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61

Dynamic Clustering and User Association inWireless Small Cell Networks with Social

ConsiderationsMuhammad Ikram Ashraf∗, Mehdi Bennis∗‡, Walid Saad†‡, Marcos Katz∗ and Choong-Seong Hong‡

∗Centre for Wireless Communications, University of Oulu, Finland,email: ikram.ashraf,mehdi.bennis,marcos.katz@oulu.fi

†Wireless@VT, Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, USA, email: walids@vt.edu‡Department of Computer Engineering, Kyung Hee University,South Korea, email: cshong@khu.ac.kr

Abstract—In this paper, a novel social network-aware userassociation in wireless small cell networks with underlaiddevice-to-device (D2D) communication is investigated. The proposedapproach exploits social strategic relationships betweenuserequipments (UEs) and their physical proximity to optimize theoverall network performance. This problem is formulated as amatching game between UEs and their serving nodes (SNs) inwhich, an SN can be a small cell base station (SCBS) or animportant UE with D2D capabilities. The problem is cast as amany-to-one matching game in which UEs and SNs rank oneanother using preference relations that capture both the wirelessaspects (i.e., received signal strength, traffic load, etc.) and users’social ties (e.g., UE proximity and social distance). Due tothecombinatorial nature of the network-wide UE-SN matching, theproblem is decomposed into a dynamic clustering problem inwhich SCBSs are grouped into disjoint clusters based on mutualinterference. Subsequently, an UE-SN matching game is carriedout per cluster. The game under consideration is shown to belongto a class of matching games withexternalities arising frominterference and peer effects due to users social distance,enablingUEs and SNs to interact with one another until reaching a stablematching. Simulation results show that the proposed social-awareuser association approach yields significant performance gains,reaching up to 26%, 24%, and 31% for 5-th, 50-th and 95-thpercentiles for UE throughputs, respectively, as comparedto theclassical social-unaware baseline.Keywords. Small cell network, matching theory, offloading,D2D, user association.

I. I NTRODUCTION

The proliferation of bandwidth intensive wireless applica-tions such as multimedia streaming and online social net-working (OSN) has led to a tremendous increase in wire-less spectral resources [1]. This increasing need for wirelesscapacity mandates novel cellular architectures for deliveringhigh quality-of-service (QoS) in a cost-effective manner.Inthis respect, small cell networks (SCNs), built on the premiseof deploying inexpensive, low-power small cell base stations(SCBSs) are seen as a key technique to boost wireless capacityand offloading traffic. Reaping the benefits of SCNs requiresovercoming a number of challenges that include user associ-ation, traffic offloading, resource management, among others[1]–[4]. Along with the rapid proliferation of SCNs, cellularsystems are moving from a base station to a user-centricarchitecture driven by the surge of user specific applications

This research was supported by TEKES grant 2364/31/2014 andtheAcademy of Finland (CARMA) and the U.S. National Science Foundationunder Grants CNS-1513697 and CNS-1460316.

[5]. It is anticipated that a large number of devices withvarying QoS requirements will interact within small coveragefootprints [6]. Hence, in conjunction with SCNs, device-to-device (D2D) communication over cellular bands has emergedas a promising technique to further improve the performanceof SCNs, in which D2D devices communicate directly bypass-ing the infrastructure yielding increased network capacity, ex-tended coverage, enhanced data offload and improved energyefficiency [6]–[11]. The 3GPP LTE Release 12 has dealt withD2D communication in order to address the ever-increasingdemands for data traffic.

The benefits of D2D communication are accompanied witha number of technical challenges that include proximityservice discovery (ProSe), resource allocation, and intercellinterference coordination between cellular and D2D links [7]–[9]. In particular, one key challenge in D2D-enabled SCN isthat of associating user equipments (UEs) to their preferredserving node (SN) that can be either a SCBS or other D2Dusers. In [10], the authors present a protocol for resourceallocation and selection of potential D2D SNs to improvethe sum rate of D2D links. In [11], an optimization problemis formulated enabling D2D links to improve their resourceutilization and aggregate link capacity. Most of the existingworks on SCNs and D2D enabled user association are focusedon conventional physical layer metrics to optimize the networkperformance [1]–[4], [7], [8], [11]. To this end, one promisingapproach for addressing the user association problem is toincorporate additional contextual information such as user’ssocial ties, network connectivity and other features to furtherboost the network performance. For example, in a footballstadium, a group of neighboring friends may like to share thestatistics of a player. Coupled with their physical proximity,the social networking relationships between these users canprovide an indication on their common interests to sharethe same content. In a conventional setting, a SCBS oftenends up serving different users with the same content usingmultiple duplicate transmissions which leads to a waste ofresources and degrades the overall QoS. Social network-aware user association, as presented in this work, is a newparadigm to boost the performance of SCNs by exploitingD2D communications.

However, incorporating different contextual informationinconjunction with conventional physical layer metrics enablesbetter resource utilization and enhanced traffic offloading

2

[12]–[14]. In [12], the authors presented a radio resourcemanagement technique which incorporates multiple contextinformation (spectrum bands, QoS, location) within the SCNswhich leads to better spectrum usage. A network utilitymaximization problem is solved by exploring contextual in-formation at UEs such as application’s foreground/backgroudstate in order to improve QoS [13]. A scheduling algorithmis developed for cellular wireless networks, which utilizesthe information captured from users’s environment (packetflow and delay requirement) to examine the throughput-delaytradeoff [14]. In this respect, the authors in [15] propose aself-organizing cluster-based load balancing scheme for trafficoffloading while, the authors in [16] propose a decentralizedcoordination mechanism with a focus on cell edge users basedon system level simulations. However, while interesting, theseworks are limited to conventional wireless systems relyingona central controller which can cause significant informationexchange, and thus will not be appropriate for dense SCNs.This motivates the need for decentralized and self-organizingresource management solutions.

The main contribution of this work is to propose a novel,dynamic clustering and social-aware user association mecha-nism in D2D-enabled SCNs. Unlike previous works [15], [17],[18], we propose a clustering approach that incorporates bothlocation and traffic load of SCBS and specifically, incorpo-rates conventional channel information and social interactionbetween users to optimize user association in D2D enabledSCNs. In order to exploit the social-ties among nodes, weutilize the notion ofsocial distanceto identify sets of sociallyimportant nodes acting as the best SNs for other UEs withinproximity range. In the proposed model, the decision of UEson whether to use a cellular or D2D link takes into account thesocial importance of the node in conjunction with the trafficload, channel conditions and interference. We formulate theproblem as a many-to-one matching game per cluster withexternalities in which the serving nodes (i.e., SCBS and/orimportant UE) and UEs are the players, which rank oneanother based on set of preferences seeking suitable and stableassociation. The use of coalition formation games for D2D sce-narios, as studied in the literature, typically seeks to maximizeresource utilization and enhance network performance suchasin [19] and [20]. In [19], the authors presented spectrum shar-ing problem as a Bayesian non-transferable utility overlappingcoalition formation (BOCF) game between a set of device-to-device (D2D) links and multiple co-located networks. In[20], the authors studied D2D coalition formation among UEsin a single cell for video sharing scenario with peak signal-to-noise ratio (PSNR) as the measurement for video quality.Unlike [19] and [20], here, we consider an equally loadedmulti-cell system such that spectrum is shared in the first timeslot, whereas full spectrum is utilized in the subsequent timeslot for D2D-transmissions. Furthermore, physically sharedlinks with interference and social interactions among UEs areincorporated into the proposed matching game.

Many works have been presented in the literature to solvenumerous matching markets in microeconomics such as [21]–[23]. Unlike previous works [21]–[26], the strategy of eachplayer in the proposed matching game is affected by the

decisions of its peers. In this regard, the works in [27] and[28] deal with network externalities, however, they considerdifferent types of optimization/game-theoretic problemsanddo not focus on matching games with externalities as studiedhere. In particular, we show that the proposed game belongsto a class of matching games with externalities (i.e., negativeexternalities) arising from interference and peer effects(i.e.,social interaction) between nodes, which distinctly differs fromthe prior works presented in [24]–[28]. To solve this game,we propose a distributed algorithm that allows UEs and SNsto self-organize and to maximize their own utilities withintheir respective clusters. In addition, the proposed algorithmis shown to converge to a stable matching in which no playerhas an incentive to match to other player, even in the presenceof externalities. We use the concept oftwo-sided pairwisematchingto prove stability, in which UEs and SNs can swaptheir association preference in order to maximize their utility.Simulation results validate the effectiveness of the proposedapproach, and show significant performance gains comparedto the baseline social-unaware user association approaches.

