Maximizing Influence in a Competitive Social Network: A Follower s Perspective

Transcription

1 Maximizing Influnc in a Comptitiv Social Ntwork: A Followr s Prspctiv [Extndd Abstract] Tim Carns, Chandrashkhar Nagarajan, Stfan M. Wild, and Ank van Zuyln School of Oprations Rsarch and Information Enginring Cornll Univrsity Ithaca, NY 4853 {tcarns, chandra, stfan, ank}@ori.cornll.du ABSTRACT W considr th problm facd by a company that wants to us viral markting to introduc a nw product into a markt whr a compting product is alrady bing introducd. W assum that consumrs will us only on of th two products and will influnc thir frinds in thir dcision of which product to us. W propos two modls for th sprad of influnc of compting tchnologis through a social ntwork and considr th influnc maximization problm from th followr s prspctiv. In particular w assum th followr has a fixd budgt availabl that can b usd to targt a subst of consumrs and show that, although it is NP-hard to slct th most influntial subst to targt, it is possibl to giv an fficint algorithm that is within 63% of optimal. Our computational xprimnts show that by using knowldg of th social ntwork and th st of consumrs targtd by th comptitor, th followr may in fact captur a majority of th markt by targting a rlativly small st of th right consumrs. Catgoris and Subjct Dscriptors F. [Analysis of Algorithms & Problm Complxity]: Nonnumrical Algorithms and Problms; G.. [Discrt Mathmatics]: Combinatorics combinatorial algorithms Gnral Trms Algorithms, Prformanc, Thory Kywords Approximation Algorithms, Social Ntworks, Viral Markting, Ntwork Analysis, Targtd Markting Rsarch supportd by NSF grants CCR-0635 & DMI Rsarch supportd by NSF grant CCF Rsarch supportd by a DOE Computational Scinc Graduat Fllowship undr grant numbr DE-FG0-97ER5308 and by NSF grant CCF Prmission to mak digital or hard copis of all or part of this work for prsonal or classroom us is grantd without f providd that copis ar not mad or distributd for profit or commrcial advantag and that copis bar this notic and th full citation on th first pag. To copy othrwis, to rpublish, to post on srvrs or to rdistribut to lists, rquirs prior spcific prmission and/or a f. ICEC 07, August 9, 007, Minnapolis, Minnsota, USA. Copyright 007 ACM /07/ $ INTRODUCTION Th sprad of a nw ida or product is oftn studid by modling a social ntwork as a graph whr th nods rprsnt individuals, and dgs rprsnt intractions btwn individuals. Ths intractions could includ th rcommndation of a particular product and such rcommndation ntworks and thir ffcts on consumr purchasing hav rcntly bn analyzd in [5] and [6]. Furthr, thr has bn rcnt statistical support that such ntwork linkag can dirctly affct product adoption [9]. Basd on ths mpirical studis, w can formulat assumptions on how popl affct th popl thy intract with. W can thn us ths graphs to answr qustions such as: If customrs influnc ach othr in thir dcisions to buy products, which customrs should b targtd to maximiz th xpctd profit of a nw product? and How larg of a consumr bas nds to b targtd for a nw tchnology, product, or ida to captur a significant shar of th markt? Motivatd by th dclining ffctivnss of traditional mass markting tchniqus [5], many rcnt paprs hav studid ths and similar typs of qustions. Th algorithmic problm of dsigning viral markting stratgis, markting tchniqus which xploit pr-xisting social ntworks to rach consumrs, was studid by Richardson and Domingos [], and Kmp, Klinbrg and Tardos [, 3]. Thir rsarch builds on a word-of-mouth approach xamind in a markting contxt by Goldnbrg t al. in [8]. In th aformntiond works, th producr of a nw product is assumd to hav th ability to influnc a particular st of consumrs within th social ntwork ithr through targtd advrtising, providing fr sampls, or adding montary incntiv to adopt th nw product. If ths popl influnc som of thir frinds to also try th product, and ths frinds in turn rcommnd it to othrs, and so forth, th producr can crat a cascad of rcommndations. Th qustion thn bcoms how to choos an initial subst of so-calld arly adoptrs to maximiz th numbr of popl that will vntually b rachd, and hnc b likly to purchas th product. Th siz of th substs allowd is assumd to b limitd du to markting budgt constraints. Kmp t al. dvlop gnral modls for th sprading of influnc, show that finding th most influntial st of nods is NP-hard, and giv an approximation algorithm for finding a st of nods that approximatly maximizs th xpctd influnc. Th modls dvlopd by Kmp t al. assum that thr 35

