When Network Effect Meets Congestion Effect: Leveraging Social Services for Wireless Services

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1 When Network Effect Meets Congeston Effect: Leveragng Socal Servces for Wreless Servces aowen Gong School of Electrcal, Computer and Energy Engeerng Arzona State Unversty Tempe, AZ 8587, USA u Chen Insttute of Computer Scence Unversty of Goettngen Goettngen 77, Germany Lngje Duan Engneerng Systems and Desgn Pllar Sngapore Unversty of Technology and Desgn Sngapore 877, Sngapore ABSTRACT The recent development of socal servces tghtens wreless users socal relatonshps and encourages them to generate more data traffc under network effect Ths boosts the demand for wreless servces yet may challenge the lmted wreless capacty To fully explot ths opportunty, we study moble users data usage behavors by jontly consderng the network effect based on ther socal relatonshps n the socal doman and the congeston effect n the physcal wreless doman Accordngly, we develop a Stackelberg game for problem formulaton: In Stage I, a wreless provder frst decdes the data prcng to all users to maxmze ts revenue, and then n Stage II users observe the prce and decde data usage subject to mutual nteractons under both network and congeston effects We analyze the two-stage game usng backward nducton For Stage II, we frst show the exstence and unqueness of a user demand equlbrum (UDE) Then we propose a dstrbuted update algorthm for users to reach the UDE Furthermore, we nvestgate the mpacts of dfferent parameters on the UDE For Stage I, we develop an optmal prcng algorthm to maxmze the wreless provder s revenue We evaluate the performance of our proposed algorthms by numercal studes usng real data, and thereby draw useful engneerng nsghts for the operaton of wreless provders Categores and Subject Descrptors C1 [Computer-Communcaton Networks]: Network Archtecture and Desgn Wreless communcaton The correspondng author s u Chen Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page Copyrghts for components of ths work owned by others than ACM must be honored Abstractng wth credt s permtted To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee Request permssons from permssons@acmorg MobHoc 15, June 5, 15, Hangzhou, Chna Copyrght c 15 ACM /15/ $15 Keywords Moble socal network, wreless data usage, network effect, congeston effect 1 INTRODUCTION The past few years have wtnessed pervasve penetraton of moble devces n people s daly lfe, thanks to the wreless technology advances Motvated by many socal applcatons on moble platforms (eg, WeChat, WhatsApp [1, ]), moble users data usage behavors have been ncreasngly nfluenced by ther socal relatonshps In 1, the number of onlne socal meda users on moble platforms has reached 1 bllon, accountng for % of moble users and 8% of onlne socal meda users [] The popularty of socal servces on moble platforms also gves opportuntes to wreless servce provders who operate the moble networks Intutvely, socal servces can encourage moble users to demand more data usage by stmulatng ther nteractons wth each other through these servces (eg, onlne socal gamng and bloggng) When a user ncreases ts actvty n a socal servce, ts socal frends would also ncrease ther actvtes Therefore, users data usage levels for socal servces present network effect to others [] Ths demand ncrease provdes a great potental for wreless provders revenue ncrease However, ths potental beneft s subject to the lmted wreless capacty n physcal communcaton networks (eg, spectrum) As users ncrease ther data usage, they also experence more congeston (eg, servce delays), whch dscourages them to use more The ncreasng congeston poses a sgnfcant challenge for wreless provders to ncrease ther revenues As a result, moble users data usage behavors are not only subject to congeston effect n the physcal network, but also network effect n the socal network (as llustrated n Fg 1), whch has been largely overlooked by tradtonal wreless provders To fully explot the potental beneft brought by socal servces, t s necessary to nvestgate users data usage behavors n these two domans, so that a wreless provder can take the best strategy n favor of ts revenue Wth ths nsght, we not only analyze users nteractons subject to both network and congeston effects,

2 1 1 Socal Doman Network Effect Physcal Doman Congeston Effect Fgure 1: Moble users experence network effect n the socal network and congeston effect n the physcal network but also study the optmal prcng strategy for the wreless provder The man contrbutons of ths paper can be summarzed as follows Stackelberg game formulaton: By jontly consderng users socal relatonshps and the wreless network s congeston, we formulate the nteracton between the wreless provder and moble users as a Stackelberg game: In Stage I, the wreless provder chooses a prce to maxmze ts revenue; n Stage II, moble users choose ther data usage levels based on the prce to maxmze ther socally-aware payoffs Equlbrum analyss for user demands n Stage II: We frst gve a general condton under whch there exsts a user demand equlbrum (UDE), and then we show that under a further general condton the game s a concave game and thus admts a unque UDE We also propose a dstrbuted algorthm for users to acheve the UDE Next we show that a user s usage can ncrease when prce ncreases We further show that f the socal network s symmetrc, the total usage always ncreases when a user s parameter (eg, socal te) mproves Provder s optmal prcng n Stage I: Fnally, by takng nto