The rest of this paper is organized as follows. In SectionII, we present the system model followed by the wirelessand social network models. The social network-aware userassociation problem is formulated in Section III. The dynamicclustering and intra-cluster coordination is detailed in SectionIV. In Section V, we study the UE-SN association as amatching game with externalities and discussed its proper-ties. Simulation results are presented in Section VI. Finally,conclusions are drawn in Section VII.

II. SYSTEM MODEL

A. Wireless Network Model

Consider thedownlinktransmission of a macrocell networkunderlaid byN SCBSs. We assume that all SCBSs transmiton the same frequency spectrum (i.e., co-channel deployment)with bandwidthB. Let N = 1, . . . , N, M = 1, . . . ,M,and I = 1, . . . , I, I ⊂ M, I 6= M, be the setsof SCBSs, UEs, and important UEs, respectively. Here, animportant UE is defined as a socially well connected nodewithin a confined coverage area serving as anchor node1 forD2D communication. We letP = 1, . . . , P be the setof SNs, which can be either SCBSs or important UEs, i.e.,P = N ∪ I. We let Ln be the set of UEs serviced bySCBSn andMi be the set of UEs serviced by the importantnode i ∈ I. Let Mu be the set ofMu non-serving UEssuch that,M = Mu ∪ I. All the symbols which are usedin the rest of the paper are summarized in Table I. Theconsidered network model is shown in Fig. 1. We assumeonly slowly-varying channel state information (CSI) at theSCBS [29]. Moreover, in our model users are not capable oftransmitting and receiving simultaneously, so half duplexUEsare considered. In the first time slotτ0, SCBS n transmitsto UE m ∈ M \ Mi, ∀i ∈ I while in the next time slotτ1,important UEi decodes and forwards its received signal to UEm ∈ Mi. Thus, the achievable rate between SCBSn ∈ N andUE m ∈ M \Mi, ∀i ∈ I at time slotτ0 is given by:

1The terms Important UE and anchor node are used interchangeably.

3

TABLE ISUMMARY OF IMPORTANT SYMBOLS

Symbol DescriptionN Set of SCBSs in the networkM Set of UEs in the networkI Set of important UEsMi Set of UEs serviced by important UEiP Set of SNsC Set of ClustersB BandwidthLn Set of UEs serviced by SCBSnRn,m Achievable rate between SCBSn and UEm

Rn,m Achievable rate between SCBSn and D2D UEm ∈Mi

pn Transmit power of SCBSnhn,m Channel gain between SCBSn and UEmS Similarity matrixA Edge betweenness centrality matrix for UEsX Weighted cost matrix for UEsW Social distance matrix for UEswm,m Social distance between UEsm andmUp,m Utility of UE m with respect to SNpρn Total load of SCBSn ∈ NUp Utility of SN pΓ(ηc) Social welfare of clusterc ∈ C for given matchingηcsdn1,n2

Gaussian distance similarity between SCBSn1, n2 ∈ Nsln1,n2

Gaussian load dissimilarity between SCBSn1, n2 ∈ ND Gaussian distance similarity matrixL Gaussian load dissimilarity matrixY Gaussian affinity matrixH Degree matrixΩ Tunable parameter control the impact of distance and

load on similarities≻ preference relation

Rn,m =τ0T

·B

|Ln|· log2

(1 +

pnhn,m

N0 +∑

n′∈N\n pn′hn′,m

),

(1)whereT is the time duration for a frame such thatT = τ0+τ1,pn is the transmission power of SCBSn, hn,m is the channelgain from SCBSn to UEm, respectively, whileN0 is the noisespectral density. The interference term in the denominatorrepresents the aggregate interference at UEm caused by thetransmissions of other SCBSsn′ ∈ N \ n. We assume thatthe important UE sends (the same) content to all D2D UEswithin the cell. Therefore, the rate between important UEi ∈ Iand UEm ∈ Mi at time slotτ1 is:

Ri,m = min∀m∈Mi

[τ1T·B·log2

(1+

pihi,m

N0 +∑

i′∈I\i pi′hi′,m

)],

(2)where pi is the transmission power of important UEi andhi,m is the channel gain from the important UEi to a givenUE m, respectively. The interference term in the denominatorrepresents the aggregate interference at UEm caused bythe transmissions of other important UEsi′ ∈ I \ i. Theachievable rate between SCBSn and D2D UEm ∈ Mi overT = τ0 + τ1 is:

Rn,m = min(Rn,i, Ri,m). (3)

Our objective is to propose an efficient and self-organizinguser association scheme for D2D-enabled SCNs. In conven-tional SCNs, each UE is associated to a SCBS based on themaximum signal-to-interference-plus-noise (SINR) or highestreceived signal strength indicator (RSSI) [1] ignoring UEs’contextual information such as proximity services, network

Important UE

SCBS

UE

m1

n1

Rn ,1 m1

i1

Rn ,1 i1

hn ,1 m1

hn ,2 m1

hn ,2 i1

m2

m3

i2

h i ,1 m2

h i ,1 m3

h i ,2 m2

h i ,2 m3

h i ,2 i1

Cellular transmission

D2D transmission

D2D Interference

Cellular Interference

n2

R i ,1 m3

~

~R i ,1 m2

hn ,1 i1

T 2T --

T : number of frames

Transmission:

Transmission:

0τ 1τ0τ 1τ

0τ 1τ

0τ 1τ

0τ 1τ 0τ 1τ

Fig. 1. Network model for D2D-enabled SCNs.

and social ties. This motivates for investigating novel social-aware user association mechanism in D2D enabled SCNs.

B. Social Network Model for UEs

For a more efficient user association, we define the notionof social tie which characterizes the strength of the socialrelationship between two nodes. Here, we assume that aD2D link between two nodes is formed if they are sociallyconnected and they are within proximity range. In order toestablish D2D links, some UEs can be selected as importantnodes to serve other UEs within proximity. For instance, inthe context of content sharing leveraging users social tiesallows the SCBS to avoid sending multiple copies of thesame content. Instead, by selecting socially important nodesas caching points, UEs communicate via D2D links withinthe same social network, thereby offloading the base station.In particular, the social network can be represented by aweighted graph whose vertices represent nodes and edgesrepresent their relationships strength based on parameters suchas friendship or common interests. We use the concept ofsocial distanceto measure the strength of a link between twonodes. LetGs = (M, E , w) be the social graph, whereM isthe set of UEs, andE is the set of edges. The social distancewm,m is the weight of the edgee ∈ E between UEsm andm, and adjacent UEs(m, m) are connected via an edgee.Moreover, the social distance matrixW is symmetric suchthatwm,m = wm,m, ∀m, m.

1) Important UE: An important UE is a UE that is sociallypopular or well-connected as compared to other UEs in thenetwork. Popularity or centrality in social network graphsquantify the importance of a peak in such graphs, or popularityof a node in social networks. Evidently, a node with high pop-ularity has high probability of having a link to other networknodes. Hence, the social importance can be characterized byhaving curtail points for data distribution in the network,sinceit has social ties/links with other nodes in the network. Thethree most popular ways to quantify the social popularity ofnodes in a social graph are degree, closeness, and betweennesscentrality [30], [31]. In this work, thesocial importanceof aUE is defined as a mixture of edge betweenness centrality,similarity, and physical distance to other peer nodes. Theconcept ofsocial distancebetween nodes is based on edge

4

betweenness and node similarity.2) Social Distance:The social distance is defined as the

social interaction parameter between communicating nodes.Important UEsin a given cell can be interpreted as the subsetof UEs with the highest social distance for data transfer. LetW be a social distance matrix where elementwm,m ∈ [0, 1]quantifies how the social distance of a user affects its util-ity which is given as the weighted sum of matricesS, A

representing respectively the similarity and edge betweennesscentrality among UEs [32]:

W = αS + βA, (4)

whereα andβ given in (4) are tunable parameters such thatα+ β = 1. The important UEselects its preferred peer basedon the composite social and physical distance captured by thefollowing weighted cost matrixX, where elementxm,m isgiven by:

xm,m = (ǫm,mwm,m)/dm,m. (5)

In (5), we combine the social distance with the actual physicaldistance between UEs, wherewm,m denote the social distance,dm,m is the Euclidean physical distance between UEm andmand ǫm,m is a normalization constant. In order to establish aD2D link between UEs, we assume that some UEs are selectedas important UEs. The social distance is used for rankingthe nodes for selecting popular (important) nodes in thesocial network. A UE is considered important if its aggregateweighted cost is larger than other UEs in the proximity ofSCBS n, such thatIn = argmax∀m∈M

∑m∈M,m 6=m xmm.

