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A Social Group Utility Maximization Framework with Applications in Database Assisted Spectrum Access

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A Social Group Utility Maximizatio Framewor with Applicatios i Database Assisted Spectrum Access Xu Che, Xiaowe Gog, Lei Yag, Jusha Zhag School of ECEE, Arizoa State Uiversity {xche179, xgog9, lyag55,
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A Social Group Utility Maximizatio Framewor with Applicatios i Database Assisted Spectrum Access Xu Che, Xiaowe Gog, Lei Yag, Jusha Zhag School of ECEE, Arizoa State Uiversity {xche179, xgog9, lyag55, Abstract I this paper, we develop a social group utility maximizatio (SGUM framewor for cooperative etworig that taes ito accout both social relatioships ad physical couplig amog users. Specifically, istead of maximizig its idividual utility or the overall etwor utility, each user aims to maximize its social group utility that higes heavily o its social ties with other users. We show that this framewor provides rich modelig flexibility ad spas the cotiuum space betwee o-cooperative game ad etwor utility maximizatio (NUM two traditioally disjoit paradigms for etwor optimizatio. Based o this framewor, we study a importat applicatio i database assisted spectrum access. We formulate the distributed spectrum access problem amog white-space users with social ties as a SGUM game. We show that the game is a potetial game ad always admits a social-aware Nash equilibrium. We also desig a distributed spectrum access algorithm that ca achieve the social-aware Nash equilibrium of the game ad quatify its performace gap. We evaluate the performace of the SGUM solutio usig real social data traces. Numerical results demostrate that the performace gap betwee the SGUM solutio ad the NUM (social welfare optimal solutio is at most 15%. I. INTRODUCTION Networ utility maximizatio (NUM has bee extesively studied for etwor optimizatio problems (see [1], [] ad the refereces therei. A ey assumptio uderpiig etwor utility maximizatio is that the iterests of all users are aliged, i.e., they act i a altruistic maer with the same social objective of maximizig the total etwor utility. Alog a differet lie, game theory has foud a wide variety of importat applicatios for distributed resource allocatio problems i various etworig applicatios [3]. May existig game-theoretic models assume that all users are selfish ad ratioally, aimig at maximizig its ow beefit. I fact, the assumptios that users are altruistic ad selfish by etwor utility maximizatio ad o-cooperative game represet two extreme cases that users are fully social-aware (i.e., carig about the beefit of all the users ad socially oblivious (i.e., carig about their ow beefits oly, respectively. I may applicatios such as mobile social etworig, however, such assumptios are ot readily applicable sice had-held devices are carried by huma beigs who have diverse social relatioships ad care about their social eighbors at differet levels This research was supported i part by the U.S. Natioal Sciece Foudatio uder Grats CNS111746, CNS , ad DoD MURI project No. FA /14/$31.00 c 014 IEEE Fig. 1. A illustratio of the social group utility maximizatio framewor. I the physical domai, users have differet physical couplig subject to physical relatioships (e.g., iterferece. I the social domai, users have heterogeeous social couplig due to the social ties amog users. [4]. Clearly, there is much room left to exploit diverse social relatioships amog users for etworig optimizatio, ad it is of great iterest to explore the cotiuum space betwee these two extreme cases. Ideed, with the explosive growth of olie social etwors such as Faceboo ad Twitter, more ad more people are actively ivolved i olie social iteractios, ad social relatioships amog people are hece extesively broadeed ad sigificatly ehaced [5]. This has opeed up a ew aveue to itegrate the social iteractios for cooperative etwor desig. With this motivatio, we advocate a ovel social group utility maximizatio (SGUM framewor that taes ito accout both the users social relatioships ad physical couplig. As illustrated i Figure 1, a ey observatio is that users are coupled ot oly i the physical domai due to the physical relatioship (e.g., iterferece, ad but also i social domai due to the social ties amog them. It would be a wi-wi case for users to help those users havig social ties with them. With this isight, we view the etwor as a overlay/uderlay system where a virtual social etwor overlays the physical commuicatio etwor (the social etwor is virtual, i the sese that the social tie structure therei results from existig huma relatioship ad olie social etwors. The, the social tie structure is leveraged to facilitate cooperative etworig. Specifically, we cast the distributed decisio maig problem for cooperative etworig amog users as a SGUM game, where each user maximizes its social group utility, defied as the sum of its ow idividual utility ad the weighted sum of the utilities of other users havig social tie with it. Oe primary objective