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Social-Aware Resource Allocation for Device-to-Device Communications Underlaying Cellular Networks Yulei Zhao, Yong Li, Member, IEEE, Yang Cao, Member, IEEE, Tao Jiang, Senior Member, IEEE, and Ning Ge, Member, IEEE Abstract—The ever-increasing demands for local area services underlaying cellular networks benefit from direct device-to-device (D2D) communications, where an efficient scheme for resource allocation is needed to increase the system capacity as the result of interference caused by spectrum sharing. Current works mainly focus on maximizing the overall transmission capacity according to interference constraints of the physical domain. However, D2D users in the social domain form different social communities, and each social community is likely to improve its own group’s data transmission cooperatively without considering other communities. Therefore, social relationships among mobile users influence the strategy of the resource allocations for the D2D communications. In this paper, we first introduce social relationships in the continuum space into the resource allocation for D2D communications, which consider the complex social connections in the social domain. Then a social group utility maximization game is formulated to maximize the social group utility of each D2D user, which quantitatively measures the joint performance of social and physical domains. We theoretically investigate the Nash Equilibrium of our proposed game and further propose a distributed algorithm based on the switch operations of the resource allocation vector. Numerical results demonstrate that our proposed solution increases the utility of overall social groups about 45% on average without loss of the fairness compared with other state-of-the-art schemes. Index Terms—Device-to-device communication, social group utility, game theory, resource allocation.

I. I NTRODUCTION

W

ITH the popularity of wireless access and mobile Internet, Cisco estimates that mobile traffic will grow at an annual rate of 81% from 2014, and reach over 15 exabytes per

Manuscript received January 11, 2015; revised May 7, 2015 and July 6, 2015; accepted July 6, 2015. Date of publication July 16, 2015; date of current version December 8, 2015. This work was supported by the National Basic Research Program of China (973 Program) under Grant 2013CB329001, by the National Nature Science Foundation of China under Grant 61132002 and Grant 61301080, and by the Creative Research Group Program from NSFC (61321061). The associate editor coordinating the review of this paper and approving it for publication was J. Huang. Y. Zhao, Y. Li, and N. Ge are with Tsinghua National Laboratory for Information Science and Technology (TNLIST), Department of Electronic Engineering, Tsinghua University, Beijing 100084, China (e-mail: [email protected] mails.tsinghua.edu.cn; [email protected]; [email protected]). Y. Cao and T. Jiang are with School of Electronics Information and Communications, Huazhong University of Science and Technology, Wuhan 430074, P.R. China (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TWC.2015.2457427

month in 2018 [1]. Local area services of popular content sharing are emerging demands for geographically proximate users. For example, friends share photos, music and videos with each other through smart phones. Current mobile content sharing is provided by traditional cellular and WiFi networks [2], which will be overloaded and congested in the near future [3]. Deviceto-Device (D2D) communications enable the proximity cellular users to communicate directly, instead of through a Base Station (BS) [3]. Local data services are important usage cases for D2D communications through peer-to-peer scheme [4]. For example, when two friends are in geographical proximity, they want to exchange photos or videos via their smart phones. Furthermore, a smart phone may connect to a television to display the photo or video. The D2D communications enable the proximity cellular users to share the same interested contents directly, which saves power consumption and improves the spectral efficiency [5]. D2D communications underlaying cellular networks, as a fairly popular choice [3], [4], occupy the spectrum resources of regular cellular users to increase the system capacity, which also causes complex interference to the existing cellular network. Considering that the D2D transmissions share the uplink spectrum resources instead of the downlink spectrum resources, where BS causes strong interference for D2D transmissions in a cell [6], [9]. In this scenario, in order to obtain the maximum system achievable transmission rate, we need to implement effective resource allocation among the regular cellular users and D2D users to manage the interference [6]–[11]. Feng et al. [7] studied the resource allocation for D2D communications to maximize the network throughput. Lee et al. [8] proposed a two-stage semi-distributed resource management framework for D2D communications. Ye et al. [9] adopted a distributed algorithm to decrease the computation complexity significantly for the uplink resource allocations. Xu et al. [11] gave a reverse iterative combinatorial auction based approach to allocate the resource between the cellular and D2D User Equipments (UEs). Furthermore, Li et al. [12] proposed a dynamic optimization framework for multihop device-to-device communication. These above studies suppose all the mobile users are altruistic and focus on increasing the overall network transmission rate cooperatively. On the other hand, hand-held wireless devices are carried by mobile users who form social networks with stable social characteristics [13]. In social networks, social community, as one of the most important characteristics, represents real social

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groupings by interests or background, and different communities are usually interested in different mobile contents [14]. Human beings in one social community have the same interested contents, i.e., content sharing occurs among friends in the same community by online social networking services, such as Facebook, Twitter and etc. Therefore, each social community will consider maximization of its own transmission rate, which can speed up the traffic offloading process. For example, when some regular cellular users in a community obtain some mobile data from the BS, other D2D users in the same community can get the same requested contents through the direct D2D communications. The existing resource allocation algorithms [6]–[11] do not consider social information of the mobile users, which influences the effectiveness of resource allocations in D2D communications underlaying cellular networks. The social-aware D2D communications gain attentions in recent studies [15]–[22]. In cognitive radio networks, Chen et al. [15] utilized the reward-based Markov decision process to enable secondary users cooperatively access the spectrum resource. Furthermore, imitation-based spectrum access mechanism was proposed to implement efficient spectrum access [16]. Li et al. [17] summarized the influence of social features on D2D communications and quantitatively analyzed the achievable gains in a social-aware D2D communication system. Cao et al. [19] provided the cooperative video multicast framework to implement the mobile user’s cooperation efficiently. Sun et al. [20] used a Bayesian model to illustrate the social ties for D2D mobile users and achieved efficient data transmission among D2D users. Zheng et al. [22] proposed a social aware algorithm for efficient multi-file disseminations under multi-hop D2D communication networks. Zhang et al. [23] utilized the social network characteristics to assist the ad-hoc peer discovery. These works focus on the influence of social relationships on overall transmission rate of D2D communication networks. However, cellular users have diverse social relationships with their neighbors at different levels [24], and the network optimization without considering these social relationships cannot maximize the overall network utility. D2D users have different strength of social connections with others in the same social community, which needs to be considered for the resource allocation to increase the network utility. Network utility maximization has been extensively studied for network optimization problems [6]–[11], and these studies are based on an assumption that all users act in an altruistic manner in which they have the same social objective to maximize the overall network utility. On the other hand, game theory has found a wide variety of important applications for distributed resource allocation problems in various networking applications [25], [26]. These game theory models can be classified by two categories: non-cooperative and cooperative models [25]. For example, Stackelberg game was utilized to implement efficient resource allocation for femtocells and D2D users, respectively [9], [27]. However, these non-cooperative game-theoretic models usually assume that all users are selfish and rational, aiming at maximizing its own benefit. On the other hand, the cooperative game-theoretic models suppose that users are altruistic and helpful. In fact, these assumptions, that users are either altruistic or selfish by network utility maximization

and non-cooperative game, represent two extreme cases of users’ relationships that are fully socially oblivious and socialaware. Social relationships among D2D users are more complex than these two conditions. In order to evaluate the diverse relationships among human beings, Chen et al. [28], [29] propose a concept of social group utility maximization (SGUM), which exploits the impact of diverse social relationships on physical domain of the wireless communication networks. The social group utility is able to establish the connection between the physical domain and the social domain of D2D communications. The regular cellular users and D2D users in the same social community have common interests and can share their contents directly. Therefore, each community in the D2D communications needs to increase its sum transmission rate cooperatively without considering the transmission rate of other social communities. Both schemes of maximizing the individual rate non-cooperatively and the sum rate of overall networks cannot exploit the characteristic of social community and offload the traffic from the BS effectively. In this paper, in order to evaluate the joint optimization performance of social and physical domains qualitatively, we investigate the resource allocation problem by maximizing the social group utility for D2D communications. Indeed, when this framework and concept are utilized to solve and implement efficient resource allocation for D2D communications, many new problems need to be solved. We propose a general cooperative game theory solution through utility definition, game establishment, existing Nash Equilibrium and solutions with convergence rate guarantee. Specifically, considering the interferences among cellular users and D2D users, we formulate the social community aware resource allocation as a social group utility maximization game. Then, we propose a distributed resource allocation algorithm to achieve the Nash Equilibrium (NE) for the proposed SGUM game. Our contributions are summarized as follows. • We define the utility of each D2D user considering both the physical interference and social characteristics of the regular cellular users and D2D users, and formulate the SGUM game to maximize the utility of each D2D user. To the best of our knowledge, this is the first study to apply the SGUM concept into the resource allocation problem for diverse social relationships of D2D communications. • For the SGUM game, we theoretically investigate the existence of NE for the proposed SGUM game and propose a distributed algorithm to implement efficient social-aware resource allocation with low computation complexity. In the solution, each D2D user independently maximizes its social group utility through switch operations. Furthermore, we obtain the convergence rate for our proposed algorithm. • We evaluate the influence of the different networking environments on the performance of proposed solution. Numerical results demonstrate that it increases the utility of overall social groups about 16% to 56% without loss of fairness compared with other state-of-the-art schemes.

ZHAO et al.: SOCIAL-AWARE RESOURCE ALLOCATION FOR DEVICE-TO-DEVICE COMMUNICATIONS

Fig. 1. Illustration of the social community aware uplink resource sharing of the D2D communications underlaying cellular networks, where there are 2 cellular users, c1 and c2 , and 4 D2D users. In physical domain, wireless links are subject to the physical interference constraints, while social domain indicates the relationships among mobile users.

The rest of this paper is organized as follows. After presenting the system model in Section II, we formulate the SGUM game in Section III. Next, distributed algorithm is proposed in Section IV. Performance evaluations are given in Section V, and finally Section VI concludes this work. II. S YSTEM OVERVIEW, M ODELS , AND P ROBLEM F ORMULATION In this section, we first give a system overview for the D2D communications underlaying cellular networks consisting of different social communities, then derive the group utility of each D2D user in terms of the channel rate and the social relationships. Finally, we formulate the resource allocation problem that we need to investigate. A. System Description In our work, social community aware D2D communication underlaying cellular networks are considered as combination of physical domain and virtual social domain, which is illustrated in Fig. 1. In the physical domain, wireless links are determined by the radio transmission distance between two mobile nodes; while the social domain indicates the relationships among mobile users with hand-held wireless devices. In the social domain, mobile users in D2D communication networks are divided into different social communities, which are defined as the groups with common traffic interests, content sharings, and so forth. As an example showing in Fig. 1, there are two communities denoted by CommunityA and CommunityB, which is represented by the solid line and dash line in the social domain, respectively. Social relationship among the cellular users can be obtained through the online scheme and offline scheme. On one hand, the social relationships can be obtained through exploring the

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content sharing and account information. For example, Wang et al. [30] have kept track of 2,223,294 users for four weeks in the online social network service Sina Weibo and obtained the list of all resharing activities for each microblog. Based on social graph, Wang et al. [30] calculate the importance of each user for propagating the micropost. On the other hand, we can also exploit the social relationships through the offline mobile social networks, which mean the realistic social network due to user mobility, e.g., meeting and groups [31]. It is discovered that people who are graphically close may have similar trends of accessing and sharing content with each other. K-CLIQUE by Palla et al. [32] can detect the social community based on the realistic mobile traces of cellular users. In the physical domains, we assume that the regular cellular users share their uplink resources with D2D communications, and one cellular user’s resource can be shared with multiple D2D users in order to maximize the spectrum efficiency. It is supposed that a D2D user shares at most one cellular user’s spectrum resource. This constraint is used to reduce the complicated interference environment brought by the D2D communications. When the distance between two D2D users satisfies the predefined distance threshold, the corresponding D2D link is established [11]. In LTE system, the prevalent resource allocation unit is physical resource block (RB), which occupies 0.5 ms in the time domain and 180 kHz in the frequency domain. In our model, we utilize RB as the resource allocation unit and regular cellular users have orthogonal RBs initially. As illustrated in Fig. 1, there are two cellular users c1 and c2 , and D2D users (d11 , d12 ) and (d21 , d22 ) occupy the spectrum resource of u1 , while D2D users, (d31 , d32 ) and (d41 , d42 ), occupy the spectrum resource of c2 . In such a system, we concentrate on assigning appropriate RBs occupied by the cellular users to D2D users in order to achieve a higher social group utility. Therefore, we suppose that the transmit power of cellular users and D2D transmitters are constant. Since D2D communications share the same spectrum resources of cellular communications at the same time slot, their interference should be limited as much as possible to optimize the system performance. Consider a set of D2D users, denoted by D, share the resource of cellular user c. As shown in Fig. 1, during the uplink period of the cellular system, all the cellular and D2D users interfere with each other as they share the same spectrum resource blocks. The received signals at the BS from cellular users c is interfered by the transmitters of D2D links, which occupy the same spectrum resource of c. The signal at the D2D receiver d is interfered by the cellular user and other D2D links, which share the same spectrum resource with d. B. System Model and Problem Description 1) Interference Graph for Physical Domain: In this subsection, we construct the interference graph for the physical domain to represent the resource sharing relations for the D2D communication networks [33], [34]. In the system, we assume there are C cellular users labelled as the set of C = {c1 , c2 , · · · , cC }, which share their uplink resources with D2D users. Besides the cellular users, there are D D2D users labelled