We use an adjacency matrixE to determine the existenceof a D2D link, where elementem,i = 1, if m is connectedto important UE i, otherwiseem,i = 0. Next, we brieflyreview the concepts of similarity and betweenness centralityto capture the social distance among UEs in a given socialnetworks.

3) Similarity Matrix: The similarity matrix is a measure ofcloseness between a pair of nodes. The degree of similaritycan be measured by the ratio of common neighbors betweenindividuals in a social network. The degree of similaritybetween UEsm and m has an important effect in terms ofdata dissemination. Nodes having lower degree of similarityare good candidates for data dissemination [33]. LetQ be aM ×M similarity matrix, such that a pair of nodes(m, m),depending on whether they are connected directly or indirectly,their corresponding similarity measuring elementqm,m of Qis defined as [33]:

qm,m =

∑

m∈ν(m)∩ν(m)

1

t(m), if m, m are connected,

0, otherwise,

(6)

whereν(m) is the set of neighbors ofm, m ∈ ν(m)∩ ν(m)are the common neighbors of UEsm andm, andt(m) is thedegree of UEm. To normalize the similarity matrix, we usethe simple additive weighting (SAW) method, in which thenormalized value of each elementqm,m of Q is:

sm,m = qm,m/qmaxm ∀m, m, (7)

Cellular link

D2D link

Data transmission

Graph = (V,E)s

m2

n1

m3

m4

m1

Important UE

Fig. 2. Illustrative example of the considered network deployment withone SCBS(n1), one important UE(m1) and three UEs(m2,m3,m4)represented as graphGs = (V , E).

where qmaxm = maxm qm,m. Consequently, we obtain the

normalized similarity matrixS of dimensionM ×M , wherethe mth row andmth column ofS, i.e., sm,m denotes thenormalized similarity between UEsm andm.

4) Edge Betweenness Centrality:Edge betweenness cen-trality is based on the idea that an edge becomes central to agraph if it lies between many other UEs, i.e., it is traversedby many of the shortest paths connecting a pair of UEs[31]. Edges with a high betweenness centrality are consideredimportant because they control information flow in the socialnetwork. LetA beM×M edge betweenness centrality matrix,where elementam,m is the edge betweenness centrality ofthe link between nodesm andm. The betweenness centralityam,m of an edgee [30] between UEs(m, m) is the sum ofthe fraction of all-pairs’ shortest paths that pass throughedgee. The normalizedam,m is:

am,m =

∑

m,m∈M

γ(m, m|e)

γ(m, m)

(M − 1)(M − 1), (8)

whereM is the number of UEs, the summationγ(m, m) isover the number of shortest(m, m)-paths, andγ(m, m|e) isthe number of those paths that traverse edgee. To providemore insights on this social model, we present the followingexample to compute the social distance (4) and weighted cost(5).

Example. Consider four UEs from the setM =m1,m2,m3,m4 and one SCBS,n1 ∈ N as shown in Fig. 2.Formally, the connectivity of UEm ∈ M can be representedby an adjacency matrixE, which is aM × M symmetricmatrix, whereM is the number of UEs in the social graphGs. The adjacency matrix is expressed as:

Em,m =

1, if there is a edge between UEsm andm,

0, otherwise.

The similarityS and edge betweenness centralityA betweenUEs computed using (6) and (8), respectively.

S =

m1 0 0.5833 0.583 0.2500m2 0.583 0 0.500 0.500m3 0.583 0.500 0 0.500m4 0.250 0.500 0.500 0

5

A =

m1 0 0.0750 0.0750 0.100m2 0.0750 0 0.0500 0m3 0.0750 0.0500 0 0m4 0.1000 0 0 0

W =

m1 0 0.3292 0.3292 0.1750m2 0.3292 0 0.2750 0.2500m3 0.3292 0.2750 0 0.2500m4 0.1750 0.2500 0.2500 0

X =

m1 0 0.0432 0.0411 0.01240m2 0.0432 0 0.0320 0.0117m3 0.0411 0.0320 0 0.0121m4 0.0124 0.0117 0.0121 0

To determine which UE is the most important UE with respectto SCBSn1, we use (4) withα = 0.5 and β = 0.5. Bylooking at the social distance matrixW and calculating therespective weighted cost matrix (5), it is clear that UEm1 ismore socially important than other UEs while,m4 is the leastimportant one.

III. PROBLEM FORMULATION

As previously mentioned, classical approaches for userassociation in SCNs, are typically based on physical layermetrics and assume a central controller which gathers allnetwork information and decisions [1]. In this section, westudy the problem of base station clustering and flexible userassociation by incorporating users’ social-ties in the network.Then, we will use the framework of matching theory [34], todevelop a distributed and self-organizing solution composed oftwo steps: 1) we cluster SCBSs in terms of mutual interferencedescribed in detail in Section IV, 2) we study a two-sidedmatching model that enables each cluster to efficiently opti-mize user association by incorporating both physical and socialaspects. Therefore, we define a two-sided matching game inwhich UE and SN acts as players. In this game each playertries to match (associate) to the most suitable serving nodebased on its own preferenceη : M → P . Next, we define thesocial-aware utility functions which capture both wireless andsocial network metrics in order to optimize the user associationmechanism.

A. UE and SN Utilities

The utility of a given UE is defined as the achievable ratetaking into account the interference from adjacent SCBSs andimportant UEs. An arbitrary UEm can either connect to aSCBS n via a cellular connection or animportant UE i ∈I, via a D2D link. The achievable rate between SNp ∈ P(important UE or SCBS) and UEm ∈ M for a given matchingη is:

Up,m(Rp,m, wp,m, η)

=

Rp,m +∑

m∈M\m

Rm,m

1− wm,mem,m, (p = n andm = i),

Rp,m, if m connected to SNp (p = n),

Rp,m, D2D UE m as per (3),(9)

wherewp,m represents the social distance between SNp andUE m in the social graphGs defined in (5). The elementem,m ∈ 0, 1, shows the existence of a D2D-link between UEm and UEm. Moreover, (9) defines the utility of a UE whenit acts as an important UEm = i serviced by SCBSp = n andforwards data to other UEs within a given social network ofUEs. Therefore, in order to capture the social impact betweena pair of UEsm and m that have social ties between themin the social network, we formalize the strength of the socialtie (social distance) aswm,m = [0, 1), with a higher valueof wm,m being a stronger social tie. It follows that the socialutility of an important UE consists of its achievable rate anda weighted sum of the achievable rates of other UEs havingsocial tie with it [35]. This will induce socially well connectedUEs to associate to one another.

To calculate the utility of SNp ∈ P , we incorporate thesocial distance of each UEm with respect to SNp [36]. Theutility of an SN p is the sum of utilities of its associated UEsm ∈ Lp, for a matchingη given by:

Up(η) =∑

m∈Lp

Up,m(Rp,m, wp,m). (10)

B. Social Welfare

We use the social welfare to define the network wideperformance expressed as the sum of the utilities of UEs andSNs [34].