2 is only on company introducing a product. Howvr, producrs of consumr tchnologis oftn must introduc a nw product into a markt whr a comptitor will offr a comparabl product. Th introduction of Nintndo s Wii, to compt with Sony s Playstation 3, and Blu-ray discs, compting with Toshiba s HD DVD, ar rcnt canonical xampls of such bhavior. Whn adoption of th tchnology is not fr, it is unlikly that a typical consumr will us both products. Furthrmor, vn if a compting product is suprior, consumrs ar oftn rluctant to switch tchnologis if thy must bar a cost of transition which may outwigh any dirct bnfits of th tchnology [6]. Th qustion whthr in this stting a compting product can surviv and will b adoptd by a significant fraction of th markt, or if it will vntually disappar, has bn studid in numrous works, including [0], [7], and [3]. It is not always th cas that th product with th largst numbr of arly adoptrs can translat this initial dg into markt dominanc. A classical xampl whr such tipping did occur is th dmis of th BETA format du to th VHS format s initial popularity. Howvr, Katz and Shapiro not that consumr htrognity coupld with distinct faturs of rival products tnds to limit tipping in markts whr consumrs car mor about a product s faturs than its ovrall prvalnc []. Hnc it is an intrsting qustion to considr how a company with a smallr markting budgt may ffctivly infiltrat a markt in which a strongr compting company is also prsnt. Historically, comptition btwn two products has largly bn addrssd from an conomic modling prspctiv and focusd on aras such as markt quilibrium. For xampl, in [] and [3], primarily ntwork-indpndnt proprtis ar mployd to modl th propagation of two tchnologis through a markt. Tomochi t al. [3] offr a mor gamthortic approach which rlis on th ntwork for spatial coordination gams. Howvr, thy do not addrss th problm of taking advantag of th social ntwork and viral markting whn introducing a nw tchnology into a markt. In this papr, w study th algorithmic problm of how to introduc a nw product into an nvironmnt whr a compting product is also bing introducd. W focus on th cas whn a company can kp itslf hiddn from a comptitor until th momnt of introduction. W assum that th company has a fixd budgt for targting consumrs and knows who its comptitor s arly adoptrs ar ithr through xtnsiv markt rsarch or industrial spionag. W first dvlop two modls for th sprad of adoption of th two products through th ntwork. W show that finding th most influntial st of a givn siz for th company to targt th st that maximizs th xpctd numbr of popl that will adopt th nw product is NP-hard undr th proposd modls in this stting. From a gam thortic point-of-viw, this can also b viwd as calculating th company s bst rspons to a comptitor s mov in a Stacklbrg gam [7]. Following Kmp t al. w show that using wll known rsults on submodular functions [8], w can giv a ( ε)-approximation algorithm for finding th most influntial st of nods. Additionally, using a rsult of Sviridnko [], w gnraliz th allowd substs to b limitd basd upon cost rathr than simply siz, hnc allowing diffrnt costs to b associatd with targting diffrnt substs of customrs. W will mpirically show that a company can obtain a largr markt shar than its unsuspcting comptitor vn if th comptitor has a much largr markting budgt. Furthr, w show that knowing who th comptitor s arly adoptrs ar, hnc bing abl to apply our algorithm, will allow th company to captur a givn prcntag of th markt using a much smallr markting budgt. In th squl w us th words tchnology and product synonymously. W discuss usful rsults from rlatd work in Sction. Building on ths rsults, w dscrib th modls w dvlopd for th sprad of two compting tchnologis in Sction 3 and th rsults drivd for ths modls in Sction 4. In Sction 5 w giv th rsults of som numrical simulations of th bhavior of ths modls, and w prsnt conclusions and furthr rsarch dirctions in Sction 6.. BACKGROUND W bgin by rcalling som xisting rsults cntral to th prsnt work. Submodular function maximization: Givn a ground st V, a function f : V R is said to b submodular if f(s {v}) f(s) f(t {v}) f(t ) for all v V and sts S T V. W furthr say that f is monoton if f(s {v}) f(s) for all v V and substs S V. For a non-ngativ, submodular, and monoton function f, and th optimization problm max {f(i) : I = k, I V }, () th grdy Hill Climbing Algorithm rpatdly adds th lmnt from V that givs th gratst improvmnt, by solving max {f(i {v}) :v V I} () until I = k. In [8] Nmhausr t al. show that hill climbing yilds a ( ) -approximation: if I is th st found by th ( Hill Climbing ) Algorithm, and I maximizs (), thn f(i) f(i ). This rsult has bn xtndd [] to show that for any ε>0, thr is a γ>0such that whn using a ( + γ)-approximation of f( ) in (), w obtain a ( ε)- approximation. Sviridnko rcntly gnralizd th rsult from Nmhausr t al. to includ problms of th form () with an additional knapsack-typ constraint []. In particular, for a st of nonngativ wights {c i : i V } and a budgt B 0, w now considr th problm: { max f(i) : } c i B,I V, (3) i I whr f is again a non-ngativ, submodular, and monoton st function. An xtnsion of hill climbing, itrativly adding to I lmnts v V I which maximiz { f(i {v}) f(i) max : c v + } c i B, (4) c v i I until c v >B i I ci for all v V I, is dscribd in [4]. Sviridnko ( ) showd that this vrsion of hill climbing yilds a -approximation to (3). Influnc maximization on a ntwork in th singl tchnology cas: Th sprad of a singl tchnology through a ntwork has bn approachd using diffrnt diffusion modls (s for xampl [, 3,, 4]). Hr w dscrib th indpndnt cascad modl introducd by Kmp t al. [] which rsmbls th modls w will dvlop in th cas of compting tchnologis. 35