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if and only if ∃c ∈ C : xc,d = 1, xc,d = 1; otherwise, yd,d = 0. From the above resource usage relationship, we have the value p of rij as xi,j , i ∈ C, j ∈ D or i ∈ D, j ∈ C, p rij = (1) yi,j , i ∈ D, j ∈ D. In this work, the channel is modeled as Rayleigh fading channel, under which the instantaneous channel taps are the function of time and spatial locations. The power or secondorder statistic of the channel, denoted by |h0 |2 , is a constant within the BS’s coverage area. Thus, under the free space propagation path-loss model [11], the received power of each link between nodes i and j, says i − j, can be expressed as −α · |h0 |2 , where Pi is the transmitted Pi,j = Pi · |hij |2 = Pi · ζi,j power of equipment i, ζi,j is the distance of the i − j link, α is the path-loss exponent, and h0 is the complex Gaussian channel coefficient that obeys the distribution of CN (0, 1). For the purpose of maximizing the network utility in terms of the transmission rate, we need to consider the signal to interference plus noise ratio (SINR) as an important indicator. The SINR of any user j receiving signal from user i can be expressed by ρj = Fig. 2. Illustration of the social graph and interference graph for the D2D communication underlaying cellular networks.

as the set of D = {d1 , d2 , · · · , dD }. Naturally, these D2D users can choose the spectrum resources of any cellular users ci , ∀ ci ∈ C. We use the undirected weighted interference graph model Gp = (Vp , Rp ) to depict the interference of physical domains due to the spectrum sharing in our system. The vertex Vp denotes the communication links of the regular cellular users and D2D users, while the edge Rp represents mutual interference relationships among the communication links. The entry p of Rp is denoted by rij , i, j = 1, · · · , D + C, which equals to 1 if the communication links i and j occupy the same spectrum p resource and incur the mutual interference. The value of rij is determined by the resource allocations for the D2D communication networks. In the lower part of Fig. 2, we construct the interference graph for the resource usage relationships in Fig. 1. For example, c1 , d1 and d2 occupy the same spectrum and the values of mutual edges equal to 1. c2 , d3 and d4 occupy the same spectrum and the values of mutual edges equal to 1. The values of other edges equal to 0 without mutual interferences and are omitted in the interference graph for clarity. In order to describe the spectrum resource usage relationship, we define xc,d ∈ {0, 1} as the indicator that whether the D2D users d shares the resource blocks of UE c, ∀ d ∈ D, ∀ c ∈ C. That is xc,d = 1 when D2D users d uses the resource blocks of UE c; otherwise, xc,d = 0. Since we limit a D2D user shares at most one cellular user’s spectrum resource, we have the xc,d ≤ 1, ∀ d ∈ D. In this resource sharing constraint of c∈C

model, in order to increase the spectrum reuse ratio, it allows multiple D2D communications occur on the same part of the frequency resources from one cellular user, which also incurs mutual interferences of the D2D users. We use yd,d to denote the interference relationships among the D2D users. yd,d = 1

−α Pi ζi,j |h0|2

Pint,j + N0

,

(2)

where Pint,j is the interference signal power received by user j, and N0 is the terminal noise at the receiver. Based on the interference graph and channel model, we are able to derive the uplink transmission rate corresponding to the SINR of the cellular and D2D users. In the uplink period, the BS receiving the signal from the cellular user suffers interference from the D2D users that share the same spectrum resource. According to the interference graph in the physical domain, the interference power at the BS for cellular user c, denoted by Pint,c is determined by the vertices having the edges with c. Therefore, we obtain Pint,c , as follows, p Pint,c = rc,d Pd |hdb |2 . (3) d∈D

Thus, the uplink channel rate of the cellular user c, denoted by Rc , is ⎛ ⎞ −α 2 P ζ |h | c c,b 0 ⎜ ⎟ (4) Rc = log2 ⎝1 + p ⎠. −α 2 rc,d Pd ζd,b |h0 | + N0 d∈D

Similarly, the interference of D2D receiver d is from the cellular user c and the other D2D users that are assigned with the same resource. Therefore, according to the interference graph, we can obtain the interference power from BS and the other D2D users for D2D receiver d, denoted by Pint,d, as follows, p p rc,d Pc |hcd |2 + rd,d Pd |hd d |2 . (5) Pint,d = c∈C

d ∈D\{d}

Thus, the channel rate for the D2D user d, denoted Rd , is −α |h0 |2 Pd ζd,d Rd = log2 1 + . Pint,d + N0

(6)

ZHAO et al.: SOCIAL-AWARE RESOURCE ALLOCATION FOR DEVICE-TO-DEVICE COMMUNICATIONS

Consider all the cellular user C and D2D users D together, we can obtain the system sum rate, denoted by R, as Rc + (7) rc,d Rd . R= c∈C

d∈D

2) Weighted Graph for Social Domain: Now, we use a weighted graph to model the social community characteristic of the cellular users in the system. In the virtual social domain of D2D communications, the regular cellular users and D2D users form different social communities. Social connections exist in each community, while there are no connections between different communities. When a regular cellular user downloads the contents from the BS, other users in the same social community, including cellular and D2D users, can obtain the interested contents from the regular cellular users directly instead of the BS. Meanwhile, D2D users in the same social community may exchange their own contents directly, which is determined by the social connections. Therefore, there are different social communities in the social domain and diverse strength of social connections exist in the same community. From the above analysis, the relational graph for the social connections is denoted by Gs = (Vs , Rs ), where Vs is the collections of all regular cellular users and D2D users, and Rs is the collections of relational edges. The matrix Rs denotes the connection of Vs in the social domain, which is the square matrix with D × (D + C) entries ωi,j , i, j = 1, · · · , D and δi,k , i = 1, · · · , D, k = 1, · · · , · · · , C. ωi,j is the closeness coefficient between D2D users i and j. δi,k is the closeness coefficient between D2D user i and its associated cellular user k. In the social network framework, we formalize the strength of social relationships between users as ωi,j ∈ [0, 1] or δik ∈ [0, 1] with a higher value of ωij or δi,k ∈ [0, 1] being a stronger social connection, which can be kinship, friendship, or colleague relationship between two users. ωi,j = 0 means the weakest connection of two D2D users and ωi,j = 1 means the strongest connection. As an example in the upper part of Fig. 2, we construct the social graph of the mobile users in Fig. 1. Bob, Brown and David have the same interested contents and the regular cellular user Bob downloads the complete contents from the BS. Therefore, D2D users Brown and David have the strongest connection with Bob, while Brown and David have weaker social connections. From the above analysis, the utility of D2D user d and regular cellular user c can be defined as follows: ⎛ ⎞ ⎜ Rd = log2 ⎜ ⎝1 + ⎛ ⎜ Rc = log2 ⎜ ⎝1 +

−α Pd ζd,d |h0 |2 −α Pj ζj,d |h0 |2 p

j∈Nds ∩Nd

−α |h0 |2 Pc ζc,b −α Pj ζj,b |h0 |2 p

⎟ ⎟ , ∀ d ∈ D, (8) + N0 ⎠ ⎞ ⎟ ⎟ , ∀ c ∈ C, + N0 ⎠

(9)

j∈Ncs ∩Nc

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same social community with d on D2D user d, and Rc indicates the impact of spectrum sharing in the same social community with c on regular cellular user c. Then, we define the social group utility of each D2D user i, denoted by Ui , as follows, Ui (X) = Ri + ωi,j Rj + δi,c Rc . (10) j∈D\{i}

c∈C

The social group utility of D2D users consists of 3 parts: its own utilities of individual rate Ri , the weighted sum of individual other D2D users having social connections with it ωi,j Rj , j∈D\{i}

and the weighted sum of utilities of cellular users in the system δi,c Rc . From the above analysis, we observe that Ui (X) is c∈C

determined by the interference graph in the physical domain and the weighted graph in the social domain. Therefore, the social group utility of each D2D user represents the coupling between physical and social domains. 3) Problem Formulation: In order to maximize the social group utility of each D2D user, we need to determine the optimal resource allocation. The social group utility Ud of each D2D user d depends on the resource allocation and social relationships between the cellular users and D2D users. The social information is more long-lasting and stable. Thus, at each time point we denote the social group utility of user d as function Ud (X), where X is the matrix of xc,d , c ∈ C, d ∈ D. In order to maximize the social group utility of each D2D user, we need to determine the optimal resource allocation policy. Thus, combing the above definitions, we formulate the optimal resource allocation problem in the D2D communications underlaying cellular networks as the following optimization problem, max Ud (X), d∈D ⎧ ⎨xc,d ∈ {0, 1}, ∀ d ∈ D, c ∈ C; s.t. ⎩ xc,d ≤ 1, ∀ d ∈ D.

(11)

c∈C

In the formulated problem, since the optimization utility function (10) has no obvious increasing or concave properties with xc,d even the constraint is linear. In contrast to the multiobjective optimization, each D2D user would like to maximize its own utility. Actually, our optimization problem maximizes the social group utility of each D2D user at some time point. D2D links in a community are changing over the time, and our proposed approach can be applied in the dynamic networks to maximize the social group utility at each time point. The distributed nature of the above problem leads to a game theory naturally, where each user d maximizes its corresponding Ud (X). The resource allocation variable xc,d indicates the strategy of each D2D user d to occupy the spectrum of the regular cellular c. Therefore, the formulated problem (11) can be viewed as a distributed decision game for each D2D user d. Base on the above analysis, we solve the above social group utility maximization problem by formulating a SGUM game.

p

which are also viewed as the interference aware rate. Nk and Nks denote the set of cellular users which occupy the same spectrum resource and have the social connection with k, respectively. Rd indicates the impact of other devices in the

III. S OCIAL G ROUP U TILITY M AXIMIZATION G AME In this section, we present the social group utility maximization game for the formulated resource allocation problem.

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Then, we theoretically prove that it has Social Nash Equilibrium (SNE).

Based on the above definitions, we define the ordinal potential function as Ud (X) P(x) =

A. SGUM Game Formulation

d∈D ⎛

From the formulated problem, we can obtain that the optimization problem aims to maximize each D2D user’s utility with the consideration of the social information. Naturally, we utilize the game theory to solve the optimization problem. Therefore, we consider the distributed decision problem among the users for maximizing their own social group utilities. Define the strategy of each D2D user is xd ∈ Xd , where Xd = {1, 2, . . . , C} is set of all feasible strategies of the D2D user d. xd can be determined by the resource allocation relationship xc,d one by one. For example, xc,d = 1 indicates that xd = c. For our formulated problem (11), let x−d = (x1 , . . . , xd−1 , xd+1 , . . . , xD ) be the set of strategies chosen by all other D2D users except user d. We also define x = (x1 , x2 , . . . , xD ) = (xd , x−d ) to denote the resource allocation vector and X is the set of all possible x. Given the other users’ strategies x−d , user d wants to choose a strategy xd ∈ Xd to maximize its social group utility, i.e., max

xd ∈Xd

=

⎝Rd +

d∈D

ωd,j Rj +

j∈D\{d}

j∈D\{d}

−

j∈D\{d}

⎞

⎛

⎜ Rk − ⎝Rd +

Ud (x−d , xd ) − Ud x−d , xd > 0

iff

P(x−d , xd ) − P x−d , xd > 0.

⎟ Rk ⎠ . p

k∈Nds ∩Nd

P(d) − P (d) ⎛ ⎞ ⎝Rd + = ωd,j Rj + δd,c Rc ⎠ d∈D

−

⎛

+

j∈D\{d}

⎝Rd +

d∈D

B. Social Aware Nash Equilibrium

and x∗ = {x∗1 , x∗2 , · · · , x∗D } is the SNE. Motivated by the method in [35], we utilize the potential game theory to verify the existence of SNE for the SGUM game [36]. We provide the definition of ordinal potential game and give its associated ordinal potential function. G is named as an ordinal potential game if it has an ordinal potential function. A function P(x) is an ordinal potential for G if for every d ∈ D and for every x−d ∈ X−d :

Next, the changing of the ordinal function is

= Ud − Ud

xd ∈Xd

(13)

c∈C

= Rd +

δd,c Rc

p

x∗d = arg max Ud (xd , x−d ), ∀ d ∈ D,

(12)

c∈C

ωd,j Rj −

k∈Nds ∩Nd

In the subsection, we first give the definition of social aware Nash Equilibrium. Then, we theoretically prove the existence of Nash Equilibrium through the potential game. The SGUM game G = (D, {Xd }d∈D , {Ud }d∈D ) has the Nash Equilibrium if no D2D user can increase its social group utility by changing its resource sharing strategy, i.e.,

δd,c Rc ⎠ .

c∈C

Theorem 1: Function P(x) is an ordinal function for SGUM game G = (D, {Xd }d∈D , {Ud }d∈D ) when the strength of social ties in the same community is 1. Proof: We suppose D2D user d changes its spectrum sharing and occupies the spectrum resource of cellular c instead of c. Therefore, the indicator xc,d changes to xc,d and yd,d0 changes to yd,d0 . Then, the changing payoff of each D2D user is Ud − Ud = Rd + ωd,j Rj + δd,c Rc − Rd

Ud (xd , x−d ), ∀ d ∈ D.