Γ(η) =∑

p∈P

∑

m∈Lp

Up,m(Rp,m, wp,m, η), (11)

whereM and P are the set of UEs and set of SNs in thenetwork, respectively. The objective is to maximize the totalnetwork wide social welfare given in (11). Unfortunately,maximizing the network-wide social welfare in a central-ized manner requires large information exchange between allSCBSs and UEs in the network, calling for a distributedsolution with minimum coordination. To address this issue,wegroup mutually-interfering SCBSs into a number of clusterssuch that SCBSs within a cluster coordinate locally amongeach other. Specifically, we consider that SCBSs are groupedinto a set of well-chosen clustersC = C1, C2, ..., C|C|.Let ηc represents the user association (matching), such thatηc(m, p) represents the matching of UEm and SNp withincluster c ∈ C. Each clusterc consists of locally-coupledSCBSs in terms of mutual interference in whichNc denotesthe number of SCBSs belonging to clusterc ∈ C. It is assumedthat, in clusterc, SCBSs efficiently offload traffic among eachother while satisfying UEs’ QoS. Moreover, the matching foreach cluster is represented by a vectorη = [η1, η2, ..., η|C|].Hereinafter, we refer toη as the “network wide matching”,which captures the utilities of all the UEs and SNs in thenetwork whereas the per cluster matching is denoted byηc.Finally, we define the social welfare per clusterc, Γc(ηc) givenmatchingηc by:

Γc(ηc) =∑

p∈Pc

∑

m∈Mc

Up,m(Rp,m, wp,m, ηc), (12)

whereMc is the set of the UEs,Nc the set of SCBSs, andPc the set of SNs belonging to clusterc ∈ C. The objective is

6

Similarity and edge

betweenness centrality

as per (6-8)

Social distance, Selection

of Important UEs (4-5)

Social network model

Utility of UE (9), Utility of SN (10), Social

Welfare (12), Cell Load (15)

Clustering between SCBSs based on

Distance and load similarities (16-18)

(Algorithm 1)

Sta

ble

ma

tchin

g

Wireless network parameters as per (1-3)

Proposed Social network-aware user matching

(Algorithm 2)

Fig. 3. An Illustration of the different steps of the proposed solution.

to maximize the social welfare for all clusters, which is givenby the following optimization problem:

maximizeη,C

∑

∀c∈C

Γc(η) (13a)

subject to |Nc| ≥ 1, ∀c ∈ C, (13b)⋃

∀c∈C

Nc = N , Nc ∩ Nc′ = ∅,

∀c, c′ ∈ C, c 6= c′, (13c)∑

∀p∈Pc

ηc(m, p) = 1, ∀m ∈ Mc, (13d)

where constraints (13b) and (13c) imply that any SCBS is partof one cluster only. The constraint given in (13d) depicts thata given UEm can be matched to only one SNp whereas, SNp can be matched to one or more UEs for a given matchingηc.Solving (13), requires global network information, which canbe complex and not practical. Therefore, in the subsequentsection, we propose a distributed solution composed of: 1)dynamic SCBS clustering, 2) flexible user association basedonintra-cluster coordination. The different steps of our proposedsolution are summarized in Fig. 3.

IV. DYNAMIC CLUSTERING

The centralized optimization problem in (13) is difficultto solve and is combinatorial in nature. Developing a de-centralized approach based on minimal coordination betweenneighboring SCBSs is needed. First, we propose a cluster-based mechanism which incorporates, both location of SCBSand their traffic load. Clustering enables coordination amongwell selected pairs of SCBSs. We propose a dynamic clus-tering approach, in which the cluster size varies dynamicallydepending on the dynamic nature of traffic (e.g., load, in-terference). Subsequently, we propose a distributed and self-organized matching algorithm to dynamically optimize theuser association per cluster. The procedure only depends onthe local information available at the cluster level. The set ofSCBSs are partitioned into|C| non-overlapping clusters, suchthat:

⋃

∀c∈C

Nc = N andNc ∩ Nc′ = ∅, ∀ c 6= c′. (14)

Let Gc = (N ,F) be the undirected connected graph, whereN is the set of SCBSsN and F ⊂ N × N is the set oflinks between locally-coupled SCBSs. In order to calculatethe cell load, let us denoteηn as a UE random association2

2Equivalently, the UE can be initially associated to the closest SCBS.

to an SCBSn and0 ≤ ρn(ηn) ≤ 1 as the normalized load ofSCBSn ∈ N , given by:

ρn(ηn) ,∑

∀m∈Ln\Mi

Rn,m

Rmaxn,m

, ∀i ∈ I, (15)

where Rmaxn,m is calculated neglecting the interference from

other SCBSs. The average load of each clusterρc for a givenmatching ηc is the arithmetic average load of its memberSCBSs, such thatρc(ηc) = 1

|Nc|

∑∀n∈Nc

ρn(ηn). The cluster-ing mechanism between SCBSs and intra-cluster coordinationare demonstrated in Fig. 4.

A. Similarity-based SCBS Clustering

In order to minimize the signalling overhead, we groupSCBSs based on similar attributes. There are numerous aspectsthat impact interference between SCBSs. Two key factors aretheir physical distance separation and traffic load condition.Having said that, we utilize location and traffic load sim-ilarities to group SCBSs and we use a spectral clusteringalgorithm [37] to identify similarities between SCBSs to formclusters. Next, we calculate the Gaussian affinity matrix [38]representing the similarities between SCBSs based on theirgeographical locations and loads.

Let vn1 andvn2 be the geographical coordinates of SCBSn1 and n2, respectively, in the Euclidean space. Here, wedefine parameterΥd to represent the presence of a link oredge f ∈ F between neighboring SCBSn1 and n2 suchthat fvn1 ,vn2

= fvn2 ,vn1= 1, ||vn1 − vn2 || ≤ Υd. To find

locally-coupled SCBSs in terms of distance, letD denotesthe Gaussian distance similarity matrix, and letsdn1,n2

bean element ofD representing the distance similarity amongSCBSsn1, n2 ∈ N given [37]:

sdn1,n2=

exp

(−||vn1−vn2 ||

2

2σ2d

), if ||vn1 − vn2 || ≤ Υd,

0, otherwise,(16)

where the parameterσd controls the impact of neighborhoodsize. For a givenΥd the range of the Gaussian distancesimilarity for any two connected SCBSs is[e−Υd/2σ

2d , 1],

whereas the lower bound is determined byσd. The rationalefor (16) is that, when the SCBSs are located far from eachother, the distance similarity is low. On the other hand, thedistance similarity increases as SCBSs come closer to oneanother and more likely to cooperate with each other.

Unlike the static distance based clustering in (16), thetraffic load of SCBSs varies over time thus, the load basedclustering provides a more dynamic manner of groupingneighboring SCBSs. Therefore, we are interested in clusteringSCBSs which have load dissimilarities. letsln1,n2

be an entryof the Gaussian load dissimilarity matrixL between SCBSn1, n2 ∈ N with respect to cell loadρn1 and ρn2 , which isgiven [37]:

sln1,n2= exp

(||ρn1 − ρn2 ||

2

2σ2l

), (17)

where the parameterσl controls the impact of load on thesimilarity. The range of load dissimilarity is[1, e1/2σ

2l ]. The

7

cCluster

n1

Important UE

Cellular Interference

D2D transmission

D2D Interference

Cellular transmission

Without Cluster SCBSs

Cluster

n1 n2

Clustered SCBSs

(a) (b)

ϵC

c ϵC

n2

cSCBS Cluster ϵC

,( )ρ2nv

2n

,( )ρ1nv

1n

,( )ρ3nv

3n

Cluster

Graph F, )c

G

ϒd

=(

n1

n3

n2

Fig. 4. (a) Graph representationGc = (N ,F) of clustering among SCBSs, (b) Intra-cell and D2D interference for uncoordinated SCBSs, cluster orcluster-based SCBSs and D2D users interfere with each otherwithin a cluster.

Algorithm 1: Spectral clustering for clustering SCBSs[37]

Input : N ,Y = yn1,n2 ,Gc the graph of SCBS,kmin, kmax

1 Compute diagonal degree matrixH with diagonal(d1, ..., dnv) wheredi =

∑nvj=1 yni,nj .

2 Z := H − Y

3 Znorm := H−1/2ZH−1/2.4 Let λ1 ≤ .. ≤ kmax be the smallest eigenvalues ofZnorm. Setk = argmaxi=kmin,...,kmax−1 ∆i where∆i = λi+1 − λi.