of this study is to establish a geeral framewor that bridges the gap betwee o-cooperative game ad etwor utility maximizatio two traditioally disjoit paradigms for etwor optimizatio. These two paradigms are captured uder the proposed framewor as two special cases where o social tie exists amog users (i.e., users are socially oblivious ad all users are coected by the strogest social ties (i.e., users are fully altruistic, respectively (as illustrated i Figure. Uder the SGUM framewor, we study a importat applicatio i database assisted spectrum access. The very recet FCC rulig requires that white-space users (i.e., secodary TV spectrum users must rely o a geo-locatio database to determie the spectrum availability [6]. Although the databaseassisted approach obviates the eed of spectrum sesig by idividual users, it remais challegig to achieve reliable shared spectrum access, because differet white-space users may choose to access the same vacat chael ad thus icur severe iterferece to each other. To stimulate effective cooperatio for chael allocatio amog whit-space users, we cast the database assisted distributed spectrum access problem amog white-space users with social ties as a SGUM game. We also desig a distributed spectrum access algorithm that ca coverge to a social-aware Nash equilibrium ad quatify its performace gap from the social welfare optimal solutio. A. Summary of Mai Cotributios The mai cotributios of this paper are as follows: Social Group Utility Maximizatio Framewor: We propose a SGUM framewor, which highlights the iterplay betwee the physical couplig subject to users physical relatioships ad the social couplig due to the social ties amog users. The SGUM framewor ca provide rich flexibility for modelig etwor optimizatio problems ad spa the cotiuum space betwee o-cooperative game ad etwor utility maximizatio two extreme paradigms based o drastically differet assumptios that users are selfish ad altruistic, respectively. Social Group Utility Maximizatio for Database Assisted Spectrum Access: We apply the SGUM framewor for database assisted spectrum access ad formulate the distributed spectrum access problem amog white-space users with social ties as a SGUM game. We prove that the game is a potetial game ad always admits a socialaware Nash equilibrium. Moreover, we show that the potetial fuctio of the game exhibits a ice structure that ca be decomposed ito two parts, capturig the impact of the physical couplig ad social couplig i spectrum access, respectively. Distributed Spectrum Access Algorithm for Achievig Social-aware Nash equilibrium: We desig a distributed spectrum access algorithm that ca achieve the socialaware Nash equilibrium of the SGUM game for database assisted spectrum access. We also derive the upperboud of the performace gap of the social-aware Nash equilibrium from the NUM solutio. We further evaluate the performace of the SGUM solutio usig real social data traces. Numerical results demostrate that the performace gap betwee the SGUM solutio ad the NUM solutio is at most 15%. B. Related Wor Although there exists a sigificat body of wor o ocooperative game ad etwor utility maximizatio, surprisigly very little attetio has bee paid to the cotiuum space betwee these two extreme paradigms. Recet wors [7], [8] have studied the impact of altruistic behavior i a routig game. [9] has recetly ivestigated a radom access game of two symmetrically altruistic players. The social aspect is emergig as a importat dimesio for commuicatio system desig. A chael recommedatio system based o cooperative social iteractios is developed for dyamic spectrum access i [10]. Social structures, such as social commuity which are derived from the user cotact patters, have bee exploited to desig efficiet data forwardig ad dissemiatio algorithms i delay tolerat etwors [11], [1]. I a recet study [13], we leveraged two ey social social pheomea of social trust ad social reciprocity to propose a coalitioal game based mechaism for cooperative DD commuicatio. We should emphasize that the SGUM framewor i this paper is quite differet from the coalitioal game solutio, sice each user i the latter aims to maximize its idividual beefit (although it is achieved by cooperatig with other users i its coalitio. Further, while a user i a coalitioal game ca oly participate i oe cooperative group (coalitio, a user i the SGUM framewor ca be i multiple social groups associated with differet users due to diverse social ties amog them. II. SOCIAL GROUP UTILITY MAXIMIZATION FRAMEWORK I this sectio we itroduce the SGUM framewor for cooperative etworig. As illustrated i Figure 1, the framewor ca be projected oto two domais: the physical domai ad the social domai. I the physical domai, differet wireless users have differet physical couplig due to their heterogeeous physical relatioships (e.g., iterferece. I the social domai, differet users have differet social couplig due to their itrisic social ties. We ext discuss both physical ad social domais i detail. A. Physical Networ Graph Model We cosider a set of wireless users N = {1,,..., N} where N is the total umber