The distributed nature of the problem above naturally leads to a formulation based on game theory such that each user aims to maximize its social group utility. The SGUM game with the social group utility for the resource allocation for D2D communication is defined by the triple G = (D, {Xd }d∈D , {Ud }d∈D ), where the set of D2D users D is the players, Xd is the set of resource sharing strategies for each D2D user d, and the social group utility function Ud is the payoff function of each D2D user d.

⎞

c∈C

ωd,j Rj +

j∈D\{d}

⎞ δd,c Rc ⎠

c∈C

⎛

⎜ ⎜ Rj − ⎝Rk + ⎝Rk +

p k∈D\{d}∩Nds∩Nd

⎞⎞

⎛

p j∈Nks ∩Nk

p = Nds ∩ Nd + 1 Ud − Ud ,

⎟⎟ Rj ⎠⎠

p j∈Nks ∩Nk

(14)

which is obtained as d and k not only are in the same social community but also occupy the same spectrum resource. Finally, we conclude that Ud − Ud > 0 iff P(d) − P (d) > 0, ∀ d ∈ D.

(15)

Theorem 2: The SGUM game G = (D, {Xd }d∈D , {Ud }d∈D ) has the pure Nash Equilibrium. Proof: From Theorem 1, we have that the SGUM game G = (D, {Xd }d∈D , {Ud }d∈D ) is the ordinal potential game with the ordinal potential function P(x). Each D2D user d maximizes its utility Ud , which is equal to maximize the potential function P(x). Therefore, each feasible strategy of D2D user d is able to increase the overall potential function P(x). At the same time, G is a finite potential game, and the number of all feasible strategies is finite. Thus, G possesses a pure strategy Nash Equilibrium [36].

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IV. D ISTRIBUTED R ESOURCE A LLOCATION A LGORITHM FOR SGUM A. Distributed Solution The key mechanism in decision making of the SGUM game formulated in (11) is to enable D2D users to choose a cellular user to occupy the spectrum resource, which is the strategy of each D2D user. Specifically, each D2D user must be able to compare and order its potential social group utility based on which cellular user this player prefers to share. According to the property of potential game, the resource allocation strategy x∗ is a Nash Equilibrium, which maximizes the value of ordinal potential function. Furthermore, the SGUM G has the finite improvement property (FIP) due to finite values of the ordinal potential function. Specifically, the order of D2D users obtaining a better resource sharing can be random and achieve the equilibrium by finite steps. Here, we introduce the concept of preference order i for any user i ∈ D as the following definition. Definition 1 (Preference Order): For any D2D users i ∈ D, the preference order i is defined as a complete, reflexive, and transitive binary relation over the set of all resource allocation strategies that D2D user i can possibly take, i.e., the set {x ⊆ X , xi ∈ Xi }. In our SUGM game for the resource sharing, the D2D users choose to joint or leave the associated regular cellular users according to its preference order. For any D2D users, i, xi i xi means i prefers sharing the spectrum of c ∈ C than c ∈ C, where xi ⊆ X and xi ⊆ X . Since the preference order depends on its social utility, in this paper, for any D2D users, i ∈ D and xi , xi , we define the following preference xi i xi ⇔ Ui xi > Ui (xi )&Uj xi ≥ Uj (xi ),

∀ j ∈ {D\i}. (16)

This definition implies that D2D user i prefers to share the spectrum resource of c over c only when i gains increasement in its social group utility while no other D2D users in D \ {i} suffers decreasing on their social group utility due to its switching. Based on the above resource sharing and preference order, we can form the resource allocation from a given initial strategy by switch operations. Suppose given a resource allocation vector x = {x1 , . . . , xj , . . . , xk , . . . , xD } of D2D users D, for ∀ i ∈ D, we suppose its current resource allocation strategy is x ∈ X . Then, a switch operation from x to x ∈ X ∪ {∅}, x = x , means the D2D users j and k switch their resource allocation strategies. A new resource allocation vector x = {x1 , . . . , xj , . . . , xk , . . . , xD } is obtained such that xj = xk and xk = xj . After repeating switch operations, the system can generate different resource allocation vectors of the D2D users D. Combining the preference order, a switch operations from x to x is allowed for any D2D user i ∈ D, if and only if xi i xi . In this mechanism, every D2D user i ∈ D can leave its current resource allocation strategy, and occupies the spectrum resource of another cellular user given that the new resource allocation vector is strictly preferred through the preference relation defined in (16).

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Algorithm 1 The Social Group Utility Maximization Algorithm for the D2D Users Resource Allocation 1: Initialize the system by any random resource allocation vector xini ; 2: Set the current resource allocation vector as xini → xcur ; 3: repeat 4: Uniformly randomly choose one D2D user i, and denote its associate resource sharing as xi ∈ xcur ; 5: Uniformly randomly search for a possible D2D user k ∈ D \ {i}, whose corresponding switch operation from xj = xk and xk = xj . 6: Calculate U(x) and U(x ) by (10). SwitchFlag = 0; 7: if xi i xi then 8: SwitchFlag = 1; 9: else 10: Calculate φx ,x = 1+exp(−(U(x1 )−U(x))/Tn ) . 11: Get λ uniform distribution in (0,1]. 12: if λ < φx ,x then 13: SwitchFlag = 1; 14: end if 15: end if 16: if SwitchFlag == 1 then 17: D2D users i and k leave their current associate cellular users, and join the new cellular users c and c, respectively; 18: Update the current resource allocation vector as x → xcur ; 19: end if 20: until The resource allocation vector converges to a final Nash-stable resource allocation vector xfin . Based on the above resource allocation vector and switch operation, we present the distributed resource allocation algorithm for the D2D users summarized in Algorithm 1. In the algorithm, D2D users make switch operation in a random order based on a random generated initial resource allocation vector xini . In each iteration, the system uniformly randomly chooses one D2D user i ∈ D in Step 4. In Step 5, the selected D2D user recognizes its resource allocation strategy xi ∈ x and uniformly randomly selects another possible D2D user k ∈ D. Then, it will request the channel information of these two D2D users and the cellular users having connections with i or k from the cellular base station by cellular direct communications. After obtaining these information, the D2D user computes the utility of the resource allocation vector x and x , and decides to whether execute the switch operation. In order to avoid the local maximum value, when the constraint is not satisfied, we also define the acceptance probability [6] 1 , (17) φx ,x = 1 + exp (− (U(x ) − U(x))/Tn ) T0 where Tn = log(n−1) with T0 as a constant value and n is the current times of switch operations. Now we give the second condition for switch operation as follows,

λ < φx ,x ,

(18)

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where λ is the uniform distribution in (0,1]. We use Barker’s sampler to enlarge the feasible space of resource allocation to prevent our proposed algorithm from obtaining the local optimal value [38]. Therefore, our proposed distributed algorithm based on switch operation can achieve the Nash Equilibrium of SGUM game.

TABLE I S IMULATED PARAMETERS

B. Convergence and Stability In this subsection, we will analyze the properties of convergence, stability and computation complexity for Algorithm 1. Indeed, in terms of the convergence, since the number of the resource allocation vector is finite and we use the history information to avoid the repeating choices, the switch operation will always terminate. According to the concept of Nash Equilibrium in Theorem 2 for the SGUM game, the stability of the final resource allocation vector xfin depends on whether a Nash-stable exists letting ∀ i ∈ D and xi i xi . The convergence and stability of our proposed resource allocations algorithm are guaranteed as follows. Theorem 3: Starting from any initial resource allocation vector xini , the proposed distributed algorithm will always converge to a Nash-stable resource allocation vector xfin . Proof: In each switch operation in Algorithm 1, it will yield a new resource allocation vector through adopting new strategy or switching existing strategies, and the maximum number of resource allocation strategies for each D2D user is C since there is only C cellular users in the system. Therefore, the number of resource allocation vector for the given D2D users set D is a Bell number [6]. Thus, the sequence of random switch operations will terminate with probability 1, and the system then converges to a final resource allocation vector xfin after finite turns with probability 1, which proves the convergence in probability for our proposed distributed algorithm. Now, we prove the stability of the proposed algorithm by contradiction. Suppose the final resource allocation vector xfin obtained from Algorithm 1 is not Nash-stable. Then, there exists a D2D user i ∈ D whose resource allocation vector is denoted by xi , and a new resource allocation vector xi ∈ xfin such that xi i xi . According to Algorithm 1, D2D user i can perform a switch operation from xi to xi , which contradicts the fact that xfin is the final resource allocation vector. Thus, we have proved that the final network resource allocation vector xfin resulting from Algorithm 1 must be Nash-stable. For the computation complexity, we note that these switch operations in the proposed algorithm only use local D2D community instead of all the users in the network. After a limited number of switching, the system resource allocation converges to the final Nash-stable resource allocation vector xfin , which is determined by the convergence rate in the following Theorem 4. As a result, the complexity is much lower than the centralize solution. Theorem 4: The convergence of SGUM to the optimal solution is the geometric rate. Proof: As the SGUM algorithm utilizes the Barker’s Sampler to accept the worse solution, we can utilize an evolving Markov chain to depict the process of switch operation [38]. The Markov chain S(Tn ) has the transition matrix Q = {qi,j}.

The strategy vector x is the current state and xn denotes the state after n iterations. From Algorithm 1, the probability from the state x to x can be depicted as: φx ,x =

1 . 1 + exp (− (U(x ) − U(x)) /Tn )

From [38], we observe that S(Tn ) has the stationary distribution π, which does not depend on Q. Following the basic notations and main idea from Ref. [38], we utilize d(xP − πP) to denote the variation between the current system state and the stationary state. Then from the Definition 7.1 of Chapter 6 [38], we obtain d(xP − πP) = δ(Q) ≤ δ(Q1 )δ(Q2 ), where Q2 = P and Q1 can be expressed as x(1) x(2) ··· Q1 = . π(1) π(2) ··· Then d(xP − πP) ≤ d(Q1 )δ(P) = 12 d(x − π)δ(P). From the definition of δ(P), we obtain that

δ(P) ≤ 1 − e T . Therefore, after n switch operation, d(xPn − πPn ) ≤

n 1 d(x − π) 1 − e T , 2

which means that the convergence of SGUM to the optimal solution is the geometric rate. V. P ERFORMANCE E VALUATION In this section, we evaluate the performance of the proposed SGUM game based D2D resource allocation scheme. We carry out the simulations in a single cell scenario. Both path-loss model and shadow fading are considered for cellular and D2D links. In our network simulation, the wireless propagation is modeled by WINNER II channel models [11]. In particular, the D2D communication channel is based on the scenario that two communicating UEs are physically in close proximity, while the cellular communication channel is simulated according to the urban microcell scenario. The path-loss exponent for free space propagation path-loss model is set to be 4. The main parameters used in our simulation are listed in Table I. As listed in Table I, we simulate the system within a 500 m × 500 m area with the base station in the center. To obtain the settings with positions of the users, we uniformly randomly distribute the cellular users and D2D users within the

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rithm could achieve the near-optimal solution of system sum rate. b) Nearest First (NF), where the system allocates the communication resources to the D2D users with the cellular users that are nearest to the D2D users in their position. c) Furthest First (FF), where the system allocates the D2D communication resources with the cellular users which is the furthest to the D2D users. d) Random Selection (RS), which uniform randomly allocates the resources to the D2D users to communicate. That is to say, for any D2D user, the system randomly selects a cellular user’s resource for sharing. A. System Sum Social Group Utility

Fig. 3. A snapshot of a final resource allocation vector resulting from the proposed algorithm for a network of 3 cellular users and 20 D2D users.

In order to demonstrate our proposed solution can achieve the social group utility maximization, we compare its performance with four other algorithms. The sum social group utility can be calculated as Ud (x), (19) Usum (x) = d∈D

coverage of the BS. The transmitter of D2D link is randomly distributed in the coverage of BS, and the receiver is randomly distributed in the circle of transmitter with the maximum distance. As an illustration for how cellular and D2D users located in the simulation, we plot the positions of the base station, cellular users and D2D users in an instance by randomly generating a network with 3 cellular users and 20 D2D users in Fig. 3. In this figure, the cellular users are represented by triangle, while the D2D users are denoted by circle. Moreover, we also show the snapshot of a final resource allocation vector resulting from our proposed algorithm under the above network setting. In this example, 20 D2D users occupy the spectrum resources of 3 cellular users, and they are marked by different symbols of 1 labeled black color, 2 labeled red color and 3 labeled blue color in Fig. 3, respectively. According to the solution of resource allocation, we evaluated the following four performance metrics: Ud (x), which is determined 1) Social group sum utility d∈D

by all the D2D users, the cellular users and the strength of the social connections between them. 2) System sum rate R, which is the aggregated communication rate of all users including D2D users and cellular users, and it is obtained by (7). 3) The Jain’s fairness measure [37], which determines whether the receivers of D2D and cellular users are receiving fair share of the system resources. In order to show the efficiency of our social group utility maximization based resource allocation, we compare the performance of our scheme, denoted as Social Group Utility Maximization (SGUM), with the four schemes: a) Coalition Game (CG), where the system allocates the system resources to the D2D users through the coalition formation game model [6]. This distributed algo-

where Ud (x) is obtained from (10). In this simulation, we suppose that each D2D user randomly selects another D2D user to establish the social connection based on the social link probability 0.5, and the strength of social connection is set to be 1. We set the number of D2D users to be 15 and vary the number of the cellular users to be 1 to 15 to obtain the results of the sum social group utility shown in Fig. 4(a), while set the number of cellular users to be 5 and vary the number D2D users from 1 to 20 to obtain the results shown in Fig. 4(b). From these two figures, we observe that the system utility increases with the number of cellular users or D2D users increasing. In Fig. 4(a), when the number of cell users is equal to 15, the sum utility of SGUM is larger than that of NF and CG about 56% and 16%, respectively. In Fig. 4(b), when the number of D2D users is equal to 20, the sum utility of SGUM is larger than that of NF and CG about 53% and 16%, respectively. NF algorithm performs worst, as it neither decreases the interferences in the physical domain nor considers the social community information in the social domain. SGUM performs better than NF about 45% averagely. Although CG algorithm achieves the near-optimal solution to decrease the interferences, CG performs worse than SGUM as CG also cannot consider the social community information. From these results, we can observe that SGUM achieves the best performance in all the four schemes, which considers both the interference relationship and social information. B. System Sum Rate Then we analyze the system sum rate of all the four schemes, which can calculated by (7). Similarly, we set the number of D2D users and cellular users to be 15 and 5, respectively, varying the number of the cellular users or D2D users to be 1 to 15 or 1 to 20, respectively. The case of 20 D2D users represents the situation of resource-lacking system in which D2D users

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Fig. 4. Performance of sum social group utility of different resource allocation algorithms with (a) different number of cell users and (b) different number of D2D users.