5 find thek smallest eigenvectorse1, ..., ek of Znorm.6 Let E be annv × k matrix with ei as columns.7 Use k-means clustering to cluster the rows of matrixE.8 Cluster set1, ...,C|C|.

upper bound of the dissimilarity is based on the choice ofσl. We use a spectral clustering algorithm (Algorithm 1) toform clusters between SCBSs based on their Gaussian affinitymatrix. The Gaussian affinity matrix effectively captures thedistance and load similarities. The Gaussian affinity matrix Y

whose elementyn1,n2 represents joint similarity between twoSCBSsn1, n2 ∈ N based on the distance and load is:

Y = DΩ · L1−Ω, (18)

where 0 ≤ Ω ≤ 1 controls the impact of distance andload similarities on the joint similarity. Here, the cooperationbetween SCBSs is only possible if a physical link betweenthem exists i.e,∀Ω ∈ [0, 1], fVn1,vn2

= 0 =⇒ yn1,n2 = 0.In our model, when the UEs are first admitted in the system,they will be associated to the SCBS based on the max-RSSIcriterion. The SCBS to which UEs associate is then referredasanchor SCBSi.e., UEm associates with anchor SCBSn ifand only if RSSIn,m ≥ RSSIn′,m for all n′ ∈ N . Moreover,we would like to stress the fact that the goal of clustering isto enable coordination among well selected pairs of SCBSswithin the same cluster. Once the clusters are formed amongSCBSs, UEm ∈ Mc will always be a part of the sameclusterc, which corresponds to its anchor SCBSn ∈ Nc. ForUEs at the edge of multiple clusters, they will simply remainassociated to the SCBS cluster that contains their originalanchor SCBS to which they associated based on the RSSIcriterion. Let n(m) be the anchor SCBS of UEm and Ln

be the set of UEs served by SCBSn. Irrespective of the factthat whether UEm is at the cluster edge or not, the followingconditions will be always satisfied.

(i) n(m) ∈ c ⇐⇒ m ∈ Mc such thatc = Nc ∪Mc,(ii) m ∈ Ln, n ∈ Nc ⇒ n(m) ∈ c ∈ C.

The above conditions imply that, a given UEm will beassociated to an SCBSn based on the max-RSSI if and onlyif m andn belong to the clusterc. Furthermore, it implies thatthe set of UEs that are serviced by SCBSn also belong to thesame clusterc. It is worth to mention that after clusteringis performed, UEs can be served by any SN (i.e., SCBS,important UE) belonging to the same clusterc based on theproposed association within clusterc.

V. PER-CLUSTER SOCIAL NETWORK-AWARE USER

ASSOCIATION AS AMATCHING GAME WITH

EXTERNALITIES

Our objective is to develop a self-organizing mechanismfor solving (13). In order to overcome the combinatorialnature of the user association problem, we make use ofthe framework of matching theory in which, the social andwireless characteristics are incorporated into the matchinggame. Such wireless and social effects motivate the need foradvanced model for matching theory that take into accountthe wireless interference and strength of social ties. Thus, wepropose a social network-aware matching game per clusterc ∈ C capturing both physical and social aspects of the networkin which each UEm ∈ Mc is associated to the best servingnodep ∈ Pc via a matchingηc : Mc → Pc.

Definition 1. A matching gameis defined by two sets ofplayers (Mc,Pc) and two preference relations≻m, ≻p foreach UEm ∈ Mc to build his preference over SNp ∈ Pc

and vice-versa in a clusterc. The outcome of the matchinggame is the association mappingηc that matches each playerm ∈ M to playerp = ηc(m) p ∈ Pc and vice versa such thatm = ηc(p) ,m ∈ Mc.

A preference relation≻ is defined as a reflexive, completeand transitive binary relation between players inMc andPc.Thus, a preference relation≻m is defined for every UEm ∈Mc over the set of SNsPc such that for any two nodes inp, p ∈ P2

c , p 6= p and two matchingsηc, η′c ∈ Mc × Pc, ηc 6=

8

η′c , p = ηc(m) , p = η′c(m):

(p, ηc,η−c) ≻m (p, η′c,η−c) ⇔

Up,m(Rp,m, wp,m, ηc,η−c) > Up,m(Rp,m, wp,m, η′c,η−c),(19)

where(p,m) ∈ ηc and (p,m) ∈ η′c. Similarly the preferencerelation≻p for SN p over the set of UEsMc is defined suchthat for any two UEsm, m ∈ Mc,m 6= m ,m = ηc(p) , m =η′c(p):

(m, ηc,η−c) ≻p (m, η′c,η−c) ⇔ Up(ηc) > Up(η′c). (20)

Hereinafter, for notational simplicity we defineUp,m(ηc) :,Up,m(Rp,m, wp,m, ηc,η−c).

Remark 1. The proposed social network-aware matchinggame has externalities and peer effects.

Each SN and UE independently rank one another based on therespective utilities in (9) and (10) that capture the interferenceand social ties among nearby UEs. However, the selectionpreferences of UEs areinterdependentand influenced by theexisting network wide matching, which leads to a many-to-one matching game. Such effects which dynamically changethe preference of each player in the network, are calledexternalities [23]. In particular, the considered game is amatching game withexternalitiesdue to mutual interferenceand social ties between nodes, which differs from classicalapplications of matching theory in wireless such as those in[24]–[26]. Thus, each playerm ∈ Mc has a preference overplayers inp ∈ Pc and vice versa and these preferences changeas the game evolves. Finally, each UE is matched (associated)to one SN, while SNs can be matched to multiple UEs whichmakes the matching game as many-to-one.

In classical matching games with no peer effects, each UEhas a strict preference over SNs and vice versa that remainunchanged for the overall game. The key premise of our workis that peer effects are often the result of an underlayingsocial network. For our work we assume that peer effects iscaptured at the important UEs due to social ties with otherUEs in the proximity as per (9). The strength of social ties(i.e, social distance) among players may change if UEs aresocially connected to other UEs within their proximity rangeand thus, impact on the preference at the UEs and importantnode (SNs). To deal with externalities and peer effects due tothe interference and social ties, the most important notionisthe stability of the solution. To solve the problem in (13), eachUE and SN defines its preference over each other using (19)and (20). The objective of each player is to maximize its ownutility, by associating to its most preferred SN.

A. Proposed Social Network-Aware User Association Algo-rithm

In order to solve the proposed matching game, usuallydeferred acceptance algorithm guarantees a stable solutionin one-to-one matching [24], [26], [39]. Nevertheless, suchapproaches do not account for externalities and peer effects,and, thus, they may yield lower utilities or may not con-verge. In fact, due to externalities and peer effects, players

Algorithm 2: Proposed Social Network-aware User Asso-ciation Algorithm

Data: Each UEm is initially associated to a randomly selected SCBSn.

Result: Convergence to a stable matchingη.Phase I - Social distance computation;

• UEs and SNs exchange social-aware information and computeS andA using (6) and (8);

• Calculation of important UEs listI based on the social distanceWusing (4) and (5);

• Node with highest rank in the sorted listIn is selected as theimportant nodei ∈ I

while t ≤ tmax doPhase II - Clustering among SCBSs;

• Compute gaussian distance and load similarity metrics in (16), (17);• Gaussian similarity matrix computed using (18);• Clusters|C| are formed among SCBSs using Algorithm 1;

Phase III - SN discovery and utility computation;• Each UEm discovers a SNp in the cluster vicinityc ∈ C;• Up,m(Rp,m, wp,m) using (9),Up using (10), and social welfare

Γc(ηc) using (12) for clusterc are updated;Phase IV - Swap-matching evaluation; whilec ≤ max(|C|) do

• Pick a random pair of UEsm, m ∈ Mc within the clusterc;while count ≤ countmax do

• Up,m(Rp,m, wp,m), Up are updated based on the currentmatchingηc;

• UEs and SNs are sorted byΓc(ηc);• swap the pair of UEsηc ⇒ η↔c• Γc(ηc,η−c) = Γbest

c (ηc,η−c)

PT = 1

1+e−ϑ(Γc(ηc,η−c)−Γc(η↔

c ,η−c))

;

(ηc,η−c)← (η′c,η−c) change the configuration withprobability PT ;if Γc(η↔c ,η−c) > Γbest

c (ηc,η−c) thenΓbestc (ηc,η−c) = Γc(η↔c ,η−c)

elseSN p refuses the proposal, and UEm sends aproposal to the next configuration atcount

count = count+ 1c = c+ 1

t = t+ 1

Phase V - Stable matching

continuously change their preference orders, in response tothe formation of other UE-SN links which renders classicaldeferred acceptance solutions such as in [24], [26], [39] notapplicable for our model. Therefore, to seek a stable userassociation an Algorithm 2 is proposed which is based onthe concept of Markov Chain Monte Carlo (MCMC) [34]. Inthis approach, instead of using the greedy way of selectingthe “best” matching, the matching is chosen based on aprobability, which depends on the swap resulting in an increaseof the social welfare for a given clusterc.