of users. We deote the set of feasible strategies for each user N as X. For istace, a strategy x X ca be choosig either a chael or a power level for wireless trasmissios. Subject to heterogeeous physical costraits, the strategy set X ca be user-specific. For example, the strategy set X ca be a set of feasible relay users that are i viciity of user for cooperative commuicatio. To capture the physical couplig amog the users i the physical domai, we itroduce the physical graph G p = {N, E p } (see Figure 1 for a example. Here the set of users N is the vertex set, ad E p {(, m :e p m =1,, m N} is the edge set where e p m =1if ad oly if users ad m have physical couplig (e.g., cause iterferece to each other. We also deote the set of users that have physical couplig with user as N p {m N : e p m =1}. Let x =(x 1,..., x N N =1 X be the strategy profile of all users. Give the strategy profile x, the idividual utility fuctio of user is deoted as U (x, which represets the payoff of user, accoutig for the physical couplig amog users. For example, U (x ca be the achieved data rate or the satisfactio of quality of service (QoS requiremet of user uder the strategy profile x. Note that i geeral the uderlyig physical graph plays a critical role i determiig the idividual utility U (x. For example, users achieved data rates are determied by the iterferece graph ad chael quality. B. Social Networ Graph Model We ext itroduce the social graph model to describe the social ties amog users. The uderlyig ratioale of cosiderig social tie is that the had-held devices are carried by huma beigs ad the owledge of huma social ties ca be utilized to ehace the performace of cooperative etworig. Specifically, we itroduce the social graph G s = {N, E s } to model the social tie amog the users. Here the vertex set is the same as the user set N ad the edge set is give as E s = {(, m :e s m =1,, m N}where e s m =1if ad oly if users ad m have social tie betwee each other, which ca be iship, friedship, or colleague relatioship betwee two users. Furthermore, for a pair of users ad m who have a social edge betwee them o the social graph, we formalize the stregth of social tie as w m [0, 1], with a higher value of w m beig a stroger social tie. We defie user s social group N s as the set of users that have social ties with user, i.e., N s = {m : e s m =1, m N}. Based o the physical ad social graph models above, users are coupled i the physical domai due to the physical relatioships, ad also coupled i social domai due to the social ties amog them. It would be a wi-wi case for users to help those users havig social ties with them. With this isight, we further defie the social group utility of each user as S (x =U (x+ w m U m (x. (1 m N s It follows that the social group utility of each user cosists of two parts: 1 its ow idividual utility ad the weighted sum of the idividual utilities of other users havig social tie with it. I a utshell, the social group utility fuctio captures the feature that each user is social-aware ad cares about the users havig social tie with it. C. Social Group Utility Maximizatio Game We ext cosider the distributed decisio maig problem amog the users for maximizig their social group utilities. Let x =(x 1,..., x 1,x +1,..., x N be the set of strategies chose by all other users except user. Give the other users strategies x, user wats to choose a strategy x X to maximize its social group utility, i.e., max S (x,x, N. x X The distributed ature of the problem above aturally leads to a formulatio based o game theory such that each user Fig.. Differet problem formulatios for etwor optimizatio with differet users social awareess levels aims to maximize its social group utility. We thus formulate the decisio maig problem amog the users as a strategic game Γ=(N, {X } N, {S } N, where the set of users N is the set of players, X is the set of strategies for each user, ad the social group utility fuctio S of each user is the payoff fuctio of player. I the sequel, we call the game Γ as the SGUM game. We ext itroduce the cocept of social-aware Nash equilibrium (SNE. Defiitio 1. A strategy profile x =(x 1,..., x N is a socialaware Nash equilibrium of the SGUM game if o player ca improve its social group utility by uilaterally chagig its strategy, i.e., x =arg max S (x,x, N. x X It is worth otig that uder differet social graphs, the proposed SGUM game formulatio ca provide rich flexibility for modelig the etwor optimizatio problem (as illustrated i Figure. Whe the social graph cosists of isolated odes with w m =0for ay, m N (i.e., all users are selfish, the SGUM game degeerates to the o-cooperative game formulatio. Whe the social graph is fully meshed with edge weight w m =1for ay, m N (i.e., each user is fully altruistic ad cares eough about other users, the SGUM game becomes the etwor utility maximizatio problem, which aims to maximize the system-wide utility. The SGUM framewor i this study is applicable to geeral social graphs ad hece ca bridge the gap betwee o-cooperative game ad etwor utility maximizatio two traditioally disjoit paradigms for etwor