Fig. 5. Performance of system sum rate of different resource allocation algorithms with (a) different number of cell users and (b) different number of D2D users.

possibly need to share the same user’s resource, while the other case represents the resource-abundant system in which D2D users may have more selections. The performance of the two system sum rate are shown in Fig. 5(a) and (b), respectively. The result indicates that the system sum rate increases with the number of cellular users increasing, because the system bandwidth resource increases. When the number of D2D users increases, the system sum rate increases as the proximity D2D users occupy the same spectrum resource of cellular users. However, the spectrum sharing also incurs interferences, which constrain the system performance. Therefore, there must be saturation in this condition. Comparing the two figures, we also note that the increasing number of cellular users may bring larger system performance enhancement than increasing the number of D2D users. Among all the five schemes, the scheme of CG always achieves the best performance for the reason that it may reduce the unnecessary interference in the system, while the NF always gets the worst performance since it does the largest interference between the D2D users and cellular users. Furthermore, we observe that SGUM algorithm heavily depends on the social link probability among the D2D users. From

Fig. 5(a) and (b), when the social link probability is set to 1, we observe that SGUM achieves the same performance with CG. This is because that each D2D user has the same target to maximize the whole network performance. When the social link probability is set to 0, we observe that SGUM achieves the same performance with FF. This is because that each D2D user aims to maximize its own rate without consideration of the strategies of other D2D users. The result indicates that our social group utility maximization algorithm increases the network utility and achieves the high system sum rate. In order to demonstrate the saturation trend with increasement of D2D users, the coverage of BS and the number of cellular users are set to 350 m × 350 m and 3, respectively. The social link probability is set to be 0.1, 0.7 and 0.9 to calculate system sum rate. From Fig. 6, we observe that the sum rate do not increase due to severe interference when the number of D2D links is above 40. However, the sum utility increases with increasement of D2D users. This is because that the social group utility is the metric to consider the interference in the physical domain and social connections in the social domain. When the number of D2D users is larger than that of cellular

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Fig. 6. Performance of (a) system sum rate and (b) sum utility of different resource allocation algorithms with different number of D2D users.

Fig. 7. Fairness of D2D and cell users transmission of different resource allocation algorithms with (a) different number of cell users and (b) different number of D2D users.

users, multiple D2D users may occupy the same spectrum resource of the same cellular user. In this condition, complex interferences exist among D2D users and cellular users. From Fig. 4(b) and Fig. 6(b), we can conclude that our proposed scheme outperforms other state-of-the-art solutions.

C. System Fairness To obtain some insights on how the data transmission is actually shared among the D2D users and cellular users, we depict the Jain’s fairness index in Fig. 7(a) and (b) with the variation of the number cellular users and D2D users, respectively. We observe that changing the number of cellular users and D2D users has a non-obvious influence on the fairness of data transmission under these schemes. Among all these compared schemes, we can obtain that CG has the best fairness resource sharing among the the cellular users and D2D users. Social link probability has the significant impact on the fairness of SGUM. This is because that SGUM considers the social relationships in

the social domain and maximizes each D2D user’s utility based on social communities.

D. The Impact of Social Link Probability In order to evaluate the social link probability on the system sum utility and sum rate. PL is used to denote the social link probability, which represents the density of social community. To observe the impact of social link probability on the system utility, we first set the number of cellular users and D2D users to be 5 and 20, respectively. In Fig. 8, we plot the sum utility varying PL from 0 to 1. It is obvious that SGUM performs best at different PL values. When PL = 1, SGUM and CG have the same sum utility, which indicates they both maximize the whole network utility. The difference of sum utility between SGUM and FF increases with PL. This is because that noncooperative scheme performs worse with PL increasing. While PL = 0.5 indicates the largest difference of sum utility between SGUM and CG. The reason is that PL = 0.5 represents the

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Fig. 8. Performance of social group utility with 5 cellular users and 20 D2D pairs varying different social link probability.

Fig. 10. System convergence rate in terms of the average number of switching operations in the case of varying the number of D2D users.

number of switch operations before Algorithm 1 converges to the final resource allocation vector xfin is showing in Fig. 10. As the number of D2D users increases, the average number of switch operations increases. In the case of 10 cellular users, which provide 10 selections for each D2D user, the average number of switch operations is around 3N to find the solution by our algorithm. While the exclusive search needs 10N iterations to find the optimal solution. Therefore, comparing with the exclusive search, our proposed algorithm decreases the computation complexity significantly. VI. C ONCLUSION AND F UTURE W ORK

Fig. 9. Performance of system sum rate with 5 cellular users and 20 D2D pairs varying different social link probability.

most random social relationships among D2D users. Therefore, for brevity, we select PL = 0.5 without loss of generality in Fig. 4. In Fig. 9, we plot the sum rate varying PL from 0 to 1. When PL = 0, the system sum rate of SGUM is equal to the value of FF. It is because that SGUM enables each D2D user to maximize its own rate when PL = 0. When PL = 1, SGUM aims to maximize the whole network performance and achieves the same sum rate with CG. When PL = 1, SGUM increases the system sum rate about 26% compared with PL = 0. The PL has no impact on the performance of FF and CG for sum rate. However, the sum rate of SUGM increases with PL. It is obvious that SGUM links the non-cooperative and cooperative resource allocation schemes for D2D communication network. In this scenario, the performance of PL = 0.5 is about the medium value between that of FF and CG. Therefore, we select PL = 0.5 in Fig. 6 to represent the typical condition. E. Convergence Rate In order to show the convergence rate of our proposed algorithm, we set the number of cellular users to be 2 and 10, and vary the number of D2D users N from 1 to 20. The average

We has studied the social-aware resource allocation problem for the D2D communication underlaying cellular system under realistic networking scenarios. By formulating it as a distributed social group utility maximization problem based on the social graph and interference graph, we propose a social group utility maximization game to solve this problem. By theoretical analysis, we prove that this scheme converges to a final Nash-stable network solution. Then, we propose a distributed resource allocation algorithm to decrease the computation complexity, which can achieve the Nash equilibrium of the social group maximization game. Through extensive simulation study, we have demonstrate the effectiveness and fairness of our proposed scheme. The results show that our scheme enhances the system sum utility rate by about 16% to 56% without loss of resource sharing fairness compared with several other strategies. On the other hand, traffic types and power control also have influences on the social group utility of each D2D user. Therefore, considerable future works are required to investigate the joint resource allocation and power control for D2D communication under different traffic types. R EFERENCES [1] “Cisco visual networking index: Global mobile data traffic forecast update,” Cisco, San Jose, CA, USA, 2013–2018, Feb. 5, 2014. [Online]. Available: http://www.cisco.com/c/en/us/solutions/collateral/serviceprovider/visual-networking-index-vni/white_paper_c11-520862.pdf [2] B. Han et al., “Cellular traffic offloading through opportunistic communications: A case study,” in Proc. 5th ACM Workshop Challenged Netw., Chicago, IL, USA, Sep. 2010, pp. 31–38.

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[28] X. Chen, X. Gong, L. Yang, and J. Zhang, “A social group utility maximization framework with applications in database assisted spectrum access,” in Proc. INFOCOM, Toronto, ON, Canada, Apr. 2014. [29] X. Gong, X. Chen, and J. Zhang, “Social group utility maximization in mobile networks: From altruistic to malicious behavior,” in Proc. CISS, Princeton, NJ, USA, Mar. 2014, pp. 1–6. [30] X. Wang, M. Chen, T. T. Kwon, L. Jin, and V. C. Leung, “Mobile traffic offloading by exploiting social network services and leveraging opportunistic device-to-device sharing,” IEEE Wireless Commun., vol. 21, no. 3, pp. 28–36, Jun. 2014. [31] P. Hui, J. Crowcroft, and E. Yoneki, “BUBBLE Rap: Social-based forwarding in delay-tolerant networks,” IEEE Trans. Mobile Comput., vol. 10, no. 11, pp. 1576–1589, Nov. 2011. [32] G. Palla, I. Derényi, I. Farkas, and T. Vicsek, “Uncovering the overlapping community structure of complex networks in nature and society,” Naure, vol. 435, no. 7043, pp. 814–818, Jun. 2005. [33] R. Zhang et al., “Interference graph based reource sharing schemes for vehicular networks,” IEEE Trans. Veh. Technol., vol. 62, no. 8, pp. 4028–4039, Oct. 2013. [34] R. Zhang, X. Cheng, Y. Yang, and B. Jiao, “Interference graph based reource allocation (InGRA) for D2D communications underlaying cellular networks,” IEEE Trans. Veh. Technol., vol. 64, no. 8, pp. 3844–3850, Aug. 2015. [35] X. Chen and J. Huang, “Database-assisted distributed spectrum sharing,” IEEE J. Sel. Areas Commun., vol. 31, no. 11, pp. 2349–2361, Nov. 2013. [36] D. Monderer and L. S. Shapley, “Potential games,” Games Economic Behavior, vol. 14, no. 1, pp. 124–143, 1996. [37] R. K. Jain, D. W. Chiu, and W. R. Hawe, “A Quantitative Measure of Fairness and Discrimination for Resource Allocation in Shared Computer Systems,” DEC Res. Rep. TR-301, Eastern Res. Lab, Digital Equipment Corporation, Hudson, MA, USA, Sep. 1984. [38] P. Brémaud, “Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues.” New York, NY, USA: Springer-Verlag, 1999.

Yulei Zhao received the B.S. and Ph.D. degrees from the Department of Communication Engineering, Zhengzhou Information Science and Technology Institute, Zhengzhou, Henan, China, in 2005 and 2008, respectively. He is pursuing the Ph.D. degree with the Department of Electronic Engineering, Tsinghua University, Beijing, China. His research interests include cooperative communications, deviceto-device communications, and social networks.

Yong Li (M’09) received the B.S. and Ph.D degree in Huazhong University of Science and Technology and Tsinghua University in 2007 and 2012, respectively. During 2012 and 2013, he was a Visiting Research Associate with Telekom Innovation Laboratories and Hong Kong University of Science and Technology respectively. During 2013 to 2014, he was a Visiting Scientist with the University of Miami. He is currently a Faculty Member of the Department of Electronic Engineering, Tsinghua University. His research interests are in the areas of Mobile Computing and Social Networks, Urban Computing and Vehicular Networks, and Network Science and Future Internet. Dr. Li has served as General Chair, Technical Program Committee (TPC) Chair, and TPC Member for several international workshops and conferences. He is currently the Associate Editor of Journal of Communications and Networking and EURASIP Journal of Wireless Communications and Networking.

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Yang Cao (S’09–M’14) received the B.S. and Ph.D. degrees in information and communication engineering from Huazhong University of Science and Technology, Wuhan, China in 2009 and 2014, respectively. He is currently an Assistant Professor in the School of Electronics Information and Communications, Huazhong University of Science and Technology, Wuhan, PR China. From 2011 to 2013, he worked in the School of Electrical, Computer, and Energy Engineering, Arizona State University, AZ, USA, as a Visiting Scholar. His research interests include resource allocation for cellular device-to-device communications and smart grids. He has coauthored more than 20 papers on refereed IEEE journals and conferences. He was awarded CHINACOM Best Paper Award in 2010 and awarded Microsoft Research Fellowship in 2011.

Tao Jiang (M’06–SM’10) received the B.S. and M.S. degrees in applied geophysics from China University of Geosciences, Wuhan, China, in 1997 and 2000, respectively, and the Ph.D. degree in information and communication engineering from Huazhong University of Science and Technology, Wuhan, PR China, in April 2004. He is currently the Chair Professor in the School of Electronics Information and Communications, Huazhong University of Science and Technology, Wuhan, PR China. From August 2004 to December 2007, he worked in some universities, such as Brunel University and University of Michigan-Dearborn, respectively. He has authored or coauthored over 200 technical papers in major journals and conferences and eight books/chapters in the areas of communications and networks. He served or is serving as symposium technical program committee member of some major IEEE conferences, including INFOCOM, GLOBECOM, and ICC, etc. He is invited to serve as TPC Symposium Chair for the IEEE GLOBECOM 2013 and IEEEE WCNC 2013. He is served or serving as Associate Editor of some technical journals in communications, including in IEEE T RANSACTIONS ON S IGNAL P ROCESSING, IEEE C OMMUNICATIONS S URVEYS AND T UTORIALS , IEEE T RANSACTIONS ON V EHICULAR T ECH NOLOGY , and IEEE I NTERNET OF T HINGS J OURNAL, etc. He is a recipient of the NSFC for Distinguished Young Scholars Award in P. R. China.