In the proposed algorithm, an important UEi and set of UEsserviced by the important UEMi seek the same content. Thepreferences of both the UEs and SNs are done locally within agiven clusterc, whereas the coordination is required betweenadjacent SCBSs. If UEm ∈ Mc is not currently served byits most preferred SNp ∈ Pc, it sends a matching proposalto another SNp. Upon receiving a proposal, SNp updates itsutility and accepts the request of the UE if the externalitiesand peer effects resulting from such swap do not yield adegradation of the social welfare of the cluster. The main goalof each UE is to maximize its own utility while associatingwith the most preferred SN or important UE. Initially, each

9

UE is associated to a randomly selected SN based on the max-RSSI criterion. In the first phase, social distance matrix iscalculated using (4) and then each SCBS compute the list ofimportant UEsIn using (5). In the next phase, clusters areformed among SCBSs based on their gaussian distance andload similarity using Algorithm 1. Then, the utilities of allplayers and social welfare of a given cluster is calculated forthe current matchingηc. In the fourth phase, UEs and SNsupdate their respective utilities and individual preferences overone another. Subsequently, at each iteration, a chosen UE pairis swapped with a probability that depends on the change inthe cluster’s social welfare: a positive change in the socialwelfare of a cluster yields a probability of swapping largerthen 1/2 and vice-versa. As a result, the algorithm does notget caught in a local optimum and the algorithm continuouslykeeps track of the “best” matchings. Algorithm 2 terminateswhen no further improvement can be achieved. After phaseIV, the algorithm converges to a local maximum of the socialwelfare for a given cluster. The Algorithm 1 continue until itreaches to stable matching.

B. Convergence and Stability

The concept of peer effect and externalities requires us toadopt a new stability concept based on the idea of “pairwisestability” [34]. Before defining the pairwise stability, wefirstdefine a swap matching.

Definition 2. A swap matching is formally defined asηm↔mc = ηc \ (p,m), (p, m) ∪ (m, p), (m, p). In each

swap two UEs change their matching with their respectiveSNs while other matchings remain fixed. Having defined swapmatching, we further define pairwise stability.

Definition 3. Given a matchingηc, a pair of UEsm, m andSNs p, p within a clusterc, a pairwise matching isstable ifand only if there does not exist a pair of UEs(m, m) suchthat:

(i) ∀y ∈ m, m, p, p, such that Uy,ηm↔mc (y)(ηc) ≥

Uy,ηc(y)(ηc) and(ii) ∃y ∈ m, m, p, p Uy,ηm↔m

c (y)(ηc) > Uy,ηc(y)(ηc).

A matchingηc is said to be pairwise stable if there does notexist any UEm or SN p, for which SN p prefers UEmover UEm or any UEm which prefers SNp over p. FromDefinition 3, we can see that if two UEs swap between twoSNs, the SNs involved in the swap must “approve” the swap.Similarly, if two SNs want to swap between two UEs, theUEs and SNs must agree to the swap. This definition is usefulfor proving the two-sided stability of our proposed matching.Next, we will show that two-sided pairwise stable matchingwill always exist in our game. For that, we assume that neitherUEs nor SNs can remain unmatched in the cluster, and werestrict ourselves to considering swap of UEs between SNs.However, before stating this result, we require the followingLemma:

Lemma 1. Any swap matching(ηm↔mc ) for which

(i) ∀y ∈ m, m, p, p, such that Uy,ηm↔mc (y)(ηc) ≥

Uy,ηc(y)(ηc) and

(ii) ∃y ∈ m, m, p, p Uy,ηm↔mc (y)(ηc) > Uy,ηc(y)(ηc),

yields Γc(ηm↔mc ) > Γc(ηc) which can be written as

Γc(η↔c ) > Γc(η).

Proof: The symmetry of social network and swap match-ing are key factors for guaranteeing pairwise stability. More-over, the approved swap in the symmetry social network resultsin a Pareto improvements for the players involved in the swap,as clearly seen from the definition of pairwise stability. For allother players, a non-negative change in utility follows fromthe symmetry of the social-graph. We assume a UE centricmatching in which UEs have preference over the SNs and notvice versa. Thus, for proving Lemma 1, we consider only one-sided matching game rather than two-sided matching. In orderto proof Lemma 1, we start by calculating the difference in thesocial welfare for the swap matchingη↔c and given matchingηc. Without loss of generality, it is assumed that the swappingof UE m strictly increases its utility. Defineηc(m) = p, andηc(m) = p, then let us start by calculating the change in theutility of UE m which is given by:

0 < Up,m(ηc)−U(η↔c ) =∑

z∈ηc(p)

Uz,m−Um,m−∑

z∈ηc(p)

Uz,m.

(21)Similarly, for the change in utility for UEm, which is givenby:

0 ≤ Up,m(ηc)−U(η↔c ) =∑

z∈ηc(p)

Uz,m−Um,m−∑

z∈ηc(p)

Uz,m.

(22)Adding the above inequalities (21), (22) we have:

0 <∑

z∈ηc(p)

(Uz,m − Uz,m

)

+∑

z∈ηc(p)

(Uz,m − Uz,m

)− 2Um,m := δc. (23)

Consider a matchingηc and a swap matchingη↔c that satisfies(i) and (ii) of Lemma 1. The total change in the utility for allUEsMc is:

Mc:=

∑

z∈Mc

Up,z(η↔c )−

∑

z∈Mc

Up,z(ηc)

:= δc +∑

z∈ηc(p)

Uz,m − Uz,m

︸ ︷︷ ︸utility gain from m associatingp

−∑

z∈ηc(p)

Uz,m

︸ ︷︷ ︸utility loss fromm leavingp

+∑

z∈ηc(p)

Uz,m − Um,m

︸ ︷︷ ︸utility gain from m associatingp

−∑

z∈ηc(p)

Uz,m.

︸ ︷︷ ︸utility loss from m leaving p

Mc:= 2δc > 0, (24)

where (24) assumes that the social graph is symmetric. Thetotal change in utility for all SNsPc we have:

0 ≤ Up(η↔c )− Up(ηc) + Up(η

↔c )− Up(ηc) := Pc

. (25)

Without loss of generality, assume that UEm strictly improvesthe utility while another UEp either improves or is indifferent

10

to the swap. It can be shown from (25) that the SNsp andp are affected by the swap with non-negative change in theirutilities. Thus the social welfare strictly increases:

Γc(η↔c )− Γc(ηc) = Mc

+Pc> 0. (26)

Expanding on the idea presented in Lemma 1, it is easy toprove the following theorem.

Theorem 1. All local maxima of the social welfare for aclusterc given in (12) are two-sided pairwise stable.

Proof: Let Γc(ηc) be a local maximum of a givenmatchingηc. Lemma 1, shows that any swap matching whichis acceptable by all players satisfies conditions (i) and (ii), andstrictly increases the social welfare of clusterc. Neverthelessthis assumption is contradictory asηc is a local maximum forcluster c. Therefore,ηc must be stable. It is worth nothingthat, not all pairwise stable matchings are local maxima3 ofΓc(ηc).

Corollary 1. The proposed Algorithm 2 is guaranteed toconverge to a two-sided stable matching.