3 W assum som st of nods I initially uss th tchnology. Th diffusion procss thn unfolds in discrt tim stps. Whn a nod u first adopts th tchnology, it gts a singl chanc to mak its nighbor v adopt th tchnology. It succds with probability p uv indpndntly of th history so far. In th nxt tim stp, th nods which just adoptd th tchnology gt a chanc to influnc thir nighbors and so on. Not that th procss is progrssiv: onc a nod has adoptd th tchnology, it will not go back to th stat of not having adoptd it. Th quantity of intrst is thn th influnc function σ(i), signifying th xpctd numbr of nods that vntually adopt th tchnology givn th initial st of adoptrs I. In [] Kmp t al. addrss how to choos an initial st I of som fixd siz k to maximiz σ(i). Kmp t al. prviously showd that solving () whn f is th influnc function σ is NP-hard but that σ is submodular. Hnc if σ can b approximatd (say with numrical simulations) arbitrarily wll, thn for any givn ε>0, hill climbing givs a ( ε) -approximation algorithm for finding an influntial initial k-st I. Our rsults: W propos two modls for th simultanous diffusion of two compting tchnologis on any ntwork givn an initial st of arly adoptrs for ach tchnology. Influnc functions σ(i A I B) ar dfind to quantify th succss of a tchnology s choic of initial adoptrs. Whil th proposd modls for diffusion ar concptually simpl, w show that maximizing such influnc functions subjct to a budgt is computationally intractabl. Howvr, in ach cas w ar abl to show that th influnc function is nonngativ, submodular, and monoton, and hnc hill climbing provids an approximation algorithm. W hav also gnralizd ths rsults to addrss htrognous costs for targting consumrs. 3. MODELING THE DIFFUSION OF TWO TECHNOLOGIES W now xtnd th indpndnt cascad modl to th cas of two compting tchnologis. In particular, w propos two modls for dscribing how two tchnologis simultanously diffus ovr a givn ntwork. Consumrs ar again modld as nods in a ntwork and links btwn nods rprsnt intraction btwn consumrs. W assum that our ntwork is an undirctd graph G = (V,E) with V = n nods and E = m dgs. Nods can tak on on of thr stats A and B rfrring to th two tchnologis of intrst, and C dnoting that nithr tchnology is adoptd. W can spcify two initial sts of nods a st of initial adoptrs of A, I A V, and a st of initial adoptrs of B, I B V (with th implicit assumption that I C = V (I A I B)). W assum that I A I B =. W assum that onc a nod has chosn a tchnology, it will not chang to anothr tchnology, but that nods that ar using on of th two tchnologis can influnc thir nighbors that ar not using ithr tchnology in thir dcisions to adopt on of th two tchnologis. As in th indpndnt cascad modl for a singl tchnology, w assum that if u has adoptd a particular tchnology, thn u influncs nighbor v with probability p uv. Hnc- Th rsults in this papr can b asily xtndd to th mor gnral dirctd cas. forth, w say that an dg is activ with probability p uv. Howvr, it is now possibl that v is influncd by multipl nighbors that us diffrnt tchnologis. W will propos two modls that govrn this diffusion of tchnologis A and B, starting from th sts of initial adoptrs, givn th st of activ dgs E a of th ntwork. In othr words, th modls w dvlop oprat on a random subgraph of th social ntwork G, whr ach dg is includd indpndntly with probability p uv. Our first modl will dscrib th diffusion of a tchnology whr th product/tchnology itslf can only b obtaind from an initial adoptr, and a consumr who bcoms intrstd in th tchnology (i.. th corrsponding nod is influncd by a nighbor via an activ dg) will pick on of th closst arly adoptrs at random. In th scond modl, th tchnology availability is not tid to th ntwork, and th consumr who bcoms intrstd in th tchnology will choos on of its nighbors and adopt th sam tchnology as this nighbor. Givn such a diffusion modl, and th assumption that initially a st of consumrs, I B, is alrady using tchnology B, companya would lik to choos a st of k consumrs, I A, to targt so as to maximiz th xpctd numbr of consumrs rachd vntually. Lt th influnc function f(i A I B) b th xpctd numbr of consumrs that will adopt tchnology A, givn that initially th st I A is using tchnology A and th st I B is using tchnology B. W ar hnc aftr a solution of th influnc maximization problm: max {f(i A I B):I A V I B, I A = k}. (5) If th cost of targting consumrs varis from consumr to consumr, a company may instad wish to maximiz its rvnu subjct to som budgt B. Givn non-ngativ costs {c i : i V }, th mor gnral from of (5) is thn: max f(ia IB) :IA V IB, c i B. (6) i I A For as of xposition, throughout