay rj ,

∑i ydi,rj = 2 if rj assists the transmissions of

two D2D pairs, otherwise∑i ydi,rj = 0. Therefore, the sum

capacity of all the relay assisted D2D pairs in the whole network,labeled as cRA, can be expressed as

cRA =1

2

∑j

∑i

ydi,rjcCR(rj , dα(j), dβ(j)). (25)

In our system, the uplink transmission of D2D users are notincluded for simplicity. However, adding it does not change thenature of the problem and can be fully supported by the aboveformulation by simply adding the links from D2D users to the BS.Thus, it is also fully supported by our proposed scheme.

3.2 Network Coding Gain

To compare the achievable capacities of D2D communicationswith and without relay assisted network coding, we define thenetwork coding gain GNC as the increased sum capacity byapplying relay aided network coding dividing by the sum capacityachieved without applying network coding. For the scenario inFig. 2, GNC can be expressed as

GNC =cCR(r, d1, d2)− c(dt1, dr1)− c(dt2, dr2)

c(dt1, dr1) + c(dt2, d

r2)

. (26)

With the above definition, GNC = 0 means that the capacitieswith and without network coding are the same, while GNC = 1means that network coding doubles the capacity.

To investigate the benefits of network coding, we evaluate thismetric in the typical scenario of Fig. 2. The coordinates of the fourD2D users in the network are as follows

dt1 (0, 20), dr1 (30, 0), dt2 (30, 20), dr2 (0, 0). (27)

We deploy relay r in different positions. Specifically, r’s x-coordinate varies from 1 to 29, while r’s y-coordinate varies from1 to 19. The setup of the simulation is explained in Section 7, andthe simulation parameters can be found in Table 1. The achievednetwork coding gains GNC for these 29× 19 network topologiesare calculated and plotted in Fig. 3. As expected, the gain is highlyrelated to the position of relay. Specifically, the capacity of relayassisted network coding increases as relay moves to the center,where the maximum system capacity is more than doubling thatof simply transmitting via direct links between D2D users. GNCdrops to around 0.1 when relay is located near the four D2D users,which corresponds to the four corners in the figure.

Based on the above results we can derive the conclusionthat relay assisted network coding is capable of enhancing thesystem performance significantly. However, such benefits highlydepend on the network topology. In particular, the capacity gainof applying network coding relies on selecting proper relays toassist the D2D pairs.

3.3 Overall System Capacity

To obtain the network sum capacity as a function of XD , XRand Y , we also need the sum capacities of ordinary D2D pairsand cellular users. For an ordinary D2D pair di transmitting via

0

10

20

30

0

10

200

0.5

1

X Coordinate of Relay

Network Coding Gain

Y Coordinate of Relay

0.2 0.4 0.6 0.8 1

Fig. 3. Network coding gain GNC with relay deployed in differentlocations.

link (dti, dri ), the interference from other D2D transmitters, relays

and cellular users can be expressed respectively as

ID(dti,dri ) =∑m

∑k 6=i

xdi,um


( λ

4πl(dtk,dri )

)), (28)

IR(dti,dri ) =∑m

∑k

xdi,um


( λ

4πl(rk,dri )

)), (29)

IU(dti,dri ) =∑m

xdi,um

(PT + 20 lg

( λ

4πl(um,dri )

)). (30)

Thus the link capacity is

c(dti, dri ) = log2

PT + 20 lg(

λ4πl(dt

i,dri)

)ID(dti,d

ri )

+IR(dti,d

ri )

+IU(dti,d

ri )

+N+1

. (31)

For ordinary D2D pair di,∑j ydi,rj = 0. For relay assisted

D2D pair dk,∑j ydk,rj = 1. Therefore, the sum capacity of

all ordinary D2D pairs in the system can be expressed as,

cD =∑i

(1−∑j

ydi,rj

)c(dti, d

ri )

. (32)

Since different cellular users occupy different resource blocks,there is no interference among cellular users. For link (u,BS),however, the interferences from D2D users and relays can beexpressed respectively as

ID(u,BS) =∑i

xdi,um

(PT + 20 lg

( λ

4πl(dti,BS)

)), (33)

IR(u,BS) =∑i

xri,um

(PT + 20 lg

( λ

4πl(rti ,BS)

)). (34)

The capacity of link (u,BS) is therefore given by

c(u,BS) = log2

PT + 20 lg(

λ4πl(u,BS)

)ID(u,BS) + IR(u,BS) +N

+ 1

, (35)



7

and the sum capacity of all cellular users can be expressed as

cU =∑u

c(u,BS). (36)

The sum capacity Csum of the network, involving all relayassisted D2D users, ordinary D2D users and cellular users, cantherefore be obtained as

csum(XD,XR,Y ) = cRA + cD + cU . (37)

3.4 Relay Selection & Resource Allocation ProblemThe joint optimization problem of relay selection and resourceallocation can be formulated as the one that maximizes the sumcapacity csum(XD,XR,Y ) with the decision variables Y andXD , XR, subject to certain system constrains.

A D2D pair can only be assisted by one relay or transmitthrough direct link as an ordinary D2D pair, while a relay eitherassists two D2D pairs to form a coding region or does not takepart in any D2D pair’s transmissions. Therefore, the constrains forrelay selection are specified by

ydi,rj ∈ {0, 1} ∀i, j, (38)∑i

ydi,rj ≤ 1 ∀j, (39)∑j

ydi,rj ∈ {0, 2} ∀i. (40)

For resource allocation, each D2D pair or relay is only allowedto share the spectrum of a single cellular user. In addition, the relayis not allowed to share the same spectrum resource with the D2Dpairs it assisted due to half duplex assumption. In the senario ofFig. 2, for example, r will not be able to transmit to dr1 and dr2 aswell as to receive from dt1 and dt2 at the same time if it occupies thesame spectrum resource as dt1 or dt2. The two relay assisted D2Dpairs in a coding region are allowed to share the same spectrumresource. The constrains for resource allocation are thus given by

xdi,uk ∈ {0, 1} ∀i, k, (41)

xrj ,uk ∈ {0, 1} ∀j, k, (42)∑i

xdi,uk = 1 ∀k, (43)∑j

xrj ,uk = 1 ∀k, (44)

xdi,ukxrj ,ukydi,rj = 0 ∀i, j, k. (45)