account users equlbrum demands, we develop an optmal prcng algorthm to maxmze the revenue of the wreless provder We evaluate the performance of total usage and revenue by smulatons, and draws useful engneerng nsghts for the wreless provder s operaton The rest of ths paper s organzed as follows In Secton, we formulate the Stackelberg game between the wreless provder and moble users Secton studes users demand equlbrum n Stage II Secton studes the provder s optmal prcng strategy n Stage I Smulaton results and dscussons are gven n Secton 5 Related work are revewed n Secton Secton 7 concludes ths paper SYSTEM MODEL 1 Socally-aware Wreless Servce Consder a set of users N {1,, N} partcpatng n a wreless servce provded by a wreless operator (eg, AT&T) Each user N consumes an amount of data usage n the wreless servce, denoted by x where x [, ) Let x (x 1,, x N ) denote the usage profle of all the users and x denote the the usage profle wthout user Affected by the other users usage subject to congeston effect due to the lmted resources n the wreless network, the payoff of user by consumng data usage x s v (x, x, p) = a x 1 b x 1 c( x j ) px, j N where a > and b > are the nternal utlty coeffcents that capture the ntrnsc value of the wreless servce to user, c > s the congeston coeffcent that s determned by the resource constrants of the wreless network, and p s the usage-based prce charged by the wreless provder 1 As n [5], the quadratc form of the nternal utlty functon not only allows for tractable analyss, but also serves as a good second-order approxmaton for a broad class of concave utlty functons In partcular, a models the maxmum nternal demand rate, and b models the nternal demand elastcty factor For the congeston model, the quadratc sum form reflects that a user s congeston experence s affected by all the users, and the margnal cost of congeston ncreases as the total usage ncreases Tradtonal wreless provders operaton does not take nto account the fact that socal servces encourage moble users to demand more data usage We thus account for ths effect n our model Then user s payoff ncludes the addton of socal utlty, e, u (x, x, p)=a x 1 b x + g j x x j 1 c( x j ) px (1) j N where g j s the socal te that quantfes the socal nfluence from user j to user As n [5], the product form g jx x j of the socal utlty functon captures that a user derves more utlty by ncreasng ts usage n socal servces, and the margnal gan of socal utlty ncreases as ts socal frends ncrease ther usage Therefore, socal servces brng n network effect among users and can ncrease ther utltes Stackelberg Game Formulaton We model the nteracton between the wreless provder and moble users for the socally-aware wreless servce as a two-stage Stackelberg game Defnton 1 (Two-Stage Prcng-Usage Game) Stage I (Prcng): The wreless provder chooses prce p to maxmze ts revenue: p = arg max t(x)p p [, ) P where t(x) x denotes the total usage under strategy profle x; Stage II (Usage): Each user N chooses ts data usage level x to maxmze ts payoff gven the prce p and the usage levels of the other users x : x = arg max x [, ) u (x, x, p) We study the two-stage prcng-usage game by backward nducton [] For Stage II, gven a prce chosen by the wreless provder n Stage I, we are nterested n the exstence of 1 Usage-based prcng s wdely used n practce by wreless operators to control the demand Here the prce s the same for all users to ensure farness

3 a stable outcome of users nteractons at whch no user wll devate Ths leads to the concept of user equlbrum Defnton (User Demand Equlbrum) For any prce p gven n Stage I, the user demand equlbrum (UDE) n Stage II s a strategy profle x such that no user can mprove ts payoff by unlaterally changng ts usage, e, x = arg max x [, ) u (x, x, p), Gven the UDE n Stage II, we wll study the optmal prcng strategy for the wreless provder n Stage I STAGE II: USER DEMAND EQUILIBRIUM In ths secton, we study users demands n Stage II Usng the concave payoff functon (1), by settng the dervatve u (x,x ) x = as the frst-order condton, we obtan the best response functon of user as ( ) a p r (x ) = max, b + c + g j c b + c x j () Accordng to (), each user s usage demand conssts of two parts: nternal demand a p b that s ndependent of P g the other users, and external demand j c b xj that depends on the other users The coeffcent g j c represents b the margnal ncrease or decrease of user s demand when user j s usage ncreases: when g j > c, user j s nfluence to user s domnated by network effect; when g j < c, t s domnated by congeston effect 1 Exstence and Unqueness of UDE We frst nvestgate the exstence of UDE n Stage II We make the followng assumpton Assumpton 1 g j c b + c < 1, g j c b Ths assumpton s for analyss tractablty It s an mportant condton to guarantee the exstence of UDE, as there can exst no UDE when t does not hold (as llustrated by an example n Fg ) Accordng to the best response functon (), Assumpton 1 mples that any user s absolute ex- P ternal demand xj s less than the maxmum usage max x j among all the other users Ths s a mld condton as the aggregate effect experenced by a user from all the other users would be less than the largest effect the user can experence from an ndvdual of the other users A smlar assumpton s made n [5] for smlar consderatons Now we can show that there always exsts a UDE n Stage II Theorem 1 Under Assumpton 1, the Stage II game admts a UDE The proof s gven n Appendx and the man dea s to show that the game has an equvalent game whch admts a UDE Next we gve another