th squl w supprss ths costs and will assum that c v = for all v V. 3. A Distanc-basd modl Our first modl is rlatd to comptitiv facility location [5] on a ntwork. In this modl, th location of a nod in th ntwork is important, as wll as th connctivity of a nod. Th ida is that a consumr will b mor likly to mimic th bhavior of an arly adoptr if thir distanc in th social ntwork is small. W assum that for ach dg (u, v) E, w ar also givn a lngth d uv. If no lngth is spcifid w assum that all dgs hav lngth. In th following w will assum all dgs hav lngth, howvr th rsults can asily b xtndd for arbitrary non-ngativ dg lngths. W lt I = I A I B b th st of all initial adoptrs. Lt d u(i,e a) dnot th shortst distanc from u to I along th dgs in E a, with th notation d u(i,e a)= if and only if u is not connctd to any nod of I using only activ dgs. If d u(i,e a) <, lt ν u(i A,d u(i,e a)) If d u(i,e a)=, wsaythatu will adopt nithr tchnology (stat C) bcaus it is not connctd to any of th initial adoptrs by activ dgs. Hncforth, w assum that any nod u undr considration is connctd to som v I in G. 353

4 I A v Figur : Givn th st of activ dgs drawn, th probability that nod v adopts tchnology A is in th distanc-basd 3 modl, and in th wav propagation modl. I B and ν u(i B,d u(i,e a)) b th numbr of nods in I A and I B, rspctivly, at distanc d u(i,e a)fromu along dgs in E a. Givn that d u(i,e a) is th shortst distanc from u to I along th activ dgs of G, w will say that nod u adopts tchnology i {A, B} with probability ν u(i i,d u(i,e a)) ν u(i A,d u(i,e a)) + ν. (7) u(i B,d u(i,e a)) Not that conditiond on st E a nodu is thus only influncd by nods in I A and I B that ar at distanc d u(i,e a). W not that ths ar wll-dfind (conditional) probabilitis that sum to on, sinc at last on of ν u(i A,d u(i,e a)) and ν u(i B,d u(i,e a)) is strictly positiv. In this modl th xpctd numbr of nods which adopt A will b dnotd by [ ρ(i A I B)=E u V ν u(i A,d u(i,e a)) ν u(i A,d u(i,e a)) + ν u(i B,d u(i,e a)) ], whr th xpctation is ovr th st of activ dgs. W fix I B and try to dtrmin I A so as to maximiz th xpctd numbr of nods that adopt tchnology A: max{ρ(i A I B):I A (V I B), I A = k}. (8) 3. Wav propagation modl Although both of th modls w propos hr rduc to th indpndnt cascad modl of Kmp t al. [] if thr is no comptition (if w lt I B = ), our scond modl for propagation is closr in spirit to th indpndnt cascad modl. W motivat this modl through th xampl shown in Figur. In this xampl, with th dgs shown bing activ, our prvious distanc-basd modl givs nod v a probability of of adopting tchnology A, vn though it has only two 3 nighbors, on of which adopts tchnology A and on of which adopts tchnology B. Undr th altrnativ modl prsntd hr a nod copis th tchnology adoption of a nighboring nod randomly chosn from th st of its nighbors that ar closst to th initial sts (I A,I B). In th xampl, and givn th st of activ dgs shown, this corrsponds to giving nod v a probability of adopting A and of adopting B. W can think of th propagation as happning in discrt stps. In stp d, all nods that ar at distanc at most d from som nod in th initial sts hav adoptd tchnology A or B, and all nods for which th closst initial nod is farthr than d do not hav a tchnology yt (whr th distanc is again with rspct to activ dgs). Th nods at a distanc d from th initial sts now choos on of thir nighbors that ar at distanc d indpndntly at random, and adopt th sam tchnology as this nighbor. As in th prvious sction, w assum that th nod undr considration is in th sam connctd componnt of at last on of th nods u I. Formally, lt P (v I A,I B,E a) b th probability that nod v adopts tchnology A whn th initial sts for tchnologis A and B ar I A and I B, rspctivly, and th st of activ dgs is E a. Lt u b a nod for which th closst nod in I = I A I B is at distanc d. Lt S b th st of nighbors of u that ar at distanc d fromi, whr all distancs ar again with rspct to activ dgs. Thn P (u I A,I B,E a)= P (v IA,IB,Ea). (9) For initial sts I A,I B,lt [ ] π(i A I B)=E P (v I A,I B,E a) v V (0) dnot th xpctd numbr of nods that adopt tchnology A. ForfixdI B, w sk a solution to: max{π(i A I B):I A (V I B), I A = k}. () 4. APPROXIMATION ALGORITHMS FOR INFLUENCE MAXIMIZATION For ach of th diffusion modls proposd in Sction 3 w now show that th dcision vrsions of (8) and () ar NP-hard but that th corrsponding influnc functions ar nonngativ, monoton and submodular. It will thn follow from [8] and [] that w can us a grdy hill-climbing algorithm to gt a ( )-approximation algorithm for ths problms. In gnral it will not b possibl to xactly solv th subproblms () and (4), as this rquirs xact valuation of ρ( I B)andπ( I B). Howvr, using sampling w can gt arbitrarily clos approximations of th valus ndd in () and (4). This thn allows us to obtain ( ε)- approximation algorithms for both modls []. Thorm. For any givn I B with V I B k, th Hill Climbing Algorithm givs a ( ε) -approximation algorithm for (8). Proof. Givn inputs (I A,I B), and a st of activ dgs E a, ρ(i A I B) can b fficintly valuatd using an algorithm which rlis on a singl all-pairs shortst paths computation and has ovrall complxity O( V 3 ). Using sampling, w can thn approximat ρ(i A I B)=E [ρ(i A I B) E a]towithin (+γ) for any γ>0 (whr th running tim dpnds on ). γ Hnc w can implmnt th grdy hill-climbing algorithm using ( + γ)-approximat valus for ρ(i A {v} I B) in polynomial tim. Monotonicity and submodularity of ρ( I B) will b shown in Lmma and Lmma 3, rspctivly. Th approximation guarant is thn an immdiat consqunc of th rsults in Sction. Thorm. For any givn I B with V I B k, th Hill Climbing Algorithm givs a ( ε) -approximation algorithm for (). Proof. Givn inputs (I A,I B) and a st of activ dgs E a, π(i A I B) can b fficintly valuatd using an algorithm which rlis on a singl all-pairs shortst path computation 354

5 and has ovrall complxity O( V 3 ). W can approximat π(i A I B)=E [π(i A I B) E a]towithin(+γ) for any γ>0, hnc w can implmnt th grdy hill-climbing algorithm using ( + γ)-approximat valus for π(i A {v} I B) in polynomial tim. Monotonicity and submodularity of π( I B) will b shown Lmma 5 and Lmma 6, rspctivly. Th approximation guarant is again an immdiat consqunc of th rsults in Sction. Bfor procding, w not that to show NP-hardnss and th dsird proprtis of th influnc functions, it suffics to considr th cas whn p uv = for all dgs (or quivalntly, E a = E): NP-hardnss of a spcial cas clarly implis NPhardnss of th mor gnral cas, and th xpctd valu of a function of E a is nonngativ, monoton, and submodular if for any E a th function is nonngativ, monoton, and submodular. For as of xposition, w thrfor rstrict ourslvs to this spcial cas in th rmaindr of this sction. 4. A Distanc-basd modl Lt th dcision vrsion of (8) b to dtrmin if thr is asti A of siz k with ρ(i A I B) M for any M Q. W thn hav th following rsult. Thorm 3. Th dcision vrsion of (8) is NP-hard 3. Proof. Givn a ground st of lmnts E = { i : i n} and a collction of sts S = {s i : i m} such that ach s i E, th dcision vrsion of th st covr problm asks if thr is a collction of k sts covring all lmnts in E. Without loss of gnrality w assum that vry lmnt is covrd by at last on st and that k<min(m, n). W rduc th NP-hard st covr dcision problm to th dcision vrsion of (8). Givn an instanc (S, E, k) of st covr, w construct a graph H as follows. W add a nod s i for ach st s i S and a nod j for ach lmnt j E. W add an dg (s i, j) if and only if j s i. W add an additional nod x and connct it to ach j through anothr nod d j (s Figur ). Lastly, for a constant κ>0 to b spcifid in th subsqunt lmma, w construct a clustr C j of κ nods for ach j =,...,n and connct ach of th nods in C j to j. Th following lmma complts th rduction by spcifying th valu of κ. Lmma. Lt κ>(k +)(m + n) and I B = {x}. Thr is a collction of k sts which covr E if and only if thr is a st I A of k nods in th graph H such that ρ(i A I B) n(κ +). Proof. If thr a collction of k sts covring E thn tak I A to b th k nods corrsponding to thos sts. This givs ρ(i A I B) n(κ + ) by th following argumnt. Each of th nods j is adjacnt to on of th nods in I A sinc th corrsponding sts form a covr. Each j and th nods in ach clustr C j adopt A with probability on sinc initial adoptrs of A ar th closst for ths nods. This rsults in at last n(κ + ) nods with tchnology A. If thr is no collction of k sts that covr E thn w prov that no st I A of k nods can b th initial adoptrs of A and still achiv ρ(i A I B) n(κ + ). Considr th nods,..., n. Thr xists a nod j which adopts tchnology B 3 Kmp t al. showd that this problm is NP-hard for th dirctd cas whn I B =. Howvr th problm is not NP-hard for th undirctd cas whn I B =. x s s s 3 s m dn d d d 3 d n C C C3 Cn C 4 Figur : Graph H for st covr rduction. with probability at last sinc any st IA of siz k cannot k+ b within a distanc of from all i (by th construction of H andlackofacovr). Thisimplisthat j and all nods in C j adopt tchnology B with probability at last k+.so ρ(i A I B) (n )(κ +) + m + n k + }{{}}{{} from s,...,s m and d,...,d n from,..., n and C,...,C n = n(κ +)+(m + n) (κ +)<n(κ +). k + Having shown th hardnss of (8), w now turn to th Lmmas rquird in Thorm and show that th influnc function ρ is both monoton and submodular. W assum without loss of gnrality that vry dg is activ with probability, and for as of notation, w will writ d u(i)instad of d u(i,e a). Furthrmor, w will drop th subscript u whn u is clar from th contxt. Lmma. For any I B, ρ(i A I B) is a monoton function of I A. Proof. For a fixd u V and initial st I B,itsufficsto show for any v V I B,andanyI A V, th probability that u adopts A givn th initial st I A is at most th probability that u adopts A whn th initial st is I A {v}, that is: ν(i A,d(I)) ν(i A,d(I)) + ν(i B,d(I)) ν(i A {v},d(i {v})) ν(i A {v},d(i {v})) + ν(i B,d(I {v})). W not that th shortst distanc from u to a nod in I is not smallr than th shortst distanc from u to a nod in I {v}, sod(i) d(i {v}). Now, if d(i {v}) <d(i), thn ν(i B,d(I {v})) = 0, so th right hand sid is, and th inquality clarly holds. Othrwis, ν(i B,d(I {v})) = ν(i B,d(I)), and ν(i A,d(I)) = 355