Therefore, the optimal relay selection and resource allocationis formulated as the following optimization problem:

max csum(XD,XR,Y ),s.t. constraints (38) to (45) hold. (46)

The above problem is NP-hard. Specifically, it is a nonlin-ear 0-1 programming problem [24]. The optimization objective(37) also has no obvious convex or concave properties with thedecision variables Y , XD and XR, and we cannot derive theoptimal solution by gradient descent. Solving such a nonlinear andnon-convex problem with numerical approximation algorithms orexhaustive search has extremely high computation complexityand may therefore be infeasible for practical cellular systems.Moreover, we hope to achieve de-centralized control in our systemwhile still maintaining high performances. This motives us to

introduce a two-level de-centralized optimization approach fromthe view point of game theory, termed NC-D2D, which employscoalition formation game and greedy algorithm to solve theproblem efficiently.

In our system we assume that the transmitting power of allthe nodes are constant. In fact, power control has the potential toimprove the system performance of D2D network as well, and isable to be interrogated in the scheme we proposed. Specifically,power control will determine the achievable transmission rateof each link. Currently, we are optimizing the system from twoaspects, relay selection and resource allocation. By calculatingthe link transmission rate under different power controls andformulating power control into the system model, it can beoptimized together with relay selection and resource allocation.However, power control is beyond the scope of this paper and weonly investigate the optimization of relay selection and resourceallocation due to limited space.

4 NC-D2D OVERVIEW

Since relay selection and resource allocation are closely coupled,the optimal solutions for the two problems must be solved jointly,as in the joint optimization (46). This is because changing thesolution of relay selection or resource allocation also changes thesolution of the other problem. For example, if two ordinary D2Dpairs switch to relay assisted mode and form a coding region with arelay, that is, the solution of relay selection changes. In this case,even though the users and relays remain at the same locations,the interference from other users and relays actually changes.Therefore, once the solution for relay selection Y changes, thesolution for resource allocation, XD and XR, need to be alteredaccordingly. Resource allocation impacts relay selection in asimilar way.

In order to solve the relay selection and resource allocationproblems jointly while maintaining low computation complex-ity, our NC-D2D utilizes a de-centralized two-level optimizationapproach, where relay selection and resource allocation takeplace alternately. It should be noted that relay selection gameis performed based on the link capacities, which requires thespectrum resources allocated to nodes. Similarly, the interferencein resource allocation game is determined by the results of relayselections. In other words, these two games each requires theresults of the other game as an input. Therefore, we performthese two games alternatively. Fig. 4 shows the operations of NC-D2D. Solving the relay selection problem to obtain Y requiresthe solutions of resource allocation XD and XR as inputs,because the link capacities cannot be calculated without allocatingspectrum resources to users and relays. Similarly, optimizing XDand XR also requires Y as an input. The operations of NC-D2Dcan be summarized as follows:(a) Resource allocation matrices XD and XR are randomlygenerated under the constrains (41) to (44) to serve as the initialinput values for relay selection.(b) Given XD and XR, the relay selection is solved by a coalitionformation game, where each coding region consisting a relayand two related D2D pairs is a coalition, while all the ordinaryD2D pairs form a coalition. D2D pairs swap coalition accordinga pre-defined preference metric. The coalition formation gamecontinues until the coalition partition reaches Nash-stable state,and it outputs the relay selection matrix Y .



8

Resource Allocation

Relay Selection

Initial XD, XR

Relay Selection is Nash-stable?

XD, XR

NO

YES Output

Y

XD, XR, Y

Fig. 4. Illustration of two-level optimization approach of NC-D2D.

(c) Given Y , the resource allocation is solved with a greedyalgorithm based game. All the D2D users and relays take turnto choose a cellular user to share its uplink spectrum resource thatachieves highest transmitting capacity, until the process convergesto Nash-stable state, with the outputs XD and XR.

(d) NC-D2D checks if the relay selection’s coalition partitionis also Nash-stable at this point. If it is, NC-D2D outputs thecurrent Y , XD and XR as the joint solution, since the wholesystem has reached stable state. Otherwise, if the relay selectionis no longer stable due to resource allocation, NC-D2D repeatssteps (b), (c) and (d), until both the relay selection and resourceallocation converge to Nash-stable state.

Each time NC-D2D execute steps (b) and (c) is called oneround. Our NC-D2D achieves de-centralized control, where eachD2D pair and relay only requires local information of the network.This reduces the overhead significantly compared to centralizedcontrol, where all the nodes need to be updated with the infor-mation and topology of the whole network. These two games andtheir definitions of Nash-stability will be introduced in detail inSections 5 and 6, respectively.

5 RELAY SELECTION COALITION FORMATIONGAME

In a coalition game, the players form coalitions to improve thesystem utility. Since there are two kinds of D2D pairs in oursystem, relay assisted D2D pairs and ordinary D2D pairs, weconsider two kinds of coalitions in the coalition formation game.The first kind of coalition is formed by relay and correspondingrelay assisted D2D pairs. Let Frj represent the coalition of thecoding region where rj is in, which also consists of D2D pairsdα(j) and dβ(j). The number of first-kind coalitions equals to thenumber of relays in the network, which is fixed. The second kindof coalition is denoted by FD, which consists of all the ordinaryD2D pairs in the network.

In this coalition formation game, the players, namely, D2Dpairs, swap coalitions in order to optimize the overall systemperformance. The decision of whether to swap coalition or notshould be made according to a pre-defined preference order thatapplies to all the players. For the sake of achieving high sumcapacity, the metric that defines the preference order in ourcoalition formation game should be related to the system sumcapacity, while each node should be able to obtain it by relyingonly on local network information.