general techncal condton under whch the game admts a unque UDE Theorem Under P Assumpton 1, the Stage II game admts a unque UDE f j c g b < 1, a p b x a 1 p b 1 r (x 1 ) r 1 (x ) Fgure : For Stage II for two users, when g 1 c b 1 < 1 and g 1 c b < 1, there does not exst a UDE The bold lnes are the best response functons and do not ntersect The proof s gven n Appendx and the man dea s s to show that the game s a concave game [7], and thus admts a unque UDE Remark: Accordng to Theorem, t s worth notng that the Stage II game admts a unque UDE when users socal tes are symmetrc (e, g j = g j, j) The symmetrc settng of socal networks s of great nterests Motvated by the dea of socal recprocty [8], a user s socal behavor to another s lkely to mtate the latter s behavor to the former As a result, two users socal tes to each other tend to be the same Computng and Achevng UDE As we have showed the exstence of UDE, we then desgn an algorthm to compute the UDE, as descrbed n Algorthm 1 The algorthm teratvely updates users strateges based on ther best response functons () and converges to the UDE Algorthm 1: Compute the UDE n Stage II 1 nput: precson threshold ϵ; x (), N ; t 1; repeat foreach N ; 5 do x (t+1) = max, 7 end 8 t t + 1; 9 untl x (t) x (t 1) ϵ; 1 return x (t) ; x 1 P a p + g j c b b x(t) j ; Theorem Algorthm 1 computes the UDE n Stage II The proof s gven n Appendx and the man dea s to show that the best response updates n the algorthm result n a contracton mappng and hence converges to a fxed pont It s desrable for users to reach the UDE n a dstrbuted manner We then propose a dstrbuted update algorthm based on Algorthm 1, as descrbed n Algorthm Note that the usage update n Algorthm s equvalent to the best response update n Algorthm 1 Each user chooses ts best response usage based on the usage of ts

4 Algorthm : Dstrbuted algorthm to acheve the UDE n Stage II 1 each user N chooses an ntal usage x () ; loop at each tme nterval t = 1,, each user N n parallel: updates ts usage by max 8 < 5 end loop :, a p b + c + 1 b + c,g j > g j x (t) j c b + c x (t) j 9 = ; g 1 > c g 1 = c x 1 +x g 1 < c a 1 b 1 a Fgure : Total usage at the UDE n Stage II for two users a 1 p socal frends who have socal nfluences to t (e, each user j wth g j > ), whch can be obtaned from the socal frends, and the total usage of all users, whch can be obtaned from the wreless provder The correctness of Algorthm follows from that of Algorthm 1 and s thus omtted Proposton 1 Algorthm acheves the UDE n Stage II Parameter Analyss at UDE We frst nvestgate the mpact of prce on the UDE To draw clean nsghts, we start wth the case for two users Wthout loss of generalty, assume that a 1 a Proposton For Stage II for two users, there exsts a prce threshold p th [, a 1] where p th = a(b1 + c) a1(c g1) b 1 + g 1 () such that the UDE x s gven as follows, dependng on the prce p: Hgh prce regme: When p a 1, x 1 = x = ; Medum prce regme: When p th p < a 1, x 1 = a 1 p b 1 and x = ; Low prce regme: When p < p th, x 1 = (a 1 p)(b ) (a p)(c g 1 ) b 1 b g 1 g 1 (b 1 +b +g 1 +g 1 ) and x = (a p)(b 1 ) (a 1 p)(c g 1 ) b 1 b g 1 g 1 (b 1 +b +g 1 +g 1 ) Due to space lmtaton, the proof s gven n our onlne techncal report [9] Accordng to (), there are three cases of the threshold p th dependng on whch effect domnates user s experence from user 1 (as llustrated n Fg ) 1 Nether effect: When g 1 = c, we have p th = a As network effect and congeston effect cancel each other, user experences nether effect from user 1 Then user s usage demand s equal to ts nternal demand, and t reaches when p = a Congeston effect: When g 1 < c, we have p th < a As user experences congeston effect from user 1, even when p s less than a such that user has a postve nternal demand, ts external demand can be suffcently negatve such that user s usage demand s negatve In partcular, when a 1 a b 1 c g 1, user s usage s even when the prce s Network effect: When g 1 > c, we have p th > a As user experences network effect from user 1, even when p s greater than a such that user has a negatve nternal demand, ts external demand can be suffcently postve such that user s usage demand s postve Next we study the general case for any number of users For convenence, let us defne B = b 1 + c c c c b + c c 7 5, G= c c b N + c g 1 g 1N g 1 g N 7 5 g N1 g N Also defne C as the N N matrx wth each entry beng c For a UDE x, let S be the set of users wth postve usage n x (e, x >, S and x =, / S) For convenence, let v S denote the S 1 vector comprsed of the entres of a vector v wth ndces n S, M S denote the S S matrx comprsed of the entres of a matrx M wth ndces n S S, and [M],S denote the 1 S vector comprsed of the entres of the th row of a matrx M wth column ndces n S Accordng to the best response functon (), x S s the soluton to the system of equatons B S x S = a S p1 S + G S x S where 1 denotes the N 1 vector of 1s We need the followng lemma: Lemma 1 Under Assumpton 1, (B S G S ) s nvertble for any set S N The proof s gven n [9] Thus we have x S = (B S G S ) 1 (a S p1 S ) () When users have the same nternal coeffcent a, we can show that the same set of users have postve equlbrum usage at dfferent prces Proposton For Stage II, when a = a,, the UDE x s gven as follows, dependng on the prce p: When p > a, x =, ; When p a, there exsts a set S N such that for any p [, a), x = [(B S G S ) 1 (a S p1 S )] >, S, and x =, / S, where [M] denotes the th row of matrx M