6 ν(i A,d(I {v})) ν(i A {v},d(i {v}) and th inquality holds sinc for ral numbrs c a 0andb>0, a a+b. c c+b Lmma 3. For any I B, ρ(i A I B) is a submodular function of I A. Proof. For a st of initial adoptrs of A, S, andanod x V (S I B), w dfin th incras in th probability that nod u adopts tchnology A whn adding x to th initial st S as: P (u, S, x) ν(s {x},d(s {x} I B)) = ν(s {x},d(s {x} I B)) + ν(i B,d(S {x} I B)) ν(s, d(s I B)) ν(s, d(s I B)) + ν(i B,d(S I. B)) W nd to show that for any nod u V and S T V, P (u, S, x) P (u, T, x). Lt d = d(s), d = d(t ), d 3 = d(i B). Sinc S T, d d. W analyz thr cass: Cas (d d d 3): If d(u, x) > d 3, adding x dos not chang th probability of u adopting A. So P (u, S, x) = P (u, T, x) =0. Ifd(u, x) <d 3, thn adding x maks u adopt A with probability. It thn follows from th monotonicity of ρ that ν(s, d(s I B)) P (u, S, x) = ν(s, d(s I B)) + ν(i B,d(S I B)) ν(t,d(t I B)) ν(t,d(t I B)) + ν(i B,d(T I B)) = P (u, T, x). If d(u, x) =d 3,thnν(S, d(s I B)) = ν(s, d 3)andν(T,d(T I B)) = ν(t,d 3) and ths both incras by if x is addd. Furthrmor ν(i B,d(X I B)) = ν(i B,d 3) for X {S, S {x},t,t {x}}. So w nd to show that ν(s, d 3)+ ν(s, d ν(s, d 3) 3)++ν(I B,d 3) ν(s, d 3)+ν(I B,d 3) ν(t,d 3)+ ν(t,d ν(t,d 3) 3)++ν(I B,d 3) ν(t,d. 3)+ν(I B,d 3) This quation can b asily chckd to b tru using th fact that ν(s, d 3) ν(t,d 3). Cas (d 3 >d d ): In this cas ν(i B,d(X I B)) = 0 for X {S, S {x},t,t {x}}. In this cas th probability that u adopts A is for initial sts X {S, S {x},t,t {x}}, so P (u, S, x) =P (u, T, x) =0. Cas 3 (d d 3 >d ): Sinc d 3 >d, u will adopt tchnology A with probability if th initial st is T or T {x}. So P (u, T, x) = 0, and P (u, S, x) 0 holds by Lmma. 4. Wav propagation modl Sinc it suffics to show that π(i A I B) is monoton and submodular in th spcial cas that vry dg in th graph is activ with probability, w will rstrict ourslvs to this cas and writ P (u, I A,I B) instad of P (u I A,I B,E a) for th probability that nod u adopts tchnology A whn th initial sts for tchnology A and B ar I A and I B, rspctivly and th st of activ dgs is E a. Lt th dcision vrsion of () b to dtrmin if thr is asti A of siz k with π(i A I B) M for any M Q. W thn hav th following rsult. Thorm 4. Th dcision vrsion of () is NP-hard. Proof. W rduc th NP-hard st covr dcision problm to th dcision vrsion of () as in Thorm 3. Givn an instanc (S, E, k) of st covr, w construct th sam graph H constructd in th proof of Thorm 3. Th following lmma complts th proof. Lmma 4. Lt κ>(m +)(m + n) and I B = {x}. Thr is a collction of k sts which covr E ifandonlyifthr is a st I A of k nods in th graph H such that π(i A I B) n(κ +). Proof. If thr a collction of k sts covring E thn tak I A to b th k nods corrsponding to thos sts. This givs π(i A I B) n(κ + ) by th sam argumnt givn in th proof of Lmma. If thr is no collction of k sts that covr E thn w prov that no st I A of k nods can b th initial adoptrs of A and still achiv π(i A I B) n(κ + ). Considr th nods,..., n. Any st I A of siz k cannot b within a distanc of from all j (by th construction of H and lack of a covr). So thr xists a nod j which adopts tchnology B with probability at last bcaus on of its nighbors m+ d j has P (d j I A,I B) = 0 and is at distanc from I and at most m + of its nighbors ar at distanc from I. So P ( j I A,I B) m. This implis that j and all th nods m+ m in C j adopt tchnology A with probability at most.so m+ for any initial st I A of siz k, π(i A I B) P (v, I A,I B) v V (m + n) }{{} for v {s,...,s m} {d,...,d n} +(n )(κ +) }{{} for v i C i,i j +(κ +)P ( j I A,I B) }{{} v j C j m (m + n)+(n )(κ +)+(κ +) m + < (n )(κ +)+(m +)(m + n) < n(κ +). W again bnfit from th valuabl proprtis of monotonicity and submodularity. Lmma 5. For any I B, π(i A I B) is a monoton function of I A. Proof. To prov monotonicity w nd to show that P (u S x, I B) P (u S, I B) for all x V I B. W mploy th sam notation as in Sction 4. and lt n(v) ={u :(u, v) E} dnot th nighbors of nod v. Not that d(u, S x I B) d(u, S I B). If d(u, S x I B) <d(u, S I B)thn P (u S x, I B)=P (u S, I B) which provs monotonicity. So th intrsting cas is whn d(u, S x I B)=d(u, S I B). W prov P (u S x, I B) P (u S, I B) for this cas by induction on th distanc d = d(u, S I B). Bas cas: d =. Ifx is not a nighbor of u thn P (u S x, I B)=P (u S, I B). If x is a nighbor of u thn P (u S x, I B)= + n(u) S n(u) S + n(u) (S I B ) n(u) (S I B = P (u S, IB). ) Induction stp: Now w prov monotonicity for nods u such that d(u, S I B)=d assuming monotonicity for all th nods v with d(v, S I B) <d.ltsbth st of nighbors 356