Two D2D pairs in different coalitions may swap coalitionsif the system metric csum(XD,XR,Y ) can be improved. Sincethere are two kinds of D2D pairs as well as two kinds of coalitions

in the system, there are two kinds of coalition swapping amongD2D pairs: swapping among ordinary D2D pairs and relay assistedD2D pairs, as well as swapping among relay assisted D2D pairs.

5.1 Swapping Among Ordinary D2D Pairs and RelayAssisted D2D PairsGiven an initialized coalition partition, for D2D pair di in Frj anddk in FD , if the system sum capacity can be increased after di anddk swap their coalitions, then di should leave the coding regionof rj to switch to ordinary D2D mode while dk should switch torelay assisted mode and form coding region with rj . The systemsum capacity defines the preference orders of the players in termsof swapping coalition. It should be noted that we do not need tocompute the system sum capacity when trying to determining thepreference order. For fixed resource allocation matrices XD andXR, such a coalition swapping does not change the transmissioncapacities of other links and coding regions except the capacitiesof Frj and dk, due to the fact that the interferences to the restof the links remains the same. Therefore, di and dk should swapcoalition if

cCR(rj , dk, dm) + c(dti, dri ) ≥ cCR(rj , di, dm) + c(dtk, d

rk)

and xdk,uxri,uydk,ri = 0, ∀u, (47)

where dm is the other D2D pair assisted by rj . Since the systemconstraints must be maintained, the swapping of coalitions cannottake place if any of the constraints is violated. It can be seen thatthis swapping process is completely decentralized and only localnetwork information is required by the users. Specifically, di onlyneeds to know the capacities of its relay assisted coding regionand its direct link, while dk only need to know the capacities ofits direct link and the potential coding region involving relay rjand D2D pair dm.

In the case that the condition (47) is not satisfied, whichmeans that the two D2D pairs prefer to stay in their currentcoalitions, a chance for swapping should also be considered. Forthis chance, we design an acceptance probability for di and dk toswap coalitions, which is specified as

φFD,Frj (TN ) =exp

(cCR(rj ,dk,dm)+c(dti,d

ri )−cCR(rj ,di,dm)−c(dtk,drk)TN

),

if xdk,uxrj ,uydk,rj = 0, ∀u,0, otherwise,

(48)

where TN = T0/ log(N−1) with T0 a constant andN the currentiteration index. The reason for allowing users to swap coalitionsby chance is that the coalition formation based on maximizingthe sum capacity is guaranteed to converge to a local optimalsolution, which may deviate from the global optimal solution,since the optimization problem is non-convex. This swappingcoalitions by chance provides a mechanism for the system toescape from local optimal solutions. The acceptance probabilitygradually approaches to zero as the number of iterations increases,which ensures that the system is able to form stable coalitions.

5.2 Swapping Among Relay Assisted D2D PairsFor relay assisted D2D pairs di in coalition Frj and dkin Frl ,where j 6= l, they swap their coalitions and form coding regionswith the other relays dm and dn if doing so increases the systemsum capacity, where dm and dn are the other two D2D pairsin coalitions Frj and Frl , respectively. More specifically, di



9

and dk will leave their current coalitions and join Frl and Frj ,respectively, if

cCR(rj , dk, dm) + cCR(rl, di, dn) ≥ cCR(rj , di, dm)

+ cCR(rl, dk, dn), and xdk,uxrj ,uydk,rj = 0, ∀u,and xdi,uxrl,uydi,rl = 0, ∀u. (49)

Similarly, a chance for di and dk to swap coalitions should beconsidered even if the new coalitions are not preferred in terms ofthe system metric. The acceptance probability is given byφFrl ,Frj (TN ) =

exp(cCR(rj ,dk,dm)+cCR(rl,di,dn)−cCR(rj ,di,dm)−cCR(rl,dk,dn)

TN

),

if xdk,uxrj ,uydk,rj = 0,∀u,0, otherwise,

(50)

with TN = T0/ log(N−1). Again only local network informationis used to decide whether to swap coalitions or not.

5.3 Switching from Network Coding to Ordinary D2DTransmissionsIt should be noted that network coding might not be the betteroption over ordinary D2D transmissions for all D2D pairs atall time. In some particular cases, the capacity gain of networkcoding might be negative for some D2D pairs, which means theyshould switch to ordinary D2D mode. To avoid negative networkcoding gain, all relay assisted D2D pairs compare the capacityachieved by network coding with the capacity by ordinary D2Dtransmission at the end of the relay selection coalition formationgame. If the latter one is higher for any coalition, then the twoD2D pairs should switch back to ordinary D2D mode and thecorresponding relay switch to idle mode. For example, for dα(j)and dβ(j) assisted by rj , if c(dtα(j), d

rα(j)) + c(dtβ(j), d

rβ(j)) >

cCR(rj , dα(j), dβ(j)), then dα(j) and dβ(j) switch back to ordi-nary D2D mode.

5.4 Nash-Stability of Coalition Formation Game in Re-lay SelectionThe iterative procedure of coalition swapping ends when thecoalition partition converges to a Nash-stable state. The definitionof Nash-stability for relay selection is as follows.Definition 1. The coalition partition for relay selection is Nash-

stable if the sum capacity decreases after di and dj swap theircoalitions for any two D2D pairs di and dj in the system [25].

Since our optimization problem is non-convex, Nash-stablepartition only guarantees a local optimal solution. The completealgorithm of coalition formation game for relay selection is sum-marized in Algorithm 1.

5.5 Stability AnalysisAccording to the concept from the hedonic games [26], thestability of the final partition depends on weather a Nash-stablesolution exists for the coalition game. Let us denote the finalpartition obtained by the coalition game as Ffin. The stabilityof our proposed game is guaranteed by Theorem 1.Theorem 1. Starting from any initial coalition partition, our

coalition formation algorithm always converges to Nash-stablepartition Ffin with probability 1.