Proof: It can be shown from Lemma 1 and Theorem 1that the algorithm converges to a stable matching, since witheach iteration the social welfare strictly improves, and all localmaxima ofΓc are stable matchings. All swaps among playersmust be agreed upon as given in Lemma 1. Moreover, UEshave limited transmission range and can be matched with afinite number of SNs in their vicinity, therefore the possibleswaps for the players arefinite. Every UE has a finite numberof choices to swap so we have a finite set of matching fora given number of the SNs. In addition, considering all thepossible swaps each UE is associated to its most preferredSN and vice versa. Algorithm 2 terminates, when no furtherimprovement in social welfare is achieved by all possibleswaps among players. Therefore, Algorithm 2 converges to atwo-sided stable matching after a finite number of iterations.

C. Complexity Analysis

In order to compute the complexity of the proposed al-gorithm, we start with the simple case in which the match-ing game has no social ties (peer effects) and UEs havestrict preference ordering. As matching is done per cluster,we compute the complexity and message overhead for thematchingηc within one clusterc ∈ C. We assume thatΦp

is the maximum number of UEs matched to SNp ∈ Pc

and Φc is the total number of UEs matched per clusterΦc =

∑∀p∈Pc

Φp such thatΦp = Φp, p 6= p. The valueof Φp depends on the available bandwidth. LetΦs(≤ Φc)denotes the number of satisfied matched UEs, which is basedon the preference ordering. For simplicity, lets assumeΦs isconstant during all the iterations such thatΦs = Φs, ∀s 6= s.

3This case can be considered when one player rejects a swap as its utilitywould decrease, but the other player gets benefit from such a swap. If forcedswap happens, the total social welfare could increase, but only at the expenseof the first player.

To compute the complexity per cluster, we consider two worstcase scenarios, when all UEsm ∈ Mc inside a clustercare matched to a single SN: (1) when the number of UEsinside the cluster are less then the total number of matchedUEs i.e.,Mc ≤ Φc, |Mc| = Mc and, (2) when the numberof UEs is greater than the total number of matched UEs i.e.,Mc > Φc. Our goal is to analyze the maximum number ofiterations required for convergence and the maximum numberof proposals sent from UEs to SN (message overhead) forboth cases. In each iterationt, UEs send proposal to theirmost preferred SN (i.e., important node or SCBS), and the SNaccepts or rejects the received proposal based on its preferenceordering and available capacity. Therefore, the number ofmatched but unsatisfied UEs at each iteration is less or equalthan the available capacity.

For the first case, when the algorithm converges, all UEs arematched to a single SN, since, SNs prefer any UE to beingunmatched. It can be observed that the worst case scenariohappens, if all UEs have the same preference ordering. Thus,at the end of each iterationt we haveMc − Φst unsatisfiedUEs. All UEs are matched and satisfied when the maximumnumber of iterationstmax is obtained. i.e.,Mc − Φstmax = 0.Hence, the complexity is of orderO(Mc). Moreover, we haveMc−Φst proposal messages at each iterationT , and the totaloverhead for sending such messages is given by:

ξmax :=

tmax∑

t=1

(Mc − Φst+Φs) =Mc(Mc +Φs)

2Φs. (27)

For the second case whenMc > Φc, once the algorithmconverges, there areMc − Φc unallocated UEs. The worstcase happens, if all UEs have same preference ordering. Henceat tmax iteration we haveMc − Φstmax UEs unallocated. Thecomplexity of the orderO(Φc), and the messaging overheadis equal to:

ξmax :=

Φc∑

t=1

(Mc − Φst). (28)

The complexity of the proposed social network-aware al-gorithm will further depend on the social distance matrixW

and the spectral clustering. The social distance matrixW iscomputed only once whereas, the spectral clustering algorithmruns for finite number of iterations.

VI. SIMULATION RESULTS

We consider a single macro-cell in which UEs and SCBSsare uniformly distributed over the area of interest. Transmis-sions are affected by distance dependent path loss accordingto 3GPP specifications [40]. We assume that there is no powercontrol, and thus the power is uniformly divided betweenUEs. It is also assumed that the bandwidthB is dividedequally between the served UEs. For simulation, we assumethat one UE is selected as important UE per SCN. Thesimulation parameters are given in Table II. The position ofUEs is assumed to be static, distance dependent path lossmodel for D2D communication of LOS and NLOS103.8 +20.9 log10(d[km]), 145.4 + 37.5 log10(d[km]) respectively, isconsidered. Furthermore, for the social network, the selection

11

TABLE IISIMULATION PARAMETERS

Parameter ValueBandwidth (MHz) 5Area (m2) 500Noise power spectral densityN0 [40](dBm/Hz)

-174

SCBS, D2D transmission radius (m) 50, 20SCBS, UE transmission powers (dBm) 23, 15Tunable parameters,α, β 0.5Inter-site distance (m) 40τ0 such thatτ0 + τ1 = 1 0.84Impact of load similarityσl 1Impact of neighborhood size (distance)σd 100Parameter controls the impact of distance andload on similarityΩ

0.5

kmin, kmax parameters for clustering 2, ⌈(N/2) + 1⌉cluster radius (m) 200

Boltzman temperatureϑ ϑ = 1−(

countcountmax

)

2 4 6 8 10 12 14 16 18 2050

100

150

200

250

300

350

400

450

500

Number of SCBS

Ave

rage

sum

rat

e [M

bps]

Social−aware UE associationRandom UE associationMax−RSSI based UE association

Fig. 5. Average sum rate for a fixed number of UEs (M = 10) per SCBS,for the proposed and baseline approaches.

of important UE is based on static social information which iscollected during the network setup phase. We use a commonfull-buffer traffic model for all UEs in our simulations. Theperformance of the social-aware approach is compared withthe baseline classical association approaches (i.e., max-RSSI(single time slot) and random association). In the randomassociation, important UEs are chosen randomly and UEsare randomly associated to SN within their D2D coverageradius. For the proposed social-aware UE-association withclustering approach a dynamic clustering method is used, inwhich the number of clusters dynamically changes. Moreover,all statistical results are averaged over a large number ofindependent runs and high dense network deployment.

A. Impact of SCBS Density

Fig. 5 shows the average sum rate as a function of thedensity of SCBSsN , and fixed number of UEs per SCBSM = 10. Fig. 5 clearly shows that, in the proposed social-aware approach, user association improves the sum rate. Inparticular, we can see that, as the number of SCBSs increases,the average sum rate increases. This is due to the fact that,an increase in the number of SCBSs increases the numberof important UEs, hence UEs associate with an important

2 4 6 8 10 12 14 16 18 20 22 24180

200

220

240

260

280

300

Number of UEs/SCBS

Ave

rage

sum

rat

e [M

bps]

Social−aware UE associationRandom UE associationMax−RSSI based UE association

Fig. 6. Average sum rate for a fixed number of SCBS (N = 8), under theconsidered approaches.

Number of UEs/SCBS2 4 6 8 10 12 14 16 18 20 22 24

Ave

rage

num

ber

of U

Es

havi

ng th

e sa

me

cont

ent

0

10

20

30

40

50

60

70

80

90

SCBS = 8 SCBS = 16

Fig. 7. Average number of UEs having the same content versus density ofUEs.

UE or SCBS based on their respective utility. Fig. 5, alsoshows that the performance gains in terms of sum rate for theproposed social-aware association approach increases whenthe number of SCBSs increases in the system. We furthernote that the proposed social-aware approach yields significantperformance gains for all network size, reaching up to23%over the max-RSSI based approach and56% over the randomUE association approach.

B. Impact of UE Density Per SCBS

Fig. 6 shows the average sum rate for a fixed numberof SCBSsN = 8 and varying density of UEs. It is worthto mention that our proposed approach is suited for densenetworks where large number of UEs per SCBS are deployed.It can be seen from the figure that, there is notable performancegain in terms of average sum rate for the proposed approachas compared to random and max-RSSI approach for varyingnumber of UEs from6 to 24 per cell. Moreover, it is alsonoted that in random UE association, UEs are associated to anySN within vicinity without consideration of RSSI and social-

12

Time line (slots)50 100 150 200 250 300 350 400 450 500

Ave

rage

rat

e pe

r U

E (

bps)

×106

0

0.5

1

1.5

2

2.5

3

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Fig. 8. Average achievable rate per UE for fixed number of SCBSs (N = 8), under the considered approaches.

ties between UEs. From Fig. 6, we can also observe that, byincreasing the number of UEs, the average sum rate increasesup to 48% and 24% compared to random UE associationand max-RSSI, respectively. Furthermore, we can see thatwith the fewer number of UEs per SCBS such as2 and 4UEs/SCBS the max-RSSI association approach outperformsover the proposed approach. This is due to the under utilizationof the second time-slotτ1 if no D2D links are formed inneighboring SCBSs.4

Fig. 7 shows the average number of UEs having the samecontent for a fixed number of SCBSsN = 8 andN = 16with the variant density of UEs. It can be seen from the figurethat, the number of UEs having the same content (i.e., size ofsocial network) increases as the density of UEs increases.