7 of u which ar at a distanc d froms I B. Lt K b th st of nighbors of u which ar at a distanc d from x but at a distanc gratr than d froms I B. Lt K = K. Notthatallv Khav P (v S x, I B)=. Th probability of u accpting tchnology A is thn: P (u S x, I B) = K + P (v S x, IB) P (v S x, IB) P (v S, IB) = P (u S, I B). Th scond inquality follows from th induction assumption that monotonicity holds for th nods v with d(v, S I B) <d and th fact that all nods in S ar at a distanc d from S I B. Lmma 6. For any I B, π(i A I B) is a submodular function of I A. Proof. W will show that for two sts S T V I B, and a nod x V I B,whavthat u V P (u S x, I B) P (u S, I B) P (u T x, I B) P (u T,I B) by induction on d = d(u, x). If d = 0, thn clarly th inquality holds. Suppos it holds for any v such that d(v, x) = d. As in th proof of Lmma 3, w considr diffrnt cass for th distanc from u to th closst nod in S, T and I B. Lt d = d(u, S),d = d(u, T ),d 3 = d(u, I B). It is asy to s that th proof of Lmma 3 also works for our altrnativ modl, xcpt for th cas whn d d d 3 and d 3 = d(u, x). Lt S b th st of nighbors of u for whom th closst nod from S I B is at distanc d sothat: P (v S, IB) P (u S, I B)=. Not that ach nighbor of u that is at distanc d from x but is at distanc gratr than d from th nods in S I B, adopts A with probability. Lt K b th numbr of such nods, thn: P (u S x, I K + P (v S x, IB) B)=. Thrfor th diffrnc in th probability of u adopting A is: P (u S x, I B) P (u S, I B) = K + P (v S x, IB) P (v S, IB) P (v S x, I B) P (v S, I B) K = + K P (v S, I B) () (P (v S x, IB) P (v S, IB)) = + K ( P (v S, IB)). Similarly, lt T b th st of nighbors of u for whom th closst nod from T I B is at distanc d, and lt L b th numbr of nighbors of u that ar at distanc d from x, and at distanc gratr than d fromt I B. Thn: P (u T x, I B) P (u T,I B) (P (v T x, IB) P (v T,IB)) = L + T + L ( P (v T,IB)) L + T T W now stablish th following thr inqualitis: K L L + T (P (v S x, IB) P (v S, IB)) ( P (v S, IB)) () (P (v T x, IB) P (v T,IB)) (3) L + T ( P (v T,IB)) T (4) Clarly ths inqualitis imply that P (u S x, I B) P (u S, I B) P (u T x, I B) P (u T,I B). To prov (), lt K and L b th st of nighbors of u that ar at distanc d fromx, and at distanc gratr than d from S I B and T I B, rspctivly. (So K = K,L= L ). Sinc S T, w hav K L and hnc K L. Now, T Lis th st of nighbors of u that ar at distanc d from T x I B,andS Kis th st of nighbors of u that ar at distanc d froms x I B,soS K T L. Sinc T L = S K=, wgtthat L + T. Combining this with K L w obtain (). To prov (3), w not that for v T S,wmusthav P (v T,I B)=P (v T x, I B) =. Sinc v S, th shortst distanc from v to any nod in I B is gratr than d, and sinc v T,thrmustbanodinT that is at distanc d fromv. Hnc: [P (v T x, I B) P (v T,I B)] = [P (v T x, I B) P (v T,I B)] [P (v S x, I B) P (v S, I B)], whr th inquality follows from induction. W stablishd abov that L + T, which complts th proof of (3). For (4), w again us th fact that P (v T,I B) = 0 for v T S and obtain: [ P (v T,I B)] = [ P (v T,I B)] [ P (v S, I B)], whr th inquality follows from monotonicity. that T givs (4). Th fact 357

8 Siz of Activ St for A Siz of Activ St for A Grdy A 00 High Dgr A Cntral A k = I A Grdy A 00 High Dgr A Cntral A k = I A Figur 3: Distanc-basd modl: high-dgr I B Figur 4: Wav propagation modl: high-dgr I B 5. NUMERICAL SIMULATIONS In this sction w analyz th bhavior of both modls and th rsulting influnc sts of ach on a ral ntwork th coauthorship graph basd on paprs in thortical highnrgy physics. Empirical vidnc suggsts that coauthorship graphs ar rprsntativ of typical social ntworks [9]. By choosing to run our xprimnts on th data from an actual social ntwork as opposd to gnrating random graphs, w ar abl to obtain rsults that ar mor spcifically applicabl to th motivations for our modls. Th spcific datast w mployd was th PROXIMITY HEP-Th databas basd on data from th arxiv archiv and th Stanford Linar Acclrator Cntr SPIRES-HEP databas providd for th 003 KDD Cup comptition with additional prparation prformd by th Knowldg Discovry Laboratory, Univrsity of Massachustts Amhrst [0]. Aftr minor prprocssing, th ntwork consistd of 839 distinct authors and 46 sparat connctd componnts (of siz at last ), th largst of which containd 7034 authors. W compard diffrnt choics for companis A and B, whr company B first chooss a crtain subst of th nods, I B, unawar that company A will also try to ntr th markt, and company A subsquntly targts a subst I A, aftr which w look at th sprading of influnc from I A and I B according to th procsss dscribd in Sction 3. W ran simulations whr th st I B was chosn according to svral diffrnt huristics. As discussd in [] th huristics of choosing high-dgr nods and cntral nods ar oftn usd in th sociology litratur to find influntial sts of nods. Hr th high-dgr huristic chooss nods in ordr of highst dgr, whil th cntral nod huristic chooss nods with low avrag distanc to othr nods. Th avrag distanc is calculatd by taking th avrag of a nod s distanc to all othr nods, whr th distanc btwn unconnctd