Algorithm 1: Coalition Formation Algorithm for RelaySelection1 Initialize the system by a random partition, N = 0;2 while Partition does not converge to Nash-stable state do3 Set N = N + 1, TN = T0/ log(N − 1);4 Uniformly randomly select di from all relay assisted

D2D pairs, and denote its coalition as Frj ; Uniformlyrandomly select dk from FD;

5 if xdk,uxrj ,uydk,rj = 0, ∀u then6 if cCR(rj , dk, dm) + c(dti, d

ri ) ≥

cCR(rj , di, dm) + c(dtk, drk) then

7 dk and di swap coalitions;8 Update the current partition as follows

Frj \ {di} ∪ {dk} → Frj ;9 FD \ {dk} ∪ {di} → FD;

10 else11 Draw a random number % uniformly distributed

in (0, 1];12 if % < φFD,Frj (TN ) then13 dk and di switch coalitions;14 Update the current partition as follows

Frj \ {di} ∪ {dk} → Frj ;15 FD \ {dk} ∪ {di} → FD;

16 Uniformly randomly select di and dk from all relayassisted D2D pairs, and denote their coalitions as Frjand Frl , respectively, where j 6= l;

17 if xdk,uxrj ,uydk,rj = 0 & xdjiuxrl,uydi,rl = 0, ∀uthen

18 if cCR(rj , dk, dm) + cCR(rl, di, dn) ≥cCR(rj , di, dm) + cCR(rl, dk, dn) then

19 dk and di swap coalitions;20 Update the current partition as follows

Frj \ {di} ∪ {dk} → Frj ;21 Frl \ {dk} ∪ {di} → Frl ;22 else23 Draw a random number % uniformly distributed

in (0, 1];24 if % < φFrl ,Frj (TN ) then25 dk and di swap coalitions;26 Update the current partition as follows

Frj \ {di} ∪ {dk} → Frj ;27 Frl \ {dk} ∪ {di} → Frl ;

28 Obtain the output Y according to Fr and FD.

Proof: The maximum number of partitions in the game isR+1, where R is the number of relays in the network. Therefore,the number of partitions for the D2D pairs set D is the Bellnumber in the coalition formation game [27]. Thus, the swappingoperations of the D2D pairs will terminate in a finite time withprobability 1, where the system converges to the final stablepartition Ffin. Suppose that the final partition Ffin is not Nash-stable. Then the system capacity will increases if certain D2Dpair di in coalition Fk and dj in Fl swap coalitions, accordingto the definition of Nash-stable. In this case, our algorithm willbe able to perform a swap operation for these two D2D pairswith probability 1, which contradicts to the fact that Ffin is the



10

final partition. This proves that the final coalition partition Ffinobtained by Algorithm 1 is Nash-stable.

6 RESOURCE ALLOCATION GAME

Given a relay selection coalition partition Y , we use a greedyalgorithm based game to solve the resource allocation problem.Similar to relay selection, our resource allocation game has a verylow computational complexity and it is capable of converging to astable local optimal solution.

6.1 Greedy Algorithm Based Resource AllocationGameIn the resource allocation game, all the relays and D2D pairs selectwhich spectrum resource to utilize in sequence. To ensure fairnessin resource allocation, the sequence is random. Each time a relayor a D2D pair is randomly selected from all the relays and D2Dpairs to choose a cellular user to share its spectrum. The selectedrelay or D2D pair selects the spectrum resource that achievesthe highest capacity contribution. If the resource allocation doesnot converge to Nash-stable state after all the D2D pairs andrelays have been allocated with the spectrum resource, a newsequence is generated, according to which each node select thespectrum resource that achieves the highest capacity contributionagain. The nodes may select different resources due to the factthat other nodes may occupy different spectrum resources andthe interference is different, which leads to variations in linkcapacities. This process is repeated until the resource allocationis Nash-stable.

The capacity contribution of a node is the contribution of thisnode to the sum capacity of the network. In selecting spectrum re-source, each D2D pair or relay computes its capacity contributionby only utilizing local network information.Definition 2. For an ordinary D2D pair di, its capacity con-

tribution is the capacity of its direct link c(dti, dri ). For a

relay assisted D2D pair di in the coding region of relayr with the other D2D pair dj , the capacity contributionof di is min

{c(dti, d

rj), c(d

ti, r)

}. For relay r, its capacity

contribution is min{c(r, dri ), c(r, d

rj)}

.

Definition 3. The resource allocation XD and XR is Nash-stableif the capacity contribution of any node d or r decreases after itchanges to share a different cellular user’s spectrum resource.

The complete algorithm of our resource allocation game issummarized in Algorithm 2.

6.2 Stability AnalysisSimilar to the relay selection game, the stability of the resourceallocation game as given in Algorithm 2 is guaranteed.Theorem 2. The proposed resource allocation game converges to

Nash-stable state with probability 1.

Proof: Although the resource allocation game is not acoalition game, the basic idea of proving its convergences issimilar to that for the relay selection game. For each D2D pairor relay, the maximum number of choices for spectrum resource isC, which is the number of cellular users. Therefore, the solutionspace of resource allocation is finite. This guarantees that theconvergence probability is 1. Suppose that the final solution ofthe resource allocation is not Nash-stable. Then there exists someD2D pair di or relay rj that is able to increase the system capacity

Algorithm 2: Resource Allocation Game

1 Initialize sets Ψ = R∪D, Υ = ∅;2 while Resource allocation does not converge to Nash-stable

state do3 Set N = N + 1;4 if Ψ = ∅ then5 Set Ψ = R∪D, Υ = ∅;6 Uniformly randomly select a node i from Ψ;7 Select the spectrum resource that achieves the highest

capacity contribution for i while satisfying constraint(45);

8 Denote the cellular user to whom the selected spectrumresource belongs to as u;

9 Update Ψ = Ψ \ {i}, Υ = Υ ∪ {i};10 Update resource allocation matrices XD and XR

accordingly;11 Output XD and XR.

by changing to share another cellular user uk’s spectrum resource.Based on our algorithm, the probability of di or relay rj to selectuk’s spectrum resource is 1, which contradicts to the fact that thecurrent solution is the final stable solution. This proves the Nash-stability of Algorithm 2.