Fig. 8 shows the change in the average achievable rate perUE under the considered approaches. In order to examine thedata rate per UE, we fixed the number of SCBSsN = 8 andvaried the number of UEs per SCBS. The average rate per UEis constant over the number of time slots in case of max-RSSIapproach (single time slot). The achievable UE rate varies as aresult of its association (matching) to SN (SCBS, or importantUE). It can be observed from Fig. 8, that as we increases thenumber of UEs per SCBS, more time slots are required toachieve higher rate per UE. This is due to the fact that, withthe increase in the number of UEs per SCBS, more D2D linkscan be exploited. Therefore, as the number of UEs increasein the system more time slots are required to find the suitableUE-SN association.

C. Cell Edge Performance for Fixed Number of UEs Per SCBS

Fig. 9 shows the cumulative density function of UE’s datarate forM = 10 UEs per SCBS and different number of SCBSi.e.,N = 8 andN = 20. In order to examine the gains in theUE’s rate we analyze different percentile of user throughput. Itcan be shown from the figure that there is an increase in the UEdata rate relative to the social unaware association approach.Table III shows the different percentiles of UE throughput fordifferent for different density of SCBSs. We can see that, for

4Note that the results are averaged over multiple realizations.

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Fig. 9. Cumulative density function of UE’s rate for fixed UEsper SCBS(M = 10), SCBSs (N = 8) and (N = 20) under the considered approaches.

TABLE IIIPERCENTILES OFUE THROUGHPUTS

Percentiles of UE throughputs 5-th 50-th 95-thAverage data rate gain as compare to max-RSSI approach (N = 8)

21% 22% 28%

Average data rate gain as compare to max-RSSI approach (N = 20)

16% 23% 27%

N = 8 SCBSs, our proposed social-aware user associationoutperforms the social-unaware user association (max-RSSI)approach by up to21%, 22% and28% for 5-th,50-th and95-thpercentiles of user throughput, respectively. Fig. 9 showsthat,in case ofN = 20 the social-aware user association approachshows significant gains compared to social-unaware approach(max-RSSI) in terms of average data rate up to16%, 23% and27% for 5-th, 50-th and95-th percentiles of user throughput,respectively.

D. Cell Edge Performance for Fixed Number of SCBS

Fig. 10 shows the cumulative density function of UE’s ratefor N = 8 SCBSs and different sets of UEs per SCBS i.e.,M = 8 andM = 16. In order to examine the gains in theUE’s rate with different density of UEs, we analyze different

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Fig. 10. Cumulative density function of UE’s rate for fixed SCBSs (N = 8),UEs per SCBS (M = 8) and (M = 16) under the considered approaches.

TABLE IVPERCENTILES OFUE THROUGHPUTS

Percentiles of UE throughputs 5-th 50-th 95-thAverage data rate gain as compare to max-RSSI approach (M = 8)

26% 24% 31%

Average data rate gain as compare to max-RSSI approach (M = 16)

19% 18% 18%

percentiles of user throughput in terms of achievable datarate. Fig. 10 shows that there is an increase in the data raterelative to the social unaware association approach. TableIVshows the percentiles of the UE throughput for different fordifferent density of UEs and fixed SCBSs. In Fig. 10, we cansee that, forM = 8 UEs per SCBS, our proposed social-aware user association outperforms the social-unaware userassociation (max-RSSI) approach by up to26%, 24% and31% for 5-th, 50-th and95-th percentiles of user throughput,respectively. Furthermore, in case ofM = 16 UEs per SCBS,the social-aware user association approach shows significantgains compared to social-unaware approach (max-RSSI) interms of data rate up to19%, 18% and 18% for 5-th, 50-thand95-th percentiles of user throughput, respectively.

E. Impact of Similarity-based Clustering

In Fig. 11, we present the average number of clusters andthe average cluster sizes of SCBSs for various approaches. Wefix the number of UEsM = 20 per SCBS and SCBSsN = 16,with the various neighborhood discovery rangeΥd from 120mto 240m. Fig. 11 demonstrates the impact of distance, load, andjoint similarity on the coordination of SCBSs to form clustersas per in (18). For the joint similarity,Ω is set to0.5. Asper (16)-(18), it can be shown that as the distance increases,all edges have non-zero weight between SCBSs increases.Therefore, clustering based on the distance similarity allows togroup more SCBSs together yielding less average number ofclusters with larger average cluster size. The increase of clustersize directly influences on the cluster load, while clusteringbased on the joint similarity which takes into account distanceand load similarities to form clusters.

Fig. 12 shows the effect ofσd andσl on SCBS clustering.For this result, we use joint similarity based clustering suchthat Ω is set to0.5. The number of UEs per SCBSM = 20

120 140 160 180 200 220 2402

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Distance based clustering Ω = 1

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Average cluster size

Average number of clusters

Fig. 11. Comparison of average number of clusters and average cluster sizewith different similarities for the fixed number of UEsM = 20 per SCBS,andN = 16 SCBSs.

120 140 160 180 200 220 2402

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αd = 100, αl = 0.01αd = 100, αl = 0.1αd = 100, αl = 1.0αd = 200, αl = 0.01αd = 200, αl = 0.1αd = 200, αl = 1.0αd = 300, αl = 0.01αd = 300, αl = 0.1αd = 300, αl = 1.0

Fig. 12. Average number of clusters with differentαd and αl values forfixed number of UEsM = 20 per SCBS, andN = 16 SCBSs.

and number of SCBSsN = 16 are fixed with the variation inthe neighborhood discoveryΥd ranging from120m to 240m.It can be shown that, by varyingαd the number of clustersdecreases for a fixed value ofαl when the neighborhooddiscovery radius is under160m. It is worth mentioning thatthe range of the Gaussian distance similarity for any two con-nected SCBS is[e−Υd/2σ

2d , 1]. Thus, as the distance similarity

increases, SCBSs come closer and more likely to cooperate.For a fixed value ofαd and varyingαl it can be observedthat, the average number of clusters increases asαl increaseswithin the load dissimilarity range given as[1, e1/2σ

2l ].

Fig. 13 shows the average number of iterations requiredto reach a stable matching as a function of a fixed numberSCBSsN = 8 andN = 16 and varying number of UEs perSCBSM . The average number of iterations are calculatedover all the SCBSs clusters, which are the average number ofiterations required per cluster to get converge. In the figure,we can observe that, as the number of UEs increases, the

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Fig. 13. Average number of iterations as a function of the number of theSCBSs N and fixed number of UEs per SCBS (M=10), under the proposedapproach.

average number of iterations increases due to the increase inthe number of players in the system. From Fig. 13 we canalso observe that, the proposed social-aware approach requiresreasonable number of iterations for convergence.

VII. C ONCLUSIONS

In this paper, we proposed a novel, social network-awareapproach for user association in D2D underlaid small cell basestations. We formulated the problem as a matching game withexternalities in which the goal of each cluster of SCBSs isto maximize the social welfare which captures the data ratesand peer effect due to social ties among nodes. A dynamicclustering approach is introduced to cluster base stationsbased on their distance and load similarities. In the proposedmatching game, each UEs and SNs build their preferencesand self-organize in their respective cluster as to choose theirown utilities and achieve two-sided pairwise stable matching.To solve the game, we proposed social network-aware algo-rithm, in which UEs and SNs reach a stable matching in areasonable number of simulation iterations. Simulations resultshave shown that the proposed social network-aware approachprovides considerable gains in terms of increased data rateswith respect to a classical social-unaware approaches.

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