5 Total usage 8 1 Prce (a) Total usage vs prce p Total revenue 1 1 Prce (b) Total revenue vs prce p Fgure : (a) Total usage at the UDE s a pece-wse lnear functon of prce; (b) Total revenue at the UDE s a pecewse quadratc functon of prce The proof s gven n Appendx Proposton shows that the set of users wth postve equlbrum usage (f they exst) does not change wth prce, and each user s postve usage decreases when prce ncreases We then show by a counterexample that f users have dfferent nternal coeffcents a, a user s equlbrum usage can ncrease when prce ncreases Consder a case for three users where a 1 =, a = a = 15, b =,, c =, p =, and g = g =, g j =, {, j} {, } We can show that there exsts a unque UDE x and t s the soluton to the system of equatons below: 5x 1 + x + x = 1 5x + x 1 x = 11 5x x + x 1 = 11 Solvng these equatons, we have x 1 = 571, x = 8, x = 8 When prce p ncreases to 5, the new UDE s the soluton to 5x 1 + x + x = 15 5x + x 1 x = 1 5x x + x 1 = 1 whch s x 1 = 71 > 571, x = 857, x = 857 Thus the usage of user 1 ncreases Remark: Intutvely, when the prce ncreases, the usage of both user and decrease and the nternal demand of user 1 decreases However, as user 1 experences strong congeston effect from both user and, user 1 s external demand ncreases due to the decrease of congeston effect, and t ncreases faster than the decrease of user 1 s nternal demand as prce ncreases, such that the total of nternal and external demand ncreases In addton, a larger nternal coeffcent a 1 of user 1 than that of user and allows user 1 to have a postve equlbrum usage x 1 = 71 even when ts external demand s negatve due to the strong congeston effect Indeed, f user 1 has the same nternal coeffcent a 1 = 15 as user and, then we can show that ts equlbrum usage s Furthermore, when users have dfferent nternal coeffcents a, we have the followng result Proposton For Stage II, the UDE x s gven as follows, dependng on the prce p: When p > max a, x =, ; Note that Assumpton 1 holds under ths settng r 1 (x ) a 1 p c g 1 a p b x x x a 1 p b 1 (a) r (x 1 ) a p c g 1 x 1 a p b x r 1 (x ) x a 1 p b 1 (b) x r (x 1 ) Fgure 5: For Stage II for two users, the unque UDE x s acheved at the ntersecton of the best response functons (bold lnes): (a) g 1 c ( 1, ), g 1 c b 1 b ( 1, ); (b) g 1 c (, 1), g 1 c b 1 b (, 1) When g1 = g1 ncreases, the UDE x moves to x x 1 When p max a, there s a set of prces p a, and for each < p 1 < < p M < p M+1 max k {,, M}, there exsts a set S k N such that for any p [p k, p k+1 ], x = [(B Sk G Sk ) 1 (a Sk p1 Sk )] >, S k and x =, / S k The proof s gven n [9] Proposton shows that each user s equlbrum usage s a pece-wse lnear functon of prce: wthn each prce nterval [p k, p k+1 ], the equlbrum usage s a lnear functon of prce p Next we nvestgate the mpacts of other parameters on the UDE We show that the total usage always ncreases when a user s parameter mproves, under the condton that users socal network s symmetrc For tractable analyss, we assume that users have the same nternal coeffcent a Proposton 5 For Stage II, when a = a, and socal tes are symmetrc (e, g j = g j, j), the total equlbrum usage ncreases when a or any g j ncreases, or any b or c decreases The proof s gven n Appendx We llustrate Proposton 5 by an example n Fg 5 As mentoned before, users socal tes tend to be symmetrc n practce due to socal recprocty [8] In Secton 5, smulaton results wll show that the performance under asymmetrc socal tes s very close to that under symmetrc socal tes STAGE I: OPTIMAL PRICING In the prevous secton, we have nvestgated the UDE n Stage II gven a prce chosen by the wreless provder In ths secton, we study the optmal prcng of the provder n Stage I We frst observe from Proposton that the total usage s a pece-wse lnear functon of prce (as llustrated n Fg (a)) As a result, the total revenue s a pecewse quadratc functon of prce (as llustrated n Fg (b)) Based on ths observaton, we develop an algorthm that computes the optmal prce to maxmze the provder s revenue, as descrbed n Algorthm The basc dea s to frst Users can stll have dfferent b

6 Algorthm : Compute the optmal prce that maxmzes revenue n Stage I 1 compute the UDE x at prce usng Algorthm 1; p ; p ; r ; S ; foreach N do f x > then 5 S S {}; end 7 end 8 whle p max a and S = do 9 S ; S ; 1 foreach S do 11 f [(B S G S) 1 ] 1 S > then 1 S S {}; p [(B S G S ) 1 ] a S ; [(B S G S ) 1 ] 1 S 1 end 1 end 15 foreach / S do 1 f [G C],S (B S G S ) 1 1 S < 1 then 17 S S {}; p [G C],S (B S G S ) 1 a S +a [G C],S (B S G S ) 1 1 S end 19 end p mn ; k arg mn ; S S S S ˆp 1T S (B S G S ) 1 a S ; 1 T S (B S G S ) 1 1 S 1 f ˆp [p, p] then p ˆp; else f ˆp < p then 5 p p; else 7 p p; 8 end 9 end end 1 end r p 1 T S (B S G S) 1 (a S p 1 S); f r > r then p p ; r r ; 5 end p p; 7 f k S then 8 S S \ {k}; 9 else S S {k}; 1 end end end return p, r ; determne the prce ntervals that characterze the pece-wse structure, such that wthn each prce nterval, the set of users wth postve usage s the same at any prce Then we fnd the optmal prce wthn each nterval that maxmzes the revenue Thus we can fnd the optmal prce wth the maxmum revenue among all the ntervals In partcular, Algorthm starts wth computng the set of users S wth postve usage at prce by usng Algorthm Probablty of socal edge Number of users Fgure : Probablty of socal edge vs number of users n real data trace [1] 1 Then ths set S serves as the ntal condton for the followng steps As the prce p ncreases from to max a (whch s the largest possble value of the optmal prce accordng to Theorem ), t teratvely fnds the crtcal prces at whch the set S changes In each teraton, gven the current