7 PERFORMANCE EVALUATION

In this section, we evaluate the performance of our NC-D2D todemonstrate that it is capable of achieving high system sum capac-ity and outperforming other existing state-of-the-art schemes. Westart by introducing our simulation setup. Then the performanceof our relay selection coalition formation game and resourceallocation game as well as the overall performance of NC-D2Dare investigated, respectively.

7.1 Simulation Setup

In our simulations, the relays, D2D pairs and cellular users aredeployed in a cell, covering a circle area with a radius of 500 m,and the BS is located in the cell center. As mentioned in Section 3,we assume Rayleigh fading, and adopt Friis transmission equationto calculate the path loss of the transmitted signal [23]. We set theuplink bandwidth of each cellular user to be 15 kHz. We assumeGaussian noise with the power of 132 dBm for all channels. Thetransmission power is assumed to be 0 dBm for all relays and D2Dusers, and 23 dBm for all cellular users. The antenna gains of allrelays, D2D users and cellular users are set to be identical to 0 dBi,while the BS’s antenna gain is set to be 14 dBi [6]. The parametersof the simulated system are also listed in Table 1. The D2D pairsand cellular users are uniformly randomly distributed in the cell.We simply assume that when two users are within the proximityof (10, 100) m, they are able to form a D2D pair. The relaysare uniformly randomly deployed on a circle with radius 250 m,centered at the BS.

We evaluate our scheme in five different network setups. Thenumbers of D2D pairs and cellular users in network setups 1 to4 are identical, which are 12 and 5, respectively. The numbers ofrelays in network setups 1 to 4 are set to 3, 4, 5 and 6, respectively.The locations of D2D pairs and cellular users in setups 1 to 4are identical. That is, we only change the numbers of relays and



11

TABLE 1Simulation Parameters.

Parameter Symbol ValueUplink Bandwidth W 15 kHz

Gaussian noise power N -132 dBmD2D, relay transmission power PD

T , PRT 0 dBm

Cellular transmission power PUT 23 dBm

User transmitter antenna gain GDT , GR

T , GUT 0 dBi

User receiver antenna gain GDR , GR

R , GUR 0 dBi

BS receiver antenna gain GBSR 14 dBi

1 2 3 4 50

5

10

15

20

Network Setup

Sum

Capacity (

kbps)

Final CapacityInitial CapacityCapacity Wo NCOptimal Solution

Fig. 5. Comparison of sum capacities in relay selection coalitiongame under network setups 1 to 5.

3 4 5 6 7 8 9 10 110

0.2

0.4

0.6

0.8

1

Sum Capacity (kbps)

Su

m C

ap

acity C

DF

Capacity Wo NC FC S1 FC S2 FC S3 FC S4 IC S1 IC S2 IC S3 IC S4

Fig. 6. Comparison of sum-capacity CDFs in relay selectioncoalition game under network setups 1 to 4.

relays’ locations in network setups 1 to 4. The network topologyof setup 5 is different from setups 1 to 4, with 6 relays, 18 D2Dpairs, and 8 cellular users.

Since neither the relay selection problem nor the resourceallocation problem is convex, NC-D2D only guarantees stablelocal optimal solutions, which is partially determined by theinitial value. Hence we simulate 100 times for each topologyand scenario, and evaluate the mean value and the cumulativedistribution function (CDF).

6 8 10 12 14 160

0.5

1

Sum Capacity (kbps)

Su

m C

ap

acity C

DF

FC Setup 5

IC Setup 5

Capacity Wo NC

Fig. 7. Comparison of sum-capacity CDFs in relay selectioncoalition game under network setup 5.

7.2 Relay Selection Coalition Formation Game Perfor-mance EvaluationThe mean values and variances of the system sum capacitiesunder network setups 1 to 5 are plotted in Fig. 5, while theCDFs of sum capacity under network setups 1 to 4 are plottedin Fig. 6, and the CDFs of sum capacity under network setup 5 aredepicted in Fig. 7. In these figures, ‘Capacity Wo NC’ indicates thesystem sum capacity achieved by standard D2D communicationswithout the assistant of either relay nodes or network coding,and ‘Final Capacity’ or ‘FC’ indicates the system sum capacityobtained after the relay selection coalition game converges toNash-stable state. The ‘Capacity Wo NC’ acts as a reference toindicate the performance gain achievable by relay selection andnetwork coding. In Algorithm 1, the coalition game needs an initialcoalition partition, which is randomly generated. The system sumcapacity of this randomly generated initial coalition partition isdenoted by ‘Initial Capacity’ or ‘IC’ in these figures. ‘OptimalSolution’ in Fig. 5 is the true optimal solution of the relay selectionproblem, obtained by exhaustive search. Also, labels ‘S1’ to ‘S4’in Fig. 6 represent network setups 1 to 4.

From the results, we observe that the relay selection gameincreases the sum capacity significantly, compared to the ‘InitialCapacity’ and the ‘Capacity Wo NC’, which indicates that therelay selection game helps the system evolving to the local optimalstate that achieves higher system capacity. The average capacitiesafter the relay selection game under 5 network setups outperformthose of the ‘Capacity Wo NC’ by 44.83%, 52.33%, 64.82%,74.28% and 70.52%, respectively. The results also show that therelay selection game is able to achieve near-optimal performance.The optimal solutions only outperform our proposed game by0.23%, 5.99%, 7.80%, 5.55%, and 5.47%, respectively, under 5network setups. As expected, the achievable capacity increaseswith the number of relays, since more relays in the networkmeans that more D2D pairs are able to utilize relay assistednetwork coding to assist their transmissions. Although our relayselection game only guarantees stable local optimal solutions, wefind that it achieves the global optimal solutions in 62%, 71%,61%, 42% and 41% of all the 100 simulations under networksetups 1 to 5, respectively. The high probability of obtaining theglobal optimal solution further indicates the effectiveness of ourproposed scheme.