Ning Ge (M’95) received the B.S. degree in 1993, and the Ph.D. in 1997, both from Tsinghua University, China. From 1998 to 2000, he worked on the development of ATM switch fabric ASIC in ADC Telecommunications, Dallas, TX, USA. Since 2000 he has been in the Department of Electronics Engineering, Tsinghua University. Currently he is a Professor and also serves as Director of Communication Institute. His current interests are in the areas of communication ASIC design, short range wireless communication, and wireless communications. Dr. Ge is a senior member of CIC and CIE. He has published more than 60 papers.

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Social-Aware Resource Allocation for Device-to-Device Communications Underlaying Cellular Networks Yulei Zhao, Yong Li, Member, IEEE, Yang Cao, Member, IEEE, Tao Jiang, Senior Member, IEEE, and Ning Ge, Member, IEEE Abstract—The ever-increasing demands for local area services underlaying cellular networks benefit from direct device-to-device (D2D) communications, where an efficient scheme for resource allocation is needed to increase the system capacity as the result of interference caused by spectrum sharing. Current works mainly focus on maximizing the overall transmission capacity according to interference constraints of the physical domain. However, D2D users in the social domain form different social communities, and each social community is likely to improve its own group’s data transmission cooperatively without considering other communities. Therefore, social relationships among mobile users influence the strategy of the resource allocations for the D2D communications. In this paper, we first introduce social relationships in the continuum space into the resource allocation for D2D communications, which consider the complex social connections in the social domain. Then a social group utility maximization game is formulated to maximize the social group utility of each D2D user, which quantitatively measures the joint performance of social and physical domains. We theoretically investigate the Nash Equilibrium of our proposed game and further propose a distributed algorithm based on the switch operations of the resource allocation vector. Numerical results demonstrate that our proposed solution increases the utility of overall social groups about 45% on average without loss of the fairness compared with other state-of-the-art schemes. Index Terms—Device-to-device communication, social group utility, game theory, resource allocation.

I. I NTRODUCTION

W

ITH the popularity of wireless access and mobile Internet, Cisco estimates that mobile traffic will grow at an annual rate of 81% from 2014, and reach over 15 exabytes per

Manuscript received January 11, 2015; revised May 7, 2015 and July 6, 2015; accepted July 6, 2015. Date of publication July 16, 2015; date of current version December 8, 2015. This work was supported by the National Basic Research Program of China (973 Program) under Grant 2013CB329001, by the National Nature Science Foundation of China under Grant 61132002 and Grant 61301080, and by the Creative Research Group Program from NSFC (61321061). The associate editor coordinating the review of this paper and approving it for publication was J. Huang. Y. Zhao, Y. Li, and N. Ge are with Tsinghua National Laboratory for Information Science and Technology (TNLIST), Department of Electronic Engineering, Tsinghua University, Beijing 100084, China (e-mail: [email protected] mails.tsinghua.edu.cn; [email protected]; [email protected]). Y. Cao and T. Jiang are with School of Electronics Information and Communications, Huazhong University of Science and Technology, Wuhan 430074, P.R. China (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TWC.2015.2457427

month in 2018 [1]. Local area services of popular content sharing are emerging demands for geographically proximate users. For example, friends share photos, music and videos with each other through smart phones. Current mobile content sharing is provided by traditional cellular and WiFi networks [2], which will be overloaded and congested in the near future [3]. Deviceto-Device (D2D) communications enable the proximity cellular users to communicate directly, instead of through a Base Station (BS) [3]. Local data services are important usage cases for D2D communications through peer-to-peer scheme [4]. For example, when two friends are in geographical proximity, they want to exchange photos or videos via their smart phones. Furthermore, a smart phone may connect to a television to display the photo or video. The D2D communications enable the proximity cellular users to share the same interested contents directly, which saves power consumption and improves the spectral efficiency [5]. D2D communications underlaying cellular networks, as a fairly popular choice [3], [4], occupy the spectrum resources of regular cellular users to increase the system capacity, which also causes complex interference to the existing cellular network. Considering that the D2D transmissions share the uplink spectrum resources instead of the downlink spectrum resources, where BS causes strong interference for D2D transmissions in a cell [6], [9]. In this scenario, in order to obtain the maximum system achievable transmission rate, we need to implement effective resource allocation among the regular cellular users and D2D users to manage the interference [6]–[11]. Feng et al. [7] studied the resource allocation for D2D communications to maximize the network throughput. Lee et al. [8] proposed a two-stage semi-distributed resource management framework for D2D communications. Ye et al. [9] adopted a distributed algorithm to decrease the computation complexity significantly for the uplink resource allocations. Xu et al. [11] gave a reverse iterative combinatorial auction based approach to allocate the resource between the cellular and D2D User Equipments (UEs). Furthermore, Li et al. [12] proposed a dynamic optimization framework for multihop device-to-device communication. These above studies suppose all the mobile users are altruistic and focus on increasing the overall network transmission rate cooperatively. On the other hand, hand-held wireless devices are carried by mobile users who form social networks with stable social characteristics [13]. In social networks, social community, as one of the most important characteristics, represents real social

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groupings by interests or background, and different communities are usually interested in different mobile contents [14]. Human beings in one social community have the same interested contents, i.e., content sharing occurs among friends in the same community by online social networking services, such as Facebook, Twitter and etc. Therefore, each social community will consider maximization of its own transmission rate, which can speed up the traffic offloading process. For example, when some regular cellular users in a community obtain some mobile data from the BS, other D2D users in the same community can get the same requested contents through the direct D2D communications. The existing resource allocation algorithms [6]–[11] do not consider social information of the mobile users, which influences the effectiveness of resource allocations in D2D communications underlaying cellular networks. The social-aware D2D communications gain attentions in recent studies [15]–[22]. In cognitive radio networks, Chen et al. [15] utilized the reward-based Markov decision process to enable secondary users cooperatively access the spectrum resource. Furthermore, imitation-based spectrum access mechanism was proposed to implement efficient spectrum access [16]. Li et al. [17] summarized the influence of social features on D2D communications and quantitatively analyzed the achievable gains in a social-aware D2D communication system. Cao et al. [19] provided the cooperative video multicast framework to implement the mobile user’s cooperation efficiently. Sun et al. [20] used a Bayesian model to illustrate the social ties for D2D mobile users and achieved efficient data transmission among D2D users. Zheng et al. [22] proposed a social aware algorithm for efficient multi-file disseminations under multi-hop D2D communication networks. Zhang et al. [23] utilized the social network characteristics to assist the ad-hoc peer discovery. These works focus on the influence of social relationships on overall transmission rate of D2D communication networks. However, cellular users have diverse social relationships with their neighbors at different levels [24], and the network optimization without considering these social relationships cannot maximize the overall network utility. D2D users have different strength of social connections with others in the same social community, which needs to be considered for the resource allocation to increase the network utility. Network utility maximization has been extensively studied for network optimization problems [6]–[11], and these studies are based on an assumption that all users act in an altruistic manner in which they have the same social objective to maximize the overall network utility. On the other hand, game theory has found a wide variety of important applications for distributed resource allocation problems in various networking applications [25], [26]. These game theory models can be classified by two categories: non-cooperative and cooperative models [25]. For example, Stackelberg game was utilized to implement efficient resource allocation for femtocells and D2D users, respectively [9], [27]. However, these non-cooperative game-theoretic models usually assume that all users are selfish and rational, aiming at maximizing its own benefit. On the other hand, the cooperative game-theoretic models suppose that users are altruistic and helpful. In fact, these assumptions, that users are either altruistic or selfish by network utility maximization

and non-cooperative game, represent two extreme cases of users’ relationships that are fully socially oblivious and socialaware. Social relationships among D2D users are more complex than these two conditions. In order to evaluate the diverse relationships among human beings, Chen et al. [28], [29] propose a concept of social group utility maximization (SGUM), which exploits the impact of diverse social relationships on physical domain of the wireless communication networks. The social group utility is able to establish the connection between the physical domain and the social domain of D2D communications. The regular cellular users and D2D users in the same social community have common interests and can share their contents directly. Therefore, each community in the D2D communications needs to increase its sum transmission rate cooperatively without considering the transmission rate of other social communities. Both schemes of maximizing the individual rate non-cooperatively and the sum rate of overall networks cannot exploit the characteristic of social community and offload the traffic from the BS effectively. In this paper, in order to evaluate the joint optimization performance of social and physical domains qualitatively, we investigate the resource allocation problem by maximizing the social group utility for D2D communications. Indeed, when this framework and concept are utilized to solve and implement efficient resource allocation for D2D communications, many new problems need to be solved. We propose a general cooperative game theory solution through utility definition, game establishment, existing Nash Equilibrium and solutions with convergence rate guarantee. Specifically, considering the interferences among cellular users and D2D users, we formulate the social community aware resource allocation as a social group utility maximization game. Then, we propose a distributed resource allocation algorithm to achieve the Nash Equilibrium (NE) for the proposed SGUM game. Our contributions are summarized as follows. • We define the utility of each D2D user considering both the physical interference and social characteristics of the regular cellular users and D2D users, and formulate the SGUM game to maximize the utility of each D2D user. To the best of our knowledge, this is the first study to apply the SGUM concept into the resource allocation problem for diverse social relationships of D2D communications. • For the SGUM game, we theoretically investigate the existence of NE for the proposed SGUM game and propose a distributed algorithm to implement efficient social-aware resource allocation with low computation complexity. In the solution, each D2D user independently maximizes its social group utility through switch operations. Furthermore, we obtain the convergence rate for our proposed algorithm. • We evaluate the influence of the different networking environments on the performance of proposed solution. Numerical results demonstrate that it increases the utility of overall social groups about 16% to 56% without loss of fairness compared with other state-of-the-art schemes.

ZHAO et al.: SOCIAL-AWARE RESOURCE ALLOCATION FOR DEVICE-TO-DEVICE COMMUNICATIONS

Fig. 1. Illustration of the social community aware uplink resource sharing of the D2D communications underlaying cellular networks, where there are 2 cellular users, c1 and c2 , and 4 D2D users. In physical domain, wireless links are subject to the physical interference constraints, while social domain indicates the relationships among mobile users.

The rest of this paper is organized as follows. After presenting the system model in Section II, we formulate the SGUM game in Section III. Next, distributed algorithm is proposed in Section IV. Performance evaluations are given in Section V, and finally Section VI concludes this work. II. S YSTEM OVERVIEW, M ODELS , AND P ROBLEM F ORMULATION In this section, we first give a system overview for the D2D communications underlaying cellular networks consisting of different social communities, then derive the group utility of each D2D user in terms of the channel rate and the social relationships. Finally, we formulate the resource allocation problem that we need to investigate. A. System Description In our work, social community aware D2D communication underlaying cellular networks are considered as combination of physical domain and virtual social domain, which is illustrated in Fig. 1. In the physical domain, wireless links are determined by the radio transmission distance between two mobile nodes; while the social domain indicates the relationships among mobile users with hand-held wireless devices. In the social domain, mobile users in D2D communication networks are divided into different social communities, which are defined as the groups with common traffic interests, content sharings, and so forth. As an example showing in Fig. 1, there are two communities denoted by CommunityA and CommunityB, which is represented by the solid line and dash line in the social domain, respectively. Social relationship among the cellular users can be obtained through the online scheme and offline scheme. On one hand, the social relationships can be obtained through exploring the

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content sharing and account information. For example, Wang et al. [30] have kept track of 2,223,294 users for four weeks in the online social network service Sina Weibo and obtained the list of all resharing activities for each microblog. Based on social graph, Wang et al. [30] calculate the importance of each user for propagating the micropost. On the other hand, we can also exploit the social relationships through the offline mobile social networks, which mean the realistic social network due to user mobility, e.g., meeting and groups [31]. It is discovered that people who are graphically close may have similar trends of accessing and sharing content with each other. K-CLIQUE by Palla et al. [32] can detect the social community based on the realistic mobile traces of cellular users. In the physical domains, we assume that the regular cellular users share their uplink resources with D2D communications, and one cellular user’s resource can be shared with multiple D2D users in order to maximize the spectrum efficiency. It is supposed that a D2D user shares at most one cellular user’s spectrum resource. This constraint is used to reduce the complicated interference environment brought by the D2D communications. When the distance between two D2D users satisfies the predefined distance threshold, the corresponding D2D link is established [11]. In LTE system, the prevalent resource allocation unit is physical resource block (RB), which occupies 0.5 ms in the time domain and 180 kHz in the frequency domain. In our model, we utilize RB as the resource allocation unit and regular cellular users have orthogonal RBs initially. As illustrated in Fig. 1, there are two cellular users c1 and c2 , and D2D users (d11 , d12 ) and (d21 , d22 ) occupy the spectrum resource of u1 , while D2D users, (d31 , d32 ) and (d41 , d42 ), occupy the spectrum resource of c2 . In such a system, we concentrate on assigning appropriate RBs occupied by the cellular users to D2D users in order to achieve a higher social group utility. Therefore, we suppose that the transmit power of cellular users and D2D transmitters are constant. Since D2D communications share the same spectrum resources of cellular communications at the same time slot, their interference should be limited as much as possible to optimize the system performance. Consider a set of D2D users, denoted by D, share the resource of cellular user c. As shown in Fig. 1, during the uplink period of the cellular system, all the cellular and D2D users interfere with each other as they share the same spectrum resource blocks. The received signals at the BS from cellular users c is interfered by the transmitters of D2D links, which occupy the same spectrum resource of c. The signal at the D2D receiver d is interfered by the cellular user and other D2D links, which share the same spectrum resource with d. B. System Model and Problem Description 1) Interference Graph for Physical Domain: In this subsection, we construct the interference graph for the physical domain to represent the resource sharing relations for the D2D communication networks [33], [34]. In the system, we assume there are C cellular users labelled as the set of C = {c1 , c2 , · · · , cC }, which share their uplink resources with D2D users. Besides the cellular users, there are D D2D users labelled