crtcal prce p, the next crtcal prce p s the mnmum prce greater than p at whch some user S wth postve usage decreases ts usage to, or some user / S wth usage ncreases ts usage to a postve value Wthn each prce nterval [p, p], as the revenue R s a quadratc functon of prce p, the optmal prce n [p, p] that maxmzes the revenue s the prce ˆp such that R(p) p p=ˆp =, f ˆp s n [p, p]; otherwse, the optmal prce s one of the endponts p and p By comparng the maxmum revenues at the optmal prces for all the prce ntervals, the algorthm fnds the optmal prce n the entre range of prce Theorem Algorthm computes the optmal prce n Stage I The proof s gven n [9] In the next secton, numercal results wll show that the computatonal complexty of Algorthm s lnear n the number of users 5 PERFORMANCE EVALUATION In ths secton, we frst use smulaton results to evaluate the performance of the two-stage game for the moble users and the wreless provder Then we dscuss the engneerng nsghts that can be drawn from the smulaton results 51 Smulaton Setup To llustrate the mpacts of dfferent parameters of moble socal networks on the performance, we consder a random settng as follows We smulate the socal graph G usng the Erdős-Rény (ER) graph model [11], where a socal edge exsts between each par of users wth probablty P S If a socal edge exsts, the socal te follows a normal dstrbuton N(µ G, ) (wth mean µ G and varance ) We assume that each a follows a normal dstrbuton N(µ A, ), and each b follows a normal dstrbuton N(µ B, ) We set default parameter values as follows: N = 1, P s = 8, µ A =, µ B = 1, µ G =, c = To evaluate the performance n practce, we also smulate the socal graph accordng to Recall that some user s equlbrum usage can ncrease when prce ncreases as llustrated by the example n Secton

7 Normalzed total usage NSU SU ER SU ER (asym) SU real 8 1 Probablty of socal edge Normalzed total usage NSU SU ER SU ER (asym) SU real 1 Mean of socal te Normalzed total usage NSU SU ER SU ER (asym) SU real 5 5 Congeston coeffcent Fgure 7: Normalzed total usage vs Fgure 8: Normalzed total usage vs Fgure 9: Normalzed total usage vs probablty of socal edge P S mean of socal te µ G congeston coeffcent c Normalzed total usage 8 NSU (c=) NSU (c=) SU real (c=) SU ER (c=) SU ER (c=1) 1 5 Number of users Normalzed optmal prce NSU (Ps=) SU real SU ER (Ps=8) SU ER (Ps=5) Number of users Normalzed optmal revenue 8 NSU (c=) NSU (c=) SU real (c=) SU ER (c=) SU ER (c=1) 1 5 Number of users Fgure 1: Normalzed total usage vs Fgure 11: Normalzed optmal prce vs Fgure 1: Normalzed optmal revenue number of users N number of users N vs number of users N the real data trace from Brghtkte [1], whch s a socal frendshp network based on moble phones For ths data trace, we plot the average number of socal tes between two users versus the number of users n Fg If a socal edge exsts based on the real data, the socal te also follows a normal dstrbuton N(µ G, ) As a benchmark, we evaluate the performance when users demand non-socally-aware usage (NSU) n comparson to our proposed socally-aware usage (SU) Snce NSU s a specal case of SU wth all socal tes beng, the UDE and optmal prcng for NSU can be computed as for SU To hghlght the performance comparson, we normalze the results wth respect to NSU We also compare the performance under SU wth ER model based socal graph (SU-ER) and wth real data based socal graph (SU-real) 5 Smulaton Results 51 Total Usage n Stage II We frst evaluate the performance of total usage n Stage II We llustrate the mpacts of P S, µ G, c on total usage n Fgs 7-9, respectvely As expected, we observe from all these fgures that SU always domnates NSU, and can perform sgnfcantly better than NSU From Fgs 7-8, we can see that the performance gan of SU over NSU ncreases as P S or µ G ncreases, and the margnal gan s also ncreasng Smlarly, we can see from Fg 9 that the performance gan of SU over NSU ncreases as congeston coeffcent c decreases, and the margnal gan s also ncreasng We also evaluate the performance under SU wth ER model based the asymmetrc socal graph We observe that ts performance s very close to that wth the symmetrc socal graph Fg 1 llustrates the mpact of N on total usage As expected, we observe that the total usage always ncreases wth the number of users However, for the case of NSU and SU-real, the margnal gan of total usage decreases wth the number of users, whle for the case of SU-ER, the margnal gan ncreases Intutvely, n the former case, when a new user jons the network, as the new user s socal tes wth the exstng users are weak, the congeston effect between the new user and the exstng users outweghs the network effect between them Furthermore, as more users exst n the network, the weght dfference between the congeston effect and the network effect ncreases, and thus the margnal gan of total usage by addng a user decreases In the latter case, as the new user s socal tes wth the exstng users are strong, the roles of the congeston effect and network effect are swtched 5 Optmal Prce n Stage I Next we evaluate the performance of the optmal prce and optmal revenue n Stage I Fg 11 llustrates the optmal prce as the number of users ncreases We observe that the optmal prce always decreases wth the number of users Intutvely, ths s because as the number of users ncreases, more users have hgher nternal demands, so that ncreasng the prce does not result n sgnfcant decrease n total usage Comparng dfferent curves, we can also see that the optmal prce decreases as P S ncreases from to and then to 8 Intutvely, ths s because that when network effect s strong, a low prce