We also investigate the number of iterations required by therelay selection game to converge to stable solutions in thesefive setups, and Table 2 shows the average values as well as



12

TABLE 2Number of iterations required by the relay selection coalition

game in network setups 1 to 5.

Network Setup Setup 1 Setup 2 Setup 3 Setup 4 Setup 5Average 71.5 75.3 104.0 177.4 226.6

Maximum 383 312 415 780 933Minimum 70 67 87 144 209

1 2 3 4 50

5

10

15

20

25

Network Setup

Sum

Capacity (

kbps)

Resource GameRandom AllocationOptimal Solution

Fig. 8. Comparison of sum capacities in resource allocationgame under network setups 1 to 5.

the maximum and minimum values over all the simulations. Asexpected, the average number of iterations increases with thescale of the network. Most striking observation is that the averagenumbers of iterations are very close to the minimum values underall the five setups. This indicates that only in very few simulationsit costs high number of iterations for the relay selection game toconverge, while most of the time the game is able obtain stablesolutions very quickly.

7.3 Resource Allocation Game Performance EvaluationWe next evaluate the performance of the resource allocationgame in these five setups. The mean values of the system sumcapacities under different network setups are plotted in Fig. 8.Since we focus on the performance of resource allocation inthis part, we omit the capacity achieved without network coding.It can be seen that our resource allocation game outperformsconsiderably the random allocation, where each D2D pair andrelay uniformly and randomly select one cellular user to share itsuplink resource. Specifically, the sum capacities achieved by theresource allocation game are 74.79%, 73.25%, 71.68%, 75.45%and 75.32% higher than those of the random allocation undernetwork setups 1 to 5, respectively. Also our resource allocationgame attains near-optimal solution, as can be seen from Fig. 8.In particular, the optimal solutions obtained by exhaustive searchin these five setups only outperform the sum capacities achievedby our resource allocation game by 1.94%, 2.21%, 4.55%, 4.29%and 8.22%, respectively. As expected, the sum capacity achievedby the resource allocation game increases with the network scale.

Table 3 depicts the average number as well as the maximumand minimum numbers of iterations for the resource allocationgame to converge. As expected, the number of iterations requiredincreases with the network scale but the maximum numbers ofiterations required in all the setups are all fewer then 200. Observe

TABLE 3Number of iterations required by the resource allocation game

in network setups 1 to 5.

Network Setup Setup 1 Setup 2 Setup 3 Setup 4 Setup 5Average 29.3 31.2 32.8 48.0 52.2

Maximum 92 114 123 169 175Minimum 26 26 24 30 28

5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Sum Capacity (kbps/Hz)

Su

m C

ap

acity C

DF

NC−D2D

Relay Selection

Resource Allocation

Capacity Wo NC

Fig. 9. Sum-capacity CDFs under network setup 2.

that the average number of iterations is only around 50 forthe largest network setup 5. Moreover, the average numbers ofiterations are very close to the minimum values under all the fivesetups. Thus, the resource allocation game converges fast, and it isable to achieve near-optimal solution with very low computationcost.

7.4 NC-D2D Performance EvaluationWe now ready to evaluate the performance of the NC-D2Dscheme. Fig. 9 plots the CDF of the sum capacity achieved bythe NC-D2D under network setup 2, in comparison with thoseof the relay selection game alone and the resource allocationgame alone as well as the sum-capacity CDF obtained withoutenabling relay assisted network coding, i.e., ‘Capcity Wo NC’.Other network setups are omitted since the results are similarto those of Fig. 9. From Fig. 9, it can clearly be seen thatboth the relay selection game alone and the resource allocationgame alone outperform the ‘Capcity Wo NC’, which is just theconfirmation of the previous evaluation results. As pointed outpreviously, the solutions of the relay selection game and theresource allocation game are inter-dependent of each other. Byiterating the two games for multiple times until both the relayselection and resource allocation converge to Nash stable solutionsin a two-level optimization, the proposed NC-D2D significantlyoutperforms the results of applying these two games separately.For example, from the CDFs of Fig. 9, we observe that the NC-D2D outperforms the relay selection alone by 135% in 88% of thesimulations.

To quantify the benefits of this two level-optimization, wedefine the capacity gain of the NC-D2D in round i as

Capacity gain of round i =

Capacity of round i − Capacity of round i− 1

Capacity of round i− 1, (51)

where the Capacity of round 0 is the system sum capacitywith a random selection of relays and the associated allocated



13

0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

Round

Ca

pa

city G

ain

Fig. 10. Average capacity gain of NC-D2D in each round.

1 2 3 4 50

10

20

30

40

50

60

Network Setup

Su

m C

ap

acity (

kb

ps)

NC−D2D

Only Relay Selection

Only Resource Allocation

Random Allocation

Optimal Solution

Capacity Wo NC

Fig. 11. Sum capacity in network setup 1 to 5.

resources. Obviously, for different network setups and differentinitial values for the two games, the NC-D2D may take differentnumbers of rounds to obtain the final stable solution. In all oursimulations, the NC-D2D obtains the final stable solutions with nomore than three rounds. Fig. 10 depicts the average capacity gainof the NC-D2 in each round. As a common feature of iterativeprocedure, the capacity increases more slowly as the systemconverges to the final stable state. Even so, round 3 increases thesum capacity by an average of 40% over the sum capacity attainedin round 2.