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if and only if ∃c ∈ C : xc,d = 1, xc,d = 1; otherwise, yd,d = 0. From the above resource usage relationship, we have the value p of rij as xi,j , i ∈ C, j ∈ D or i ∈ D, j ∈ C, p rij = (1) yi,j , i ∈ D, j ∈ D. In this work, the channel is modeled as Rayleigh fading channel, under which the instantaneous channel taps are the function of time and spatial locations. The power or secondorder statistic of the channel, denoted by |h0 |2 , is a constant within the BS’s coverage area. Thus, under the free space propagation path-loss model [11], the received power of each link between nodes i and j, says i − j, can be expressed as −α · |h0 |2 , where Pi is the transmitted Pi,j = Pi · |hij |2 = Pi · ζi,j power of equipment i, ζi,j is the distance of the i − j link, α is the path-loss exponent, and h0 is the complex Gaussian channel coefficient that obeys the distribution of CN (0, 1). For the purpose of maximizing the network utility in terms of the transmission rate, we need to consider the signal to interference plus noise ratio (SINR) as an important indicator. The SINR of any user j receiving signal from user i can be expressed by ρj = Fig. 2. Illustration of the social graph and interference graph for the D2D communication underlaying cellular networks.

as the set of D = {d1 , d2 , · · · , dD }. Naturally, these D2D users can choose the spectrum resources of any cellular users ci , ∀ ci ∈ C. We use the undirected weighted interference graph model Gp = (Vp , Rp ) to depict the interference of physical domains due to the spectrum sharing in our system. The vertex Vp denotes the communication links of the regular cellular users and D2D users, while the edge Rp represents mutual interference relationships among the communication links. The entry p of Rp is denoted by rij , i, j = 1, · · · , D + C, which equals to 1 if the communication links i and j occupy the same spectrum p resource and incur the mutual interference. The value of rij is determined by the resource allocations for the D2D communication networks. In the lower part of Fig. 2, we construct the interference graph for the resource usage relationships in Fig. 1. For example, c1 , d1 and d2 occupy the same spectrum and the values of mutual edges equal to 1. c2 , d3 and d4 occupy the same spectrum and the values of mutual edges equal to 1. The values of other edges equal to 0 without mutual interferences and are omitted in the interference graph for clarity. In order to describe the spectrum resource usage relationship, we define xc,d ∈ {0, 1} as the indicator that whether the D2D users d shares the resource blocks of UE c, ∀ d ∈ D, ∀ c ∈ C. That is xc,d = 1 when D2D users d uses the resource blocks of UE c; otherwise, xc,d = 0. Since we limit a D2D user shares at most one cellular user’s spectrum resource, we have the xc,d ≤ 1, ∀ d ∈ D. In this resource sharing constraint of c∈C

model, in order to increase the spectrum reuse ratio, it allows multiple D2D communications occur on the same part of the frequency resources from one cellular user, which also incurs mutual interferences of the D2D users. We use yd,d to denote the interference relationships among the D2D users. yd,d = 1

−α Pi ζi,j |h0|2

Pint,j + N0

,

(2)

where Pint,j is the interference signal power received by user j, and N0 is the terminal noise at the receiver. Based on the interference graph and channel model, we are able to derive the uplink transmission rate corresponding to the SINR of the cellular and D2D users. In the uplink period, the BS receiving the signal from the cellular user suffers interference from the D2D users that share the same spectrum resource. According to the interference graph in the physical domain, the interference power at the BS for cellular user c, denoted by Pint,c is determined by the vertices having the edges with c. Therefore, we obtain Pint,c , as follows, p Pint,c = rc,d Pd |hdb |2 . (3) d∈D

Thus, the uplink channel rate of the cellular user c, denoted by Rc , is ⎛ ⎞ −α 2 P ζ |h | c c,b 0 ⎜ ⎟ (4) Rc = log2 ⎝1 + p ⎠. −α 2 rc,d Pd ζd,b |h0 | + N0 d∈D

Similarly, the interference of D2D receiver d is from the cellular user c and the other D2D users that are assigned with the same resource. Therefore, according to the interference graph, we can obtain the interference power from BS and the other D2D users for D2D receiver d, denoted by Pint,d, as follows, p p rc,d Pc |hcd |2 + rd,d Pd |hd d |2 . (5) Pint,d = c∈C

d ∈D\{d}

Thus, the channel rate for the D2D user d, denoted Rd , is −α |h0 |2 Pd ζd,d Rd = log2 1 + . Pint,d + N0

(6)

ZHAO et al.: SOCIAL-AWARE RESOURCE ALLOCATION FOR DEVICE-TO-DEVICE COMMUNICATIONS

Consider all the cellular user C and D2D users D together, we can obtain the system sum rate, denoted by R, as Rc + (7) rc,d Rd . R= c∈C

d∈D

2) Weighted Graph for Social Domain: Now, we use a weighted graph to model the social community characteristic of the cellular users in the system. In the virtual social domain of D2D communications, the regular cellular users and D2D users form different social communities. Social connections exist in each community, while there are no connections between different communities. When a regular cellular user downloads the contents from the BS, other users in the same social community, including cellular and D2D users, can obtain the interested contents from the regular cellular users directly instead of the BS. Meanwhile, D2D users in the same social community may exchange their own contents directly, which is determined by the social connections. Therefore, there are different social communities in the social domain and diverse strength of social connections exist in the same community. From the above analysis, the relational graph for the social connections is denoted by Gs = (Vs , Rs ), where Vs is the collections of all regular cellular users and D2D users, and Rs is the collections of relational edges. The matrix Rs denotes the connection of Vs in the social domain, which is the square matrix with D × (D + C) entries ωi,j , i, j = 1, · · · , D and δi,k , i = 1, · · · , D, k = 1, · · · , · · · , C. ωi,j is the closeness coefficient between D2D users i and j. δi,k is the closeness coefficient between D2D user i and its associated cellular user k. In the social network framework, we formalize the strength of social relationships between users as ωi,j ∈ [0, 1] or δik ∈ [0, 1] with a higher value of ωij or δi,k ∈ [0, 1] being a stronger social connection, which can be kinship, friendship, or colleague relationship between two users. ωi,j = 0 means the weakest connection of two D2D users and ωi,j = 1 means the strongest connection. As an example in the upper part of Fig. 2, we construct the social graph of the mobile users in Fig. 1. Bob, Brown and David have the same interested contents and the regular cellular user Bob downloads the complete contents from the BS. Therefore, D2D users Brown and David have the strongest connection with Bob, while Brown and David have weaker social connections. From the above analysis, the utility of D2D user d and regular cellular user c can be defined as follows: ⎛ ⎞ ⎜ Rd = log2 ⎜ ⎝1 + ⎛ ⎜ Rc = log2 ⎜ ⎝1 +

−α Pd ζd,d |h0 |2 −α Pj ζj,d |h0 |2 p

j∈Nds ∩Nd

−α |h0 |2 Pc ζc,b −α Pj ζj,b |h0 |2 p

⎟ ⎟ , ∀ d ∈ D, (8) + N0 ⎠ ⎞ ⎟ ⎟ , ∀ c ∈ C, + N0 ⎠

(9)

j∈Ncs ∩Nc

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same social community with d on D2D user d, and Rc indicates the impact of spectrum sharing in the same social community with c on regular cellular user c. Then, we define the social group utility of each D2D user i, denoted by Ui , as follows, Ui (X) = Ri + ωi,j Rj + δi,c Rc . (10) j∈D\{i}

c∈C

The social group utility of D2D users consists of 3 parts: its own utilities of individual rate Ri , the weighted sum of individual other D2D users having social connections with it ωi,j Rj , j∈D\{i}

and the weighted sum of utilities of cellular users in the system δi,c Rc . From the above analysis, we observe that Ui (X) is c∈C

determined by the interference graph in the physical domain and the weighted graph in the social domain. Therefore, the social group utility of each D2D user represents the coupling between physical and social domains. 3) Problem Formulation: In order to maximize the social group utility of each D2D user, we need to determine the optimal resource allocation. The social group utility Ud of each D2D user d depends on the resource allocation and social relationships between the cellular users and D2D users. The social information is more long-lasting and stable. Thus, at each time point we denote the social group utility of user d as function Ud (X), where X is the matrix of xc,d , c ∈ C, d ∈ D. In order to maximize the social group utility of each D2D user, we need to determine the optimal resource allocation policy. Thus, combing the above definitions, we formulate the optimal resource allocation problem in the D2D communications underlaying cellular networks as the following optimization problem, max Ud (X), d∈D ⎧ ⎨xc,d ∈ {0, 1}, ∀ d ∈ D, c ∈ C; s.t. ⎩ xc,d ≤ 1, ∀ d ∈ D.

(11)

c∈C

In the formulated problem, since the optimization utility function (10) has no obvious increasing or concave properties with xc,d even the constraint is linear. In contrast to the multiobjective optimization, each D2D user would like to maximize its own utility. Actually, our optimization problem maximizes the social group utility of each D2D user at some time point. D2D links in a community are changing over the time, and our proposed approach can be applied in the dynamic networks to maximize the social group utility at each time point. The distributed nature of the above problem leads to a game theory naturally, where each user d maximizes its corresponding Ud (X). The resource allocation variable xc,d indicates the strategy of each D2D user d to occupy the spectrum of the regular cellular c. Therefore, the formulated problem (11) can be viewed as a distributed decision game for each D2D user d. Base on the above analysis, we solve the above social group utility maximization problem by formulating a SGUM game.

p

which are also viewed as the interference aware rate. Nk and Nks denote the set of cellular users which occupy the same spectrum resource and have the social connection with k, respectively. Rd indicates the impact of other devices in the

III. S OCIAL G ROUP U TILITY M AXIMIZATION G AME In this section, we present the social group utility maximization game for the formulated resource allocation problem.

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Then, we theoretically prove that it has Social Nash Equilibrium (SNE).

Based on the above definitions, we define the ordinal potential function as Ud (X) P(x) =

A. SGUM Game Formulation

d∈D ⎛

From the formulated problem, we can obtain that the optimization problem aims to maximize each D2D user’s utility with the consideration of the social information. Naturally, we utilize the game theory to solve the optimization problem. Therefore, we consider the distributed decision problem among the users for maximizing their own social group utilities. Define the strategy of each D2D user is xd ∈ Xd , where Xd = {1, 2, . . . , C} is set of all feasible strategies of the D2D user d. xd can be determined by the resource allocation relationship xc,d one by one. For example, xc,d = 1 indicates that xd = c. For our formulated problem (11), let x−d = (x1 , . . . , xd−1 , xd+1 , . . . , xD ) be the set of strategies chosen by all other D2D users except user d. We also define x = (x1 , x2 , . . . , xD ) = (xd , x−d ) to denote the resource allocation vector and X is the set of all possible x. Given the other users’ strategies x−d , user d wants to choose a strategy xd ∈ Xd to maximize its social group utility, i.e., max

xd ∈Xd

=

⎝Rd +

d∈D

ωd,j Rj +

j∈D\{d}

j∈D\{d}

−

j∈D\{d}

⎞

⎛

⎜ Rk − ⎝Rd +

Ud (x−d , xd ) − Ud x−d , xd > 0

iff

P(x−d , xd ) − P x−d , xd > 0.