8 Number of teratons 5 1 SU ER SU real 1 5 Number of users Fgure 1: Computatonal complexty of Algorthm vs number of users N s desrable, snce t encourages users nternal usage whch further stmulate sgnfcantly more usage by the network effect; when congeston effect s strong, a hgh prce s desrable, snce decreasng the prce cannot encourage sgnfcantly more usage due to the congeston effect Fg 1 llustrates the optmal revenue acheved at the optmal prce as the number of users ncreases As expected, we can make smlar observatons as for Fg 1: when network effect domnates congeston effect, the margnal gan of optmal revenue by addng more users s ncreasng; otherwse, the margnal gan s decreasng Fg 1 llustrates the computatonal complexty of Algorthm as the number of users ncreases The number of teratons s equal to the number of prce ntervals that determne the pece-wse structure of total usage and revenue as a functon of prce We observe that the complexty s O(N) 5 Further Dscussons Based on the smulaton results, we can draw the followng engneerng nsghts for the operaton of wreless provders The observatons from Fgs 7-9 suggest that as users socal tes become stronger (whch can be promoted by socal servces), the wreless provder can receve an ncreasng total usage and thus revenue, and also an ncreasng margnal gan In addton, the wreless provder can also receve an ncreasng margnal return by ncorporatng more resources for the wreless servce to mtgate congeston The observatons from Fgs 1 and 1 suggest that the wreless provder should be aware of whether the network effect determned by users socal tes domnates the congeston effect If the network effect domnates, t receves an ncreasng margnal gan by takng n more users; otherwse, the margnal gan s decreasng and the total usage wll saturate when the number of users s suffcently large The observatons from Fgs 11 suggest that the wreless provder should set a low prce when users socal tes are strong (evdenced by the popularty of socal servces), as the decrease of prce wll be outweghed by the ncrease of total usage resulted from the network effect, so that the total revenue ncreases Otherwse, the wreless provder should set a hgh prce, as cuttng the prce cannot stmulate suffcently more usage due to the congeston effect to compensate the prce decrease RELATED WORK There have been many studes on users behavors and the provder s prcng strategy when ether network effect (also known as postve externalty) or congeston effect s present, respectvely [5, 1, 1] In [5], dfferent prcng strateges of a provder have been studed where users behavors are only subject to network effect When users experence both network effect and congeston effect as consdered n ths paper, the couplng among users s very dfferent and more complex than when only network effect s present as n [5] Very few work have studed the case where both network effect and congeston effect are present [1] has studed users behavors when they experence both network effect and congeston effect However, t assumes that the network effect s the same for all users, whch does not capture the fact that users experence dfferent levels of network effect based on ther dverse socal tes as consdered n ths paper The socal aspect of moble networkng s an emergng paradgm for network desgn and optmzaton Socal contact patterns have been exploted for effcent data forwardng and dssemnaton n delay tolerant networks [15,1] Socal trust and socal recprocty have been leveraged n [17] to enhance cooperatve DD communcaton based on a coaltonal game A socal group utlty maxmzaton (SGUM) framework has been recently studed n [18 ], whch captures the mpact of moble users dverse socal tes on the nteractons of ther moble devces subject to dverse physcal relatonshps 7 CONCLUSION In ths paper, we have formulated the nteracton between moble users and a wreless provder as a Stackelberg game, by jontly consderng the network effect n the socal doman and the congeston effect n the physcal wreless doman For Stage II, we have analyzed users demand equlbrum gven a prce chosen by the wreless provder For Stage I, we have developed an algorthm to compute the optmal prce to maxmze the wreless provder s revenue We have also conducted smulatons usng real data to evaluate the performance, and drawn useful engneerng nsghts for the operaton of wreless provders For future work, we can examne other utlty functons, eg, a logarthmc functon for nternal utlty, yet the major engneerng nsghts should reman the same Another nterestng drecton s to study the provder s prcng strategy when t s allowed to dfferentate the prce for dfferent users In ths case, the prce offered to each user wll depend on ts socal nfluences to others based on the socal network ACKNOWLEDGEMENT The work of aowen Gong was supported n part by the US NSF grants CNS-177, ECCS-189, CNS-1117, DTRA grant HDTRA The work of Lngje Duan was supported n part by SUTD-ZJU Research Collaboraton Grant (Project no SUTD-ZJU/RES//1) The work of u Chen was supported n part by the fundng from Alexander von Humboldt Foundaton