Next we plot the average sum capacities and the variances ofthe NC-D2D in Fig. 11 for network setups 1 to 5, in comparisonwith the average sum capacities of only applying relay selection,only applying resource allocation, randomly selecting relays andallocating resources, the optimal solution, and the capacity withoutnetwork coding. As expected the NC-D2D outperforms both therelay selection game alone and the resource allocation game alonesignificantly on all the five setups. Moreover, the performanceof the NC-D2D is very closed to those of the optimal solution.Specifically, the optimal solution only outperforms the NC-D2Dby 2.4%, 9.6%, 10.2%, 6.2%, and 5.3%, respectively, for network

TABLE 4Average running times by the NC-D2D and the optimal solution.

Algorithm Setup 1 Setup 2 Setup 3 Setup 4 Setup 5NC-D2D 2.8 s 3.1 s 3.9 s 4.5 s 7.3 sOptimal 368.6 s 416.0 s 677.3 s 943.5 s 2757.1 s

Fig. 12. Percentages of numbers of rounds for the NC-D2D toconverge in network setups 1 to 5.

setups 1 to 5. Compared with the capacity without network coding,NC-D2D achieves the capacity gain of 270%, 351%, 457%, 456%,and 374%.

In the simulation, the optimal solution is obtained by solvingthe optimization problem (46) based on exhaustive search withpruning. Our simulations were run on Matlab, in a laptop withMac OS X Yosemite, 2.3 GHz Intel Core i7 processor and 16 GB1600 MHz DDR3 RAM. Table 4 lists the average times consumedby the NC-D2D and the optimal solution in network setups 1 to5. Although the sum capacity achieved by the NC-D2D is slightlylower than the optimal solution, the computing time required bythe optimal solution is 1000 folds higher that required by the NC-D2D. We also observe that the running time of the NC-D2D scalesup linearly with the number of nodes in the network, while therunning time of the optimal solution increases exponentially withthe network scale. A key reason for the NC-D2D’s high efficiencyis that it obtains the final solution with only a few rounds. In allour simulations, the NC-D2D converges with no more than threerounds. The percentages of numbers of rounds for the NC-D2Dto converge in setups 1 to 5 are plotted in Fig. 12. We can seethat about 80% of the simulations in network setups 1 to 4, theNC-D2D converges in two rounds, while in setup 5, it convergesin two rounds for around 57% of the simulations.

8 CONCLUSIONS

In this paper, we have introduced network coding to enhancethe performance of D2D communications underlying cellularnetworks, where one relay aids two D2D pairs’ transmissionsby performing inter-session network coding. Specifically, we havefirst formulated the joint problem of relay selection and resourceallocation under realistic system constraints for the network cod-ing assisted D2D communications underlying cellular network.



14

Since this optimization is NP-hard, we have proposed a highlyefficient near-optimal scheme to solve this joint optimization,referred to as the NC-D2D, which is a two-level de-centralizedoptimization scheme that solves the relay selection game and theresource allocation game alternatively. In particular, the relay se-lection problem is solved with a coalition formation game, whereD2D pairs and relays form and swap coalitions for the purposeof enhancing system capacity. The resource allocation game isbased on a greedy algorithm to allocate the spectral resourceof cellular users to the D2D pairs and relays appropriately. Ourextensive simulations have verified that the proposed NC-D2Dscheme attains a near-optimal system sum capacity, while onlyimposing a fraction of the computational complexity required bythe optimal solution. The results thus have demonstrated that theNC-D2D can easily be implemented in practical cellular systems.A future study is to extend the NC-D2D to dynamic networks,where the mobility patterns of users may be utilized. Future workmay include optimizing power control and the formation of codingregion to further improve the system utility.

REFERENCES

[1] Y. Gu, Y. Zhang, M. Pan, and Z. Han, “Matching and cheating in deviceto device communications underlying cellular networks,” IEEE J. Sel.Areas Commun., vol. 33, no. 10, pp. 2156–2166, Oct. 2015.

[2] X. Lin, J. G. Andrews, A. Ghosh, and R. Ratasuk, “An overview of 3GPPdevice-to-device proximity services,” IEEE Commun. Mag., vol. 52,no. 4, pp. 40–48, Apr. 2014.

[3] L. Lei, Z. Zhong, C. Lin, and X. Shen, “Operator controlled device-to-device communications in LTE-advanced networks,” IEEE WirelessCommun., vol. 19, no. 3, p. 96–104, Jun. 2012.

[4] Y. Li, D. Jin, J. Yuan, and Z. Han, “Coalitional games for resourceallocation in the device-to-device uplink underlaying cellular networks,”IEEE Trans. Wireless Commun., vol. 13, no. 7, pp. 3965–3977, Jul. 2014.

[5] C.-H. Yu, K. Doppler, C. B. Ribeiro, and O. Tirkkonen, “Resourcesharing optimization for device-to-device communication underlayingcellular networks,” IEEE Trans. Wireless Commun., vol. 10, no. 8,pp. 2752–2763, Aug. 2011.

[6] C. Xu, L. Song, Z. Han, Q. Zhao, X. Wang, X. Cheng, and B. Jiao, “Effi-ciency resource allocation for device-to-device underlay communicationsystems: a reverse iterative combinatorial auction based approach,” IEEEJ. Sel. Areas Commun., vol. 31, no. 9, pp. 348–358, Sep. 2013.

[7] N. Lee, X. Lin, J. G. Andrews, and R. W. Heath, “Power control for D2Dunderlaid cellular networks: modeling, algorithms, and analysis,” IEEEJ. Sel. Areas Commun., vol. 33, no. 1, pp. 1–13, Jan. 2015.

[8] J. Liu, M. Sheng, X. Wang, Y. Zhang, H. Sun, and J. Li, “Analysis oftransmission capacity region in D2D integrated cellular networks withpower control,” in Proc. ICC 2015 (London), Jun. 8-12, 2015, pp. 2103–2109.

[9] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, “Network informa-tion flow,” IEEE Trans. Inform. Theory, vol. 46, no. 4, pp. 1204–1216,Jul. 2000.