optimizatio. The SGUM is a geeral framewor that ca be applied for may etworig applicatios. To get a more cocrete sese of the framewor, i the followig sectios, we will focus o studyig its applicatio i database assisted spectrum access. III. SOCIAL GROUP UTILITY MAXIMIZATION FOR DATABASE ASSISTED SPECTRUM ACCESS I this sectio we apply the SGUM framewor for the database assisted spectrum access. A. Social Group Utility Maximizatio Game Formulatio We cosider a set of white-space users N = {1,,..., N} where N is the total umber of users. We deote the set of TV chaels as M = {1,,..., M}. Accordig to the recet rulig by FCC [6], to protect the icumbet primary TV users, each white-space user N will first sed a spectrum access request message cotaiig its geo-locatio iformatio to a Geo-locatio database (see Figure 3 for a illustratio. The database the feeds bac the set of vacat chaels Fig. 3. A illustratio of database assisted spectrum access M M ad the allowable trasmissio power level P to user. The rulig by FCC idicates that the allowable trasmissio power limit for persoal/portable white-space devices (e.g., mobile phoes is 100 mw [6]. For ease of expositio, we hece assume that each user accesses the white-space spectrum with the same power level. Each user the chooses a feasible chael a from the vacat chael set M for data trasmissio. Although the database-assisted approach obviates the eed of spectrum sesig by idividual users, it remais challegig to achieve reliable distributed spectrum access, because may differet white-space users may choose to access the same vacat chael ad thus icur severe iterferece to each other [14], [15]. To stimulate effective cooperatio amog users for iterferece mitigatio, we leverage the social ties amog users ad apply the SGUM approach. To capture the physical couplig, we costruct the iterferece graph G p = {N, E p } based o the iterferece relatioships amog users. Here the set of white-space users N is the vertex set, ad E p {(, m : e p m =1,, m N}is the edge set where e p m =1if ad oly if users ad m ca geerate sigificat iterferece ad affect the data trasmissios of each other. For example, we ca costruct the iterferece graph G p based o spatial relatioships of the users [16]. Let δ deote the trasmissio rage of each user. We the have e p m =1if ad oly if the distace d m betwee user ad m is ot greater tha the threshold δ, i.e., d m δ. Let a =(a 1,..., a N N =1 M be the chael selectio profile of all users. Give the chael selectio profile a, the iterferece received by user ca be computed as γ (a = m N p mi {a =a m } + ω a. ( Here α is the path loss factor ad I {A} is a idicator fuctio with I {A} =1if the evet A is true ad I {A} =0otherwise. Furthermore, ωa deotes the oisy power icludig the iterferece from primary TV users o the chael a.we the defie the idividual utility fuctio U (a as U (a = γ (a = mi {a =a m } ωa. (3 m N p Here the egative sig comes from the covetio that utility fuctios are typically the oes to be maximized. The idividual utility of user reflects the fact that each user has iterest to reduce its ow received iterferece. To capture the social couplig i the social graph G s, we further itroduce the social group utility of each white-space user accordig to (1 as S (a =U (a+ w m U m (a. (4 m N s We the formulate the database assisted spectrum access problem as a SGUM game Γ=(N, {M } N, {S } N, where the set of white-space users N is the set of players, the set of vacat chaels M is the set of strategies for each player, ad the social group utility fuctio S of each user is the payoff fuctio of player. B. Properties of SGUM game We ext study the existece of SNE of the SGUM game for database assisted spectrum access. Here we resort to a useful tool of potetial game [17]. Defiitio. A game is called a potetial game if it admits a potetial fuctio Φ(a such that for every N ad a i M i, for ay a,a M, S (a,a S (a,a =Φ(a,a Φ(a,a. (5 A appealig property of the potetial game is that it always admits a Nash equilibrium, ad ay strategy profile that maximizes the potetial fuctio Φ(a is a Nash equilibrium [17]. For the SGUM game Γ for database assisted spectrum access, we ca show that it is a potetial game. For ease of expositio, we first itroduce the physical-social graph G sp = {N, E sp } to capture both physical couplig ad social couplig simultaeously. Here the vertex set is the same as the user set N ad the edge set is give as E sp = {(, m : e sp m e s m e p m =1,, m N}where e sp m =1if ad oly if users ad m have social tie betwee each other (i.e., e s m = 1 ad ca also geerate iterferece to each other (i.e., e p m =1. We deote the set of users that have social ties ad ca also geerate iterferece to user as N sp = {m : e sp m =1, m N}. Based o the physical-social graph G sp, we show i Theorem 1 that the SGUM game Γ is a potetial game with the followig potetial fuctio Φ(a = 1 N N mi {a =a m } ωa =1 m N p =1 }{{} Φ 1 (a: due to Physical Couplig 1 N w m mi {a=a m}. (6 =1 m N } sp {{} Φ (a: due to Social Couplig The potetial fuctio i (6 ca be decomposed ito two parts: Φ 1
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