9 8 REFERENCES [1] WeChat: The new way to connect [Onlne] Avalable: [] WhatsApp: Smple Personal Real tme messagng [Onlne] Avalable: [] Global Dgtal Statstcs [Onlne] Avalable: [] B Brscoe, A Odlyzko, and B Tlly, Metcalfe s law s wrong-communcatons networks ncrease n value as they add members-but by how much? IEEE Spectrum, vol, no 7, pp 9, [5] O Candogan, K Bmpks, and A Ozdaglar, Optmal prcng n networks wth externaltes, INFORMS Operaton Research, vol, no, pp 88 95, 1 [] D Fudenberg and J Trole, Game Theory MIT Press, 1991 [7] J B Rosen, Exstence and unqueness of equlbrum ponts for concave N-person games, Econometrca, vol, no, pp 5 5, 195 [8] F Ernst and S Gächter, Farness and retalaton: The economcs of recprocty, Journal of Economc Perspectves, vol 1, no, pp , [9] Gong, L Duan, and Chen, When network effect meets congeston effect: Leveragng socal servces for wreless servces, Techncal Report [Onlne] Avalable: [1] SNAP: Network datasets: Brghtkte [Onlne] Avalable: [11] P Erdos and A Reny, On the evoluton of random graphs, Publcatons of the Mathematcal Insttute of the Hungaran Academy of Scences, pp 17 1, 19 [1] J Nar, A Werman, and B Zwart, Explotng network effects n the provsonng of large scale systems, n Proc of IFIP Performance 11 [1] W Wu, T B Ma, and C S Lu, Explorng bundlng sale strategy n onlne servce markets wth network effects, n Proc of IEEE INFOCOM 1 [1] R Johar and S Kumar, Congestble servces and network effects, n Proc ACM Conference on Electronc Commerce 1 [15] P Costa, C Mascolo, M Musoles, and G P Pcco, Socally-aware routng for publsh-subscrbe n delay-tolerant moble ad hoc networks, IEEE JSAC, vol, no 5, pp 78 7, 8 [1] W Gao, Q L, B Zhao, and G Cao, Multcastng n delay tolerant networks: a socal network perspectve, n ACM MOBIHOC, 9 [17] Chen, B Proulx, Gong, and J Zhang, Socal trust and socal recprocty based cooperatve DD communcatons, n Proc ACM MOBIHOC 1 [18] Gong, Chen, and J Zhang, Socal group utlty maxmzaton game wth applcatons n moble socal networks, n Proc IEEE Allerton Conference on Communcaton, Control, and Computng 1 [19] Chen, Gong, L Yang, and J Zhang, A socal group utlty maxmzaton framework wth applcatons n database asssted spectrum access, n Proc IEEE INFOCOM 1 [] Gong, Chen, K ng, D-H Shn, M Zhang, and J Zhang, Personalzed locaton prvacy n moble networks: A socal group utlty approach, n Proc IEEE INFOCOM 15 [1] B Debreu, A socal equlbrum exstence theorem, Proceedngs of the Natonal Academy of Scences of the Unted States of Amerca, vol 8, no 1, pp 88 89, 195 [] I Glcksberg, A further generalzaton of the Kakutan fxed pont theorem, wth applcaton to Nash equlbrum, Proceedngs of the Amercan Mathematcal Socety, vol, no 1, pp 17 17, 195 [] K Fan, Fxed-pont and mnmax theorems n locally convex topologcal lnear spaces, Proceedngs of the Natonal Academy of Scences of the Unted States of Amerca, vol 8, no, pp 11 1, 195 [] R Horn and C Johnson, Matrx Analyss Cambrdge Unversty Press, 1985 APPENDI Proof of Theorem 1 To show the exstence of UDE, we make use of the followng lemma, whch shows that the Stage II game wth unbounded usage range s equvalent to that wth bounded usage range Lemma Under Assumpton 1, the Stage II game G {N, {u }, [, ) N } admts the same set of UDEs as the game G {N, {u }, [, x] N }, where x P s any number that satsfes x > max a p /(b + c g j c ) Proof: Let x be any UDE of game G and x be the largest n x, e, x x j, j If x >, usng the best response functon (), we have x = a p b + c + g j c b + c x a p j b + c + Usng Assumpton 1, t follows from (5) that x a p /(b + c g j c ) < x g j c b + c x Snce x s the largest n x, we have x j [, x], j N, and thus x [, x] N Therefore, as game G and game G have the same set of payoff functons and the strategy spaces n both games contan [, x] N, they have the same set of UDEs Usng a celebrated result n [1 ], the nfnte game G admts a UDE f the strategy space [, x] N s compact and convex, the payoff functon u (x, x ) s contnuous n x and x, and the payoff functon u (x, x ) s concave n x It s easy to check that all these condtons hold, and thus the game G admts a UDE Then t follows from Lemma that the Stage II game G admts a UDE Proof of Theorem We wll show that the UDE s unque by showng that the game G defned n Lemma s a concave game The Jacoban matrx u(x) of the payoff functon profle u(x) (5)

10 (u 1 (x),, u N (x)) of game G s gven by u(x) = = u 1 (x) x 1 u (x) x x 1 u N (x) x N x 1 u 1 (x) x 1 x u (x) x u N (x) x N x u 1 (x) x 1 x N u (x) x x N u N (x) x N b 1 c g 1 c g 1N c g 1 c b c g N c g N1 c g N c b N c = (B G) Usng Assumpton 1, t follows that [B G] [B G] j, where [M] j denotes the entry n the th row and jth column of matrx M Therefore, B G s strctly P dagonal domnant g [] It follows from the condton j c b < 1, that (B G) T s also strctly dagonal domnant Then we have that u(x) + u(x) T = (B G) (B G) T s strctly dagonal domnant Also observe that t s symmetrc It s known that a symmetrc matrx that s strctly dagonally domnant wth real nonnegatve dagonal entres s postve defnte [] Therefore, u(x) + u(x) T s negatve defnte It follows from [7, Theorem ] that u(x) s dagonally strctly concave Therefore, usng [7, Theorem ], game G has a unque UDE Proof of Theorem Let x (t) x (t) x, For any N, accordng to step n Algorthm 1, we have x (t+1) g j c b + c x(t) j g j c b + c x(t) j () Let x (t) be the l -norm of vector ( x (t),, x (t) N ), e, x (t) max x (t) Then, usng Assumpton 1 and (), we have x (t+1) max max = max g j c b + c x(t) g j c b + c g j c b + c j!! max! x (t) x (t) j g j Š c b P Accordng to Assumpton 1, we have max < 1 Then t follows that the algorthm results n a contracton mappng of x (t), and thus converges to the UDE Proof of Proposton If the UDE s postve, e, x 1 > and x >, accordng to, we have x > s the soluton to x 1 = a 1 p b 1 + c + g 1 c b 1 + c x, x = a p b + c + g 1 c b + c x 1 Solvng t, we have the expresson gven n the low prce regme Then observe that x 1 and x are both