⎟ Rk ⎠ . p

k∈Nds ∩Nd

P(d) − P (d) ⎛ ⎞ ⎝Rd + = ωd,j Rj + δd,c Rc ⎠ d∈D

−

⎛

+

j∈D\{d}

⎝Rd +

d∈D

B. Social Aware Nash Equilibrium

and x∗ = {x∗1 , x∗2 , · · · , x∗D } is the SNE. Motivated by the method in [35], we utilize the potential game theory to verify the existence of SNE for the SGUM game [36]. We provide the definition of ordinal potential game and give its associated ordinal potential function. G is named as an ordinal potential game if it has an ordinal potential function. A function P(x) is an ordinal potential for G if for every d ∈ D and for every x−d ∈ X−d :

Next, the changing of the ordinal function is

= Ud − Ud

xd ∈Xd

(13)

c∈C

= Rd +

δd,c Rc

p

x∗d = arg max Ud (xd , x−d ), ∀ d ∈ D,

(12)

c∈C

ωd,j Rj −

k∈Nds ∩Nd

In the subsection, we first give the definition of social aware Nash Equilibrium. Then, we theoretically prove the existence of Nash Equilibrium through the potential game. The SGUM game G = (D, {Xd }d∈D , {Ud }d∈D ) has the Nash Equilibrium if no D2D user can increase its social group utility by changing its resource sharing strategy, i.e.,

δd,c Rc ⎠ .

c∈C

Theorem 1: Function P(x) is an ordinal function for SGUM game G = (D, {Xd }d∈D , {Ud }d∈D ) when the strength of social ties in the same community is 1. Proof: We suppose D2D user d changes its spectrum sharing and occupies the spectrum resource of cellular c instead of c. Therefore, the indicator xc,d changes to xc,d and yd,d0 changes to yd,d0 . Then, the changing payoff of each D2D user is Ud − Ud = Rd + ωd,j Rj + δd,c Rc − Rd

Ud (xd , x−d ), ∀ d ∈ D.

The distributed nature of the problem above naturally leads to a formulation based on game theory such that each user aims to maximize its social group utility. The SGUM game with the social group utility for the resource allocation for D2D communication is defined by the triple G = (D, {Xd }d∈D , {Ud }d∈D ), where the set of D2D users D is the players, Xd is the set of resource sharing strategies for each D2D user d, and the social group utility function Ud is the payoff function of each D2D user d.

⎞

c∈C

ωd,j Rj +

j∈D\{d}

⎞ δd,c Rc ⎠

c∈C

⎛

⎜ ⎜ Rj − ⎝Rk + ⎝Rk +

p k∈D\{d}∩Nds∩Nd

⎞⎞

⎛

p j∈Nks ∩Nk

p = Nds ∩ Nd + 1 Ud − Ud ,

⎟⎟ Rj ⎠⎠

p j∈Nks ∩Nk

(14)

which is obtained as d and k not only are in the same social community but also occupy the same spectrum resource. Finally, we conclude that Ud − Ud > 0 iff P(d) − P (d) > 0, ∀ d ∈ D.

(15)

Theorem 2: The SGUM game G = (D, {Xd }d∈D , {Ud }d∈D ) has the pure Nash Equilibrium. Proof: From Theorem 1, we have that the SGUM game G = (D, {Xd }d∈D , {Ud }d∈D ) is the ordinal potential game with the ordinal potential function P(x). Each D2D user d maximizes its utility Ud , which is equal to maximize the potential function P(x). Therefore, each feasible strategy of D2D user d is able to increase the overall potential function P(x). At the same time, G is a finite potential game, and the number of all feasible strategies is finite. Thus, G possesses a pure strategy Nash Equilibrium [36].

ZHAO et al.: SOCIAL-AWARE RESOURCE ALLOCATION FOR DEVICE-TO-DEVICE COMMUNICATIONS

IV. D ISTRIBUTED R ESOURCE A LLOCATION A LGORITHM FOR SGUM A. Distributed Solution The key mechanism in decision making of the SGUM game formulated in (11) is to enable D2D users to choose a cellular user to occupy the spectrum resource, which is the strategy of each D2D user. Specifically, each D2D user must be able to compare and order its potential social group utility based on which cellular user this player prefers to share. According to the property of potential game, the resource allocation strategy x∗ is a Nash Equilibrium, which maximizes the value of ordinal potential function. Furthermore, the SGUM G has the finite improvement property (FIP) due to finite values of the ordinal potential function. Specifically, the order of D2D users obtaining a better resource sharing can be random and achieve the equilibrium by finite steps. Here, we introduce the concept of preference order i for any user i ∈ D as the following definition. Definition 1 (Preference Order): For any D2D users i ∈ D, the preference order i is defined as a complete, reflexive, and transitive binary relation over the set of all resource allocation strategies that D2D user i can possibly take, i.e., the set {x ⊆ X , xi ∈ Xi }. In our SUGM game for the resource sharing, the D2D users choose to joint or leave the associated regular cellular users according to its preference order. For any D2D users, i, xi i xi means i prefers sharing the spectrum of c ∈ C than c ∈ C, where xi ⊆ X and xi ⊆ X . Since the preference order depends on its social utility, in this paper, for any D2D users, i ∈ D and xi , xi , we define the following preference xi i xi ⇔ Ui xi > Ui (xi )&Uj xi ≥ Uj (xi ),

∀ j ∈ {D\i}. (16)

This definition implies that D2D user i prefers to share the spectrum resource of c over c only when i gains increasement in its social group utility while no other D2D users in D \ {i} suffers decreasing on their social group utility due to its switching. Based on the above resource sharing and preference order, we can form the resource allocation from a given initial strategy by switch operations. Suppose given a resource allocation vector x = {x1 , . . . , xj , . . . , xk , . . . , xD } of D2D users D, for ∀ i ∈ D, we suppose its current resource allocation strategy is x ∈ X . Then, a switch operation from x to x ∈ X ∪ {∅}, x = x , means the D2D users j and k switch their resource allocation strategies. A new resource allocation vector x = {x1 , . . . , xj , . . . , xk , . . . , xD } is obtained such that xj = xk and xk = xj . After repeating switch operations, the system can generate different resource allocation vectors of the D2D users D. Combining the preference order, a switch operations from x to x is allowed for any D2D user i ∈ D, if and only if xi i xi . In this mechanism, every D2D user i ∈ D can leave its current resource allocation strategy, and occupies the spectrum resource of another cellular user given that the new resource allocation vector is strictly preferred through the preference relation defined in (16).

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Algorithm 1 The Social Group Utility Maximization Algorithm for the D2D Users Resource Allocation 1: Initialize the system by any random resource allocation vector xini ; 2: Set the current resource allocation vector as xini → xcur ; 3: repeat 4: Uniformly randomly choose one D2D user i, and denote its associate resource sharing as xi ∈ xcur ; 5: Uniformly randomly search for a possible D2D user k ∈ D \ {i}, whose corresponding switch operation from xj = xk and xk = xj . 6: Calculate U(x) and U(x ) by (10). SwitchFlag = 0; 7: if xi i xi then 8: SwitchFlag = 1; 9: else 10: Calculate φx ,x = 1+exp(−(U(x1 )−U(x))/Tn ) . 11: Get λ uniform distribution in (0,1]. 12: if λ < φx ,x then 13: SwitchFlag = 1; 14: end if 15: end if 16: if SwitchFlag == 1 then 17: D2D users i and k leave their current associate cellular users, and join the new cellular users c and c, respectively; 18: Update the current resource allocation vector as x → xcur ; 19: end if 20: until The resource allocation vector converges to a final Nash-stable resource allocation vector xfin . Based on the above resource allocation vector and switch operation, we present the distributed resource allocation algorithm for the D2D users summarized in Algorithm 1. In the algorithm, D2D users make switch operation in a random order based on a random generated initial resource allocation vector xini . In each iteration, the system uniformly randomly chooses one D2D user i ∈ D in Step 4. In Step 5, the selected D2D user recognizes its resource allocation strategy xi ∈ x and uniformly randomly selects another possible D2D user k ∈ D. Then, it will request the channel information of these two D2D users and the cellular users having connections with i or k from the cellular base station by cellular direct communications. After obtaining these information, the D2D user computes the utility of the resource allocation vector x and x , and decides to whether execute the switch operation. In order to avoid the local maximum value, when the constraint is not satisfied, we also define the acceptance probability [6] 1 , (17) φx ,x = 1 + exp (− (U(x ) − U(x))/Tn ) T0 where Tn = log(n−1) with T0 as a constant value and n is the current times of switch operations. Now we give the second condition for switch operation as follows,

λ < φx ,x ,

(18)

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where λ is the uniform distribution in (0,1]. We use Barker’s sampler to enlarge the feasible space of resource allocation to prevent our proposed algorithm from obtaining the local optimal value [38]. Therefore, our proposed distributed algorithm based on switch operation can achieve the Nash Equilibrium of SGUM game.

TABLE I S IMULATED PARAMETERS

B. Convergence and Stability In this subsection, we will analyze the properties of convergence, stability and computation complexity for Algorithm 1. Indeed, in terms of the convergence, since the number of the resource allocation vector is finite and we use the history information to avoid the repeating choices, the switch operation will always terminate. According to the concept of Nash Equilibrium in Theorem 2 for the SGUM game, the stability of the final resource allocation vector xfin depends on whether a Nash-stable exists letting ∀ i ∈ D and xi i xi . The convergence and stability of our proposed resource allocations algorithm are guaranteed as follows. Theorem 3: Starting from any initial resource allocation vector xini , the proposed distributed algorithm will always converge to a Nash-stable resource allocation vector xfin . Proof: In each switch operation in Algorithm 1, it will yield a new resource allocation vector through adopting new strategy or switching existing strategies, and the maximum number of resource allocation strategies for each D2D user is C since there is only C cellular users in the system. Therefore, the number of resource allocation vector for the given D2D users set D is a Bell number [6]. Thus, the sequence of random switch operations will terminate with probability 1, and the system then converges to a final resource allocation vector xfin after finite turns with probability 1, which proves the convergence in probability for our proposed distributed algorithm. Now, we prove the stability of the proposed algorithm by contradiction. Suppose the final resource allocation vector xfin obtained from Algorithm 1 is not Nash-stable. Then, there exists a D2D user i ∈ D whose resource allocation vector is denoted by xi , and a new resource allocation vector xi ∈ xfin such that xi i xi . According to Algorithm 1, D2D user i can perform a switch operation from xi to xi , which contradicts the fact that xfin is the final resource allocation vector. Thus, we have proved that the final network resource allocation vector xfin resulting from Algorithm 1 must be Nash-stable. For the computation complexity, we note that these switch operations in the proposed algorithm only use local D2D community instead of all the users in the network. After a limited number of switching, the system resource allocation converges to the final Nash-stable resource allocation vector xfin , which is determined by the convergence rate in the following Theorem 4. As a result, the complexity is much lower than the centralize solution. Theorem 4: The convergence of SGUM to the optimal solution is the geometric rate. Proof: As the SGUM algorithm utilizes the Barker’s Sampler to accept the worse solution, we can utilize an evolving Markov chain to depict the process of switch operation [38]. The Markov chain S(Tn ) has the transition matrix Q = {qi,j}.

The strategy vector x is the current state and xn denotes the state after n iterations. From Algorithm 1, the probability from the state x to x can be depicted as: φx ,x =

1 . 1 + exp (− (U(x ) − U(x)) /Tn )

From [38], we observe that S(Tn ) has the stationary distribution π, which does not depend on Q. Following the basic notations and main idea from Ref. [38], we utilize d(xP − πP) to denote the variation between the current system state and the stationary state. Then from the Definition 7.1 of Chapter 6 [38], we obtain d(xP − πP) = δ(Q) ≤ δ(Q1 )δ(Q2 ), where Q2 = P and Q1 can be expressed as x(1) x(2) ··· Q1 = . π(1) π(2) ··· Then d(xP − πP) ≤ d(Q1 )δ(P) = 12 d(x − π)δ(P). From the definition of δ(P), we obtain that

δ(P) ≤ 1 − e T . Therefore, after n switch operation, d(xPn − πPn ) ≤

n 1 d(x − π) 1 − e T , 2

which means that the convergence of SGUM to the optimal solution is the geometric rate. V. P ERFORMANCE E VALUATION In this section, we evaluate the performance of the proposed SGUM game based D2D resource allocation scheme. We carry out the simulations in a single cell scenario. Both path-loss model and shadow fading are considered for cellular and D2D links. In our network simulation, the wireless propagation is modeled by WINNER II channel models [11]. In particular, the D2D communication channel is based on the scenario that two communicating UEs are physically in close proximity, while the cellular communication channel is simulated according to the urban microcell scenario. The path-loss exponent for free space propagation path-loss model is set to be 4. The main parameters used in our simulation are listed in Table I. As listed in Table I, we simulate the system within a 500 m × 500 m area with the base station in the center. To obtain the settings with positions of the users, we uniformly randomly distribute the cellular users and D2D users within the

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rithm could achieve the near-optimal solution of system sum rate. b) Nearest First (NF), where the system allocates the communication resources to the D2D users with the cellular users that are nearest to the D2D users in their position. c) Furthest First (FF), where the system allocates the D2D communication resources with the cellular users which is the furthest to the D2D users. d) Random Selection (RS), which uniform randomly allocates the resources to the D2D users to communicate. That is to say, for any D2D user, the system randomly selects a cellular user’s resource for sharing. A. System Sum Social Group Utility

Fig. 3. A snapshot of a final resource allocation vector resulting from the proposed algorithm for a network of 3 cellular users and 20 D2D users.