[10] P. Pahlevani, M. Hundebøll, M. V. Pedersen, D. E. Lucani, H. Charaf,F. H. Fitzek, H. Bagheri, and M. Katz, “Novel concepts for device-to-device communication using network coding,” IEEE Commun. Mag.,vol. 52, no. 4, pp. 32–39, Apr. 2014.

[11] T. Ho and H. Viswanathan, “Dynamic algorithms for multicast with intra-session network coding,” IEEE Trans. Inform. Theory, vol. 55, no. 2,pp. 797–815, Feb. 2009.

[12] S. Cai, S. Zhang, G. Wu, Y. Dong, and T. Znati, “Minimum costopportunistic routing with intra-session network coding,” in Proc. ICC2014 (Sydney, NSW), Jun. 10-14, 2014, pp. 502–507.

[13] S. Katti, H. Rahul, W. Hu, D. Katabi, M. Medard, and J. Crowcroft,“XORs in the air: practical wireless network coding,” IEEE/ACM Trans.Networking, vol. 16, no. 3, pp. 497–510, Jun. 2008.

[14] Z. Liu, J. Wang, W. Xu, S. Li, and J. Lin, “Inter-session inter-layernetwork coding-based dual distributed control for heterogeneous-servicenetworks,” in Proc. WCNC 2014 (Istanbul, Turkey), Apr. 6-9, 2014,pp. 3230–3235.

[15] H. Seferoglu, A. Markopoulou, and K. K. Ramakrishnan, “I2NC: intra-and inter-session network coding for unicast flows in wireless networks,”in Proc. INFOCOM 2011 (Shanghai, China), Apr. 10-15, 2011, pp. 1035–1043.

[16] J. Krigslund, J. Hansen, M. Hundeboll, D. E. Lucani, and F. H. Fitzek,“CORE: cope with more in wireless meshed networks,” in Proc. VTC-Spring 2013 (Dresden, Germany), Jun. 2-5, 2013, pp. 1–6.

[17] A. Bhorkar, C. Ibars, and P. Zong, “Algorithm design and analysis fornetwork coding in D2D underlay communications,” in Proc. WCNCW2015 (New Orleans, LA), Mar. 9-12, 2015, pp. 317–322.

[18] E. M. Maher and K. S. Hassan, “Network coding gain in device-to-deviceunderlaying primary communications,” in Proc. CCS 2014 (Rhine River,Germany), Sep. 2-4, 2014, pp. 1–5.

[19] Y. Wu, W. Liu, S. Wang, W. Guo, and X. Chu, “Network coding indevice-to-device (D2D) communications underlaying cellular networks,”in Proc. ICC 2015 (London), Jun. 8-12, 2015, pp. 2072–2077.

[20] L. Wei, G. Wu, and R. Q. Hu, “Multi-pair device-to-device communi-cations with space-time analog network coding,” in Proc. WCNC 2015(New Orleans, LA), 2015, pp. 920–925.

[21] S. Wen, X. Zhu, Y. Lin, Z. Lin, X. Zhang, and D. Yang, “Achievabletransmission capacity of relay-assisted device-to-device (D2D) commu-nication underlay cellular networks,” in Proc. VTC Fall-2013 (Las Vegas,NV), Sep. 2-5, 2013, pp. 1–5.

[22] M.-H. Han, B.-G. Kim, and J.-W. Lee, “Subchannel and transmissionmode scheduling for D2D communication in OFDMA networks,” inProc. VTC Fall-2012 (Quebec, QC), Sep. 3-6, 2012, pp. 1–5.

[23] H. T. Friis, “A note on a simple transmission formula,” Proceedings ofthe IRE, vol. 34, no. 5, pp. 254–256, May 1946.

[24] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introductionto Algorithms. MIT Press, 2001.

[25] A. Bogomolnaia and M. O. Jackson, “The stability of hedonic coalitionstructures,” Games and Economic Behavior, vol. 38, no. 2, pp. 201–230,Feb. 2002.

[26] E. Hossain, D. I. Kim, and V. K. Bhargava, eds., Cooperative CellularWireless Networks. Cambridge University Press, 2011.

[27] W. Saad, Z. Han, M. Debbah, A. Hjørungnes, and T. Basar, “Coalitionalgame theory for communication networks,” IEEE Signal ProcessingMag., vol. 26, no. 5, pp. 77–97, Sep. 2009.

Chuhan Gao is currently working toward theB.E. degree with Tsinghua University, Beijing,China. His research interests include millimeter-wave wireless communications, wireless net-works, and software-defined networks.

Yong Li (M’09-SM’16) received the B.S. degreein electronics and information engineering fromHuazhong University of Science and Technol-ogy, Wuhan, China, in 2007 and the Ph.D. de-gree in electronic engineering from TsinghuaUniversity, Beijing, China, in 2012. He is cur-rently a Faculty Member of the Department ofElectronic Engineering, Tsinghua University.

Yulei Zhao received his B.S. and M.S. degreesfrom the Department of Communication Engi-neering, Zhengzhou Information Science andTechnology Institute, Zhengzhou, Henan, China,in 2005 and 2008, respectively. He is currentlyworking toward the Ph.D degree in the Depart-ment of Electronic Engineering, Tsinghua Uni-versity, Beijing, China. His research interestsinclude cooperative communications, device-to-device communications, and social networks.

Sheng Chen (M’1990-SM’1997-F’2008) ob-tained his BEng degree from the East ChinaPetroleum Institute, Dongying, China, in January1982, and his PhD degree from the City Univer-sity, London, in September 1986, both in controlengineering. In 2005, he was awarded the higherdoctorate degree, Doctor of Sciences (DSc),from the University of Southampton, Southamp-ton, UK.

Documents

A Two-Level Game Theory Approach for Joint Relay Selection ...chuhan/wp-content/uploads/chuhan_TMC_NCD2D.pdfmulti-antenna relay employing space-time analog network coding. However,