postve when p =, and decrease when p ncreases Also observe that x 1 = when p = p 1 a 1(b ) a (c g 1 ) b +g 1, and x = when p = p a (b 1 ) a 1 (c g 1 ) b 1 +g 1 We can check that p 1 p Therefore, when p > p = p th, we have x 1 > and x = Thus x 1 = a 1 p b 1 accordng to () Then we further observe that x 1 = x = when p > a 1 Proof of Lemma 1 We only prove the case when S = N, snce the case when S N can be proved smlarly Let Ḡ= B = b 1 + c b + c b N + c 7 5, g 1 c g 1N c g 1 c g N c 7 5 g N1 c g N c Snce B s a dagonal matrx wth postve dagonal entres, t s nvertble Let λ be any egenvalue of B 1 Ḡ wth v beng the correspondng egenvector Let v be the largest entry of v n absolute value, e, v v j, j Snce ( B 1 Ḡ)v = λv, t follows that λv = [ B 1 Ḡ] v j N v [ B 1 Ḡ] j v j j N g j c b + c < v where the last nequalty follows from Assumpton 1 It follows that the spectral radus of B 1 Ḡ s strctly less than 1 Snce each egenvalue of I B 1 Ḡ s equal to 1 λ where λ s an egenvalue of B 1 Ḡ, where I denotes the N N dentty matrx, t follows that I B 1 Ḡ has no egenvalue of, and thus s nvertble Thus B G = B Ḡ = B(I B 1 Ḡ) s also nvertble Proof of Proposton We frst show part 1) Suppose p > a and x > s the largest n x, e, x x j, j Usng the best response functon (), we have x = a p b + c + g j c b + c x j g j c b + c x < x where the last nequalty follows from Assumpton 1 Ths shows a contradcton Thus we have x =, Next we show part ) Let S be the set of users wth postve usage n x at prce For any / S, usng (), we

11 have x = a b + c + a b + c [G C],S(B S G S ) 1 1 S (7) For any p (, a], we next show that x wth x S = (a p)(b S G S ) 1 1 S and x =, / S s the UDE at prce p We observe that for any S, x s ts best response at x For any / S, usng (7), we have a p b + c + a p b + c [G C],S(B S G S ) 1 1 S = x, and thus s user s best response at x Proof of Proposton The proof of part 1) s the same as the proof of part 1) of Proposton except that a should change to max a Now we show part ) For any prce p [, max a ], the usage of the set of users S wth postve usage at the UDE (f they exst) P s gven by () Observe that the usage demand a p + g j c b b x j of any user at the UDE s a contnuous functon of prce p and other users usage x j Therefore, when the prce p ncreases by a suffcently small amount to p, the set of user wth postve usage at the UDE s stll the set S, and thus ther usage s stll gven by () except wth p replaced by p Therefore, the set of user wth postve usage s the same at any prce n a contnuous prce nterval Then the desred result follows Proof of Proposton 5 We only prove the case when any g j ncreases, snce the cases when a ncreases, any b decreases, or c ncreases can be proved smlarly Then t suffces to prove the case when any g j ncreases by any small amount Let G be a symmetrc matrx Let x be the UDE under G and S be the set of users wth postve usage n x It s easy to check that the UDE s a contnuous functon of the matrx G Then we can always fnd a symmetrc matrx G wth [G ] j [G] j,, j and at least one strct nequalty, such that the set of users wth postve usage at the UDE x under G s also S Therefore, usng the best response functons (), we have B S x S = (a p)1 S + G S x S (8) B S x S = (a p)1 S + G Sx S (9) Subtractng (8) from (9), we have B S(x S x S) = G S(x S x S) + G Sx S (1) where G S G S G S Accordng to Lemma 1, B S G S s nvertble Then t follows from (1) that x S x S = (B S G S ) 1 G S x S (11) On the other hand, t follows from (8) that x S = (a p)(b S G S ) 1 1 S (1) Usng (11) and (1), we have t(x ) t(x ) = 1 T S (x S x S) = 1 T S (B S G S ) 1 G S x S = [(B S G S ) 1 1 S ] T G S x S = 1 a p (x S) T G Sx S where the thrd equalty s due to the fact that (B S G S ) 1 s symmetrc snce B S G S s symmetrc Snce a > p and x S, G, x S only have nonnegatve entres, t follows that t(x ) t(x ) Proof of Theorem We wll show that gven the current crtcal prce p and the set of users S wth postve usage at the UDE at the prce p, each teraton from step 8 to step fnds the next crtcal prce p, and the optmal prce and revenue n the prce nterval [p, p] For any S wth x >, t follows from () that x = [(B S G S ) 1 ] a S p[(b S G S ) 1 ] 1 S > (1) Therefore, x decreases when the prce p ncreases f [(B S G S ) 1 ] 1 S > (1) For any / S wth x =, t follows from () that the usage demand of user s no greater than such that [G C],S(B S G S) 1 (a S p1 S) + a p = [G C],S (B S G S ) 1 a S + a p([g C],S (B S G S ) 1 1 S + 1) (15) Therefore, the usage demand of user / S ncreases when the prce p ncreases f [G C],S (B S G S ) 1 1 S < 1 (1) Usng (1), the prce at whch a user S changes ts usage from postve to s p [(B S G S ) 1 ] a S [(B S G S ) 1 ] 1 S where S s the set of users such that (1) holds Usng (15), the prce at whch a user / S changes ts usage from to postve s p [G C],S(B S G S ) 1 a S + a [G C],S (B S G S ) 1 1 S + 1 where S s the set of users such that (1) holds Therefore, the next crtcal prce p s p = mn p S S Usng (), the revenue R s gven by R(p) = p1 T S x S = p1 T S (B S G S ) 1 (a S p1 S ) whch s a concave quadratc functon of p By settng R(p) =, we obtan that the optmal prce p n the prce p nterval [p, p] that maxmzes the revenue s p = ˆp 1T S (B S G S) 1 a S 1 T S (B S G S ) 1 1 S (17) f ˆp [p, p] If ˆp / [p, p], the optmal prce s p = p f ˆp < p, or p = p f ˆp > p Thus the optmal revenue r n the prce nterval [p, p] s r = p 1 T S (B S G S ) 1 (a S p 1 S )

12 Then the optmal prce and revenue n the entre range [, max a ] s found by comparng the optmal revenue for all the teratons from step 8 to step