In order to demonstrate our proposed solution can achieve the social group utility maximization, we compare its performance with four other algorithms. The sum social group utility can be calculated as Ud (x), (19) Usum (x) = d∈D

coverage of the BS. The transmitter of D2D link is randomly distributed in the coverage of BS, and the receiver is randomly distributed in the circle of transmitter with the maximum distance. As an illustration for how cellular and D2D users located in the simulation, we plot the positions of the base station, cellular users and D2D users in an instance by randomly generating a network with 3 cellular users and 20 D2D users in Fig. 3. In this figure, the cellular users are represented by triangle, while the D2D users are denoted by circle. Moreover, we also show the snapshot of a final resource allocation vector resulting from our proposed algorithm under the above network setting. In this example, 20 D2D users occupy the spectrum resources of 3 cellular users, and they are marked by different symbols of 1 labeled black color, 2 labeled red color and 3 labeled blue color in Fig. 3, respectively. According to the solution of resource allocation, we evaluated the following four performance metrics: Ud (x), which is determined 1) Social group sum utility d∈D

by all the D2D users, the cellular users and the strength of the social connections between them. 2) System sum rate R, which is the aggregated communication rate of all users including D2D users and cellular users, and it is obtained by (7). 3) The Jain’s fairness measure [37], which determines whether the receivers of D2D and cellular users are receiving fair share of the system resources. In order to show the efficiency of our social group utility maximization based resource allocation, we compare the performance of our scheme, denoted as Social Group Utility Maximization (SGUM), with the four schemes: a) Coalition Game (CG), where the system allocates the system resources to the D2D users through the coalition formation game model [6]. This distributed algo-

where Ud (x) is obtained from (10). In this simulation, we suppose that each D2D user randomly selects another D2D user to establish the social connection based on the social link probability 0.5, and the strength of social connection is set to be 1. We set the number of D2D users to be 15 and vary the number of the cellular users to be 1 to 15 to obtain the results of the sum social group utility shown in Fig. 4(a), while set the number of cellular users to be 5 and vary the number D2D users from 1 to 20 to obtain the results shown in Fig. 4(b). From these two figures, we observe that the system utility increases with the number of cellular users or D2D users increasing. In Fig. 4(a), when the number of cell users is equal to 15, the sum utility of SGUM is larger than that of NF and CG about 56% and 16%, respectively. In Fig. 4(b), when the number of D2D users is equal to 20, the sum utility of SGUM is larger than that of NF and CG about 53% and 16%, respectively. NF algorithm performs worst, as it neither decreases the interferences in the physical domain nor considers the social community information in the social domain. SGUM performs better than NF about 45% averagely. Although CG algorithm achieves the near-optimal solution to decrease the interferences, CG performs worse than SGUM as CG also cannot consider the social community information. From these results, we can observe that SGUM achieves the best performance in all the four schemes, which considers both the interference relationship and social information. B. System Sum Rate Then we analyze the system sum rate of all the four schemes, which can calculated by (7). Similarly, we set the number of D2D users and cellular users to be 15 and 5, respectively, varying the number of the cellular users or D2D users to be 1 to 15 or 1 to 20, respectively. The case of 20 D2D users represents the situation of resource-lacking system in which D2D users

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Fig. 4. Performance of sum social group utility of different resource allocation algorithms with (a) different number of cell users and (b) different number of D2D users.

Fig. 5. Performance of system sum rate of different resource allocation algorithms with (a) different number of cell users and (b) different number of D2D users.

possibly need to share the same user’s resource, while the other case represents the resource-abundant system in which D2D users may have more selections. The performance of the two system sum rate are shown in Fig. 5(a) and (b), respectively. The result indicates that the system sum rate increases with the number of cellular users increasing, because the system bandwidth resource increases. When the number of D2D users increases, the system sum rate increases as the proximity D2D users occupy the same spectrum resource of cellular users. However, the spectrum sharing also incurs interferences, which constrain the system performance. Therefore, there must be saturation in this condition. Comparing the two figures, we also note that the increasing number of cellular users may bring larger system performance enhancement than increasing the number of D2D users. Among all the five schemes, the scheme of CG always achieves the best performance for the reason that it may reduce the unnecessary interference in the system, while the NF always gets the worst performance since it does the largest interference between the D2D users and cellular users. Furthermore, we observe that SGUM algorithm heavily depends on the social link probability among the D2D users. From

Fig. 5(a) and (b), when the social link probability is set to 1, we observe that SGUM achieves the same performance with CG. This is because that each D2D user has the same target to maximize the whole network performance. When the social link probability is set to 0, we observe that SGUM achieves the same performance with FF. This is because that each D2D user aims to maximize its own rate without consideration of the strategies of other D2D users. The result indicates that our social group utility maximization algorithm increases the network utility and achieves the high system sum rate. In order to demonstrate the saturation trend with increasement of D2D users, the coverage of BS and the number of cellular users are set to 350 m × 350 m and 3, respectively. The social link probability is set to be 0.1, 0.7 and 0.9 to calculate system sum rate. From Fig. 6, we observe that the sum rate do not increase due to severe interference when the number of D2D links is above 40. However, the sum utility increases with increasement of D2D users. This is because that the social group utility is the metric to consider the interference in the physical domain and social connections in the social domain. When the number of D2D users is larger than that of cellular

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Fig. 6. Performance of (a) system sum rate and (b) sum utility of different resource allocation algorithms with different number of D2D users.

Fig. 7. Fairness of D2D and cell users transmission of different resource allocation algorithms with (a) different number of cell users and (b) different number of D2D users.

users, multiple D2D users may occupy the same spectrum resource of the same cellular user. In this condition, complex interferences exist among D2D users and cellular users. From Fig. 4(b) and Fig. 6(b), we can conclude that our proposed scheme outperforms other state-of-the-art solutions.

C. System Fairness To obtain some insights on how the data transmission is actually shared among the D2D users and cellular users, we depict the Jain’s fairness index in Fig. 7(a) and (b) with the variation of the number cellular users and D2D users, respectively. We observe that changing the number of cellular users and D2D users has a non-obvious influence on the fairness of data transmission under these schemes. Among all these compared schemes, we can obtain that CG has the best fairness resource sharing among the the cellular users and D2D users. Social link probability has the significant impact on the fairness of SGUM. This is because that SGUM considers the social relationships in

the social domain and maximizes each D2D user’s utility based on social communities.

D. The Impact of Social Link Probability In order to evaluate the social link probability on the system sum utility and sum rate. PL is used to denote the social link probability, which represents the density of social community. To observe the impact of social link probability on the system utility, we first set the number of cellular users and D2D users to be 5 and 20, respectively. In Fig. 8, we plot the sum utility varying PL from 0 to 1. It is obvious that SGUM performs best at different PL values. When PL = 1, SGUM and CG have the same sum utility, which indicates they both maximize the whole network utility. The difference of sum utility between SGUM and FF increases with PL. This is because that noncooperative scheme performs worse with PL increasing. While PL = 0.5 indicates the largest difference of sum utility between SGUM and CG. The reason is that PL = 0.5 represents the

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Fig. 8. Performance of social group utility with 5 cellular users and 20 D2D pairs varying different social link probability.

Fig. 10. System convergence rate in terms of the average number of switching operations in the case of varying the number of D2D users.

number of switch operations before Algorithm 1 converges to the final resource allocation vector xfin is showing in Fig. 10. As the number of D2D users increases, the average number of switch operations increases. In the case of 10 cellular users, which provide 10 selections for each D2D user, the average number of switch operations is around 3N to find the solution by our algorithm. While the exclusive search needs 10N iterations to find the optimal solution. Therefore, comparing with the exclusive search, our proposed algorithm decreases the computation complexity significantly. VI. C ONCLUSION AND F UTURE W ORK

Fig. 9. Performance of system sum rate with 5 cellular users and 20 D2D pairs varying different social link probability.

most random social relationships among D2D users. Therefore, for brevity, we select PL = 0.5 without loss of generality in Fig. 4. In Fig. 9, we plot the sum rate varying PL from 0 to 1. When PL = 0, the system sum rate of SGUM is equal to the value of FF. It is because that SGUM enables each D2D user to maximize its own rate when PL = 0. When PL = 1, SGUM aims to maximize the whole network performance and achieves the same sum rate with CG. When PL = 1, SGUM increases the system sum rate about 26% compared with PL = 0. The PL has no impact on the performance of FF and CG for sum rate. However, the sum rate of SUGM increases with PL. It is obvious that SGUM links the non-cooperative and cooperative resource allocation schemes for D2D communication network. In this scenario, the performance of PL = 0.5 is about the medium value between that of FF and CG. Therefore, we select PL = 0.5 in Fig. 6 to represent the typical condition. E. Convergence Rate In order to show the convergence rate of our proposed algorithm, we set the number of cellular users to be 2 and 10, and vary the number of D2D users N from 1 to 20. The average

We has studied the social-aware resource allocation problem for the D2D communication underlaying cellular system under realistic networking scenarios. By formulating it as a distributed social group utility maximization problem based on the social graph and interference graph, we propose a social group utility maximization game to solve this problem. By theoretical analysis, we prove that this scheme converges to a final Nash-stable network solution. Then, we propose a distributed resource allocation algorithm to decrease the computation complexity, which can achieve the Nash equilibrium of the social group maximization game. Through extensive simulation study, we have demonstrate the effectiveness and fairness of our proposed scheme. The results show that our scheme enhances the system sum utility rate by about 16% to 56% without loss of resource sharing fairness compared with several other strategies. On the other hand, traffic types and power control also have influences on the social group utility of each D2D user. Therefore, considerable future works are required to investigate the joint resource allocation and power control for D2D communication under different traffic types. R EFERENCES [1] “Cisco visual networking index: Global mobile data traffic forecast update,” Cisco, San Jose, CA, USA, 2013–2018, Feb. 5, 2014. [Online]. Available: http://www.cisco.com/c/en/us/solutions/collateral/serviceprovider/visual-networking-index-vni/white_paper_c11-520862.pdf [2] B. Han et al., “Cellular traffic offloading through opportunistic communications: A case study,” in Proc. 5th ACM Workshop Challenged Netw., Chicago, IL, USA, Sep. 2010, pp. 31–38.

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Yulei Zhao received the B.S. and Ph.D. degrees from the Department of Communication Engineering, Zhengzhou Information Science and Technology Institute, Zhengzhou, Henan, China, in 2005 and 2008, respectively. He is pursuing the Ph.D. degree with the Department of Electronic Engineering, Tsinghua University, Beijing, China. His research interests include cooperative communications, deviceto-device communications, and social networks.

Yong Li (M’09) received the B.S. and Ph.D degree in Huazhong University of Science and Technology and Tsinghua University in 2007 and 2012, respectively. During 2012 and 2013, he was a Visiting Research Associate with Telekom Innovation Laboratories and Hong Kong University of Science and Technology respectively. During 2013 to 2014, he was a Visiting Scientist with the University of Miami. He is currently a Faculty Member of the Department of Electronic Engineering, Tsinghua University. His research interests are in the areas of Mobile Computing and Social Networks, Urban Computing and Vehicular Networks, and Network Science and Future Internet. Dr. Li has served as General Chair, Technical Program Committee (TPC) Chair, and TPC Member for several international workshops and conferences. He is currently the Associate Editor of Journal of Communications and Networking and EURASIP Journal of Wireless Communications and Networking.

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Yang Cao (S’09–M’14) received the B.S. and Ph.D. degrees in information and communication engineering from Huazhong University of Science and Technology, Wuhan, China in 2009 and 2014, respectively. He is currently an Assistant Professor in the School of Electronics Information and Communications, Huazhong University of Science and Technology, Wuhan, PR China. From 2011 to 2013, he worked in the School of Electrical, Computer, and Energy Engineering, Arizona State University, AZ, USA, as a Visiting Scholar. His research interests include resource allocation for cellular device-to-device communications and smart grids. He has coauthored more than 20 papers on refereed IEEE journals and conferences. He was awarded CHINACOM Best Paper Award in 2010 and awarded Microsoft Research Fellowship in 2011.

Tao Jiang (M’06–SM’10) received the B.S. and M.S. degrees in applied geophysics from China University of Geosciences, Wuhan, China, in 1997 and 2000, respectively, and the Ph.D. degree in information and communication engineering from Huazhong University of Science and Technology, Wuhan, PR China, in April 2004. He is currently the Chair Professor in the School of Electronics Information and Communications, Huazhong University of Science and Technology, Wuhan, PR China. From August 2004 to December 2007, he worked in some universities, such as Brunel University and University of Michigan-Dearborn, respectively. He has authored or coauthored over 200 technical papers in major journals and conferences and eight books/chapters in the areas of communications and networks. He served or is serving as symposium technical program committee member of some major IEEE conferences, including INFOCOM, GLOBECOM, and ICC, etc. He is invited to serve as TPC Symposium Chair for the IEEE GLOBECOM 2013 and IEEEE WCNC 2013. He is served or serving as Associate Editor of some technical journals in communications, including in IEEE T RANSACTIONS ON S IGNAL P ROCESSING, IEEE C OMMUNICATIONS S URVEYS AND T UTORIALS , IEEE T RANSACTIONS ON V EHICULAR T ECH NOLOGY , and IEEE I NTERNET OF T HINGS J OURNAL, etc. He is a recipient of the NSFC for Distinguished Young Scholars Award in P. R. China.

Ning Ge (M’95) received the B.S. degree in 1993, and the Ph.D. in 1997, both from Tsinghua University, China. From 1998 to 2000, he worked on the development of ATM switch fabric ASIC in ADC Telecommunications, Dallas, TX, USA. Since 2000 he has been in the Department of Electronics Engineering, Tsinghua University. Currently he is a Professor and also serves as Director of Communication Institute. His current interests are in the areas of communication ASIC design, short range wireless communication, and wireless communications. Dr. Ge is a senior member of CIC and CIE. He has published more than 60 papers.