nods is th numbr of nods in th graph. In addition to ths two huristics, w also ran simulations whr th nods of I B wr chosn according to th grdy Hill Climbing Algorithm for th singl tchnology cas []. W usd ach of ths thr huristics to choos an initial st I B of fixd siz I B = 00 corrsponding to a littl mor than % of th nods in th ntwork. For ach of ths sts, and for both diffusion modls from Sction 3, w I B ran th grdy Hill Climbing Algorithm to dtrmin th most influntial I A st, whr I A rangd from to 00 nods. Sinc th problm of finding th bst I A of a fixd siz is NP-complt, w compar th rsults of th algorithm against two huristics for choosing th I A st from V I B: high-dgr nods and cntral nods. As in Kmp t al. [], w bgin by giving ach dg (u, v) in th ntwork a probability p uv =. of bing activ. W supprss th dtails of th simulation procdur mployd, but not hr that whn p uv (0, ), many random subgraphs must b gnratd to both obtain a nod with th largst marginal xpctd influnc and to valuat th ovrall influnc of all mthods. Figur 3 compars th siz of th markt which product A capturs for incrasing valus of k = I A in th distancbasd modl from Sction 4. if A uss diffrnt huristics, whr th 00 nods of I B ar chosn according to th high dgr huristic. Figur 4 is similar to Figur 3 but uss th wav-propagation modl from Sction 3.. Du to spac limitations, w only prsnt rsults for th wav propagation rsult in th squl. In all of th xprimnts which w conductd, th Hill Climbing Algorithm for company A outprformd th othr huristics. This can b attributd to th fact that th Hill Climbing Algorithm taks into account th ffct of both th nods in I B and th nods alrady slctd in I A. In Figur 5, w fix company A s stratgy to th grdy Hill Climbing Algorithm and compar th stratgis of company B. W s that for larg nough I A, companyb s markt shar is smallst whn B usd th Hill Climbing Algorithm for a singl tchnology, vn though Kmp t al. [] xprimntally showd that in th absnc of comptition, this algorithm prformd bst. This can b xplaind by th fact that th grdy algorithm for a singl tchnology itrativly adds th nod to I B that maximizs th xpctd numbr of additional nods influncd whn thr is no comptition, which inhrntly dos not mak th solution vry robust whn an unxpctd comptitor also tris to influnc nods. Whn I A > 5, th high dgr huristic hlpd company B maintain th largst possibl markt shar in all of our xprimnts. Figur 6 shows th prcntag of th markt capturd by A and B, rspctivly and in total, for incrasing sizs 358

9 Grdy B High Dgr B Cntral B Siz of Activ St for B k = I A Prcntag of Total Markt A 0.0 B A+B k = I A Figur 5: B s shar of th markt against a grdy I A (Wav propagation modl) Figur 6: Prcntag of th total markt capturd (Wav propagation modl: high-dgr I B) of I A whn using our grdy algorithm for choosing I A in th wav propagation modl. Sinc th growth in th total numbr of adoptrs of A and B is slowr than th growth of A s influnc, this figur shows that A s incras in markt shar is du to both raching nw consumrs and drawing consumrs away from its comptitor. For I B chosn using th high-dgr huristic, w s that B maintains a largr markt shar vn whn I A = I B = 00. Whn B chooss I B with ithr of th othr two mthods considrd, A is abl to obtain a lad with a considrably smallr initial st. Figur 7 shows how much largr an initial st ach of th huristics rquir rlativ to th grdy algorithm s initial st to attain som spcifid lvl of influnc. Hr w s that to influnc consumrs, th high-dgr nods huristic rquirs an initial st which is approximatly 5% largr than that rquird by th grdy algorithm. In particular, this quantifis how much bttr th grdy algorithm prforms rlativ to th popular huristics. This information could also b usd by A to dtrmin th valu of knowing prcisly what consumrs B will targt, sinc th othr huristics do not rquir this knowldg. Figur 8 shows th marginal gain in influnc which A njoys from targting an additional consumr vrsus th thr diffrnt stratgis for B. Hr w obsrv that simply grdily targting th most influntial consumr yilds an approximat xpctd rturn of mor than 90 vntual adoptrs if B chos grdily and mor than 0 vntual adoptrs if B usd th high-dgr huristic. W not that givn costs for targting consumrs, this figur could hlp a company dcid at what point th cost of targting an additional consumr outwighs th marginal rturn xpctd from this action. 6. CONCLUDING REMARKS In this papr w studid th sprading of two compting tchnologis, A and B, in a social ntwork. W addrssd th qustion of finding an initial st of nods to targt for tchnology A, givn that th initial st of nods adopting tchnology B is known. To our knowldg, this work rprsnts th first tratmnt of such qustions. W proposd two basic modls for th sprading of tchnologis through a ntwork, in which th two tchnologis propagat in xactly th sam way. Ths modls and our rsults can b asily xtndd to handl additional comptitors. Furthrmor, by adding dummy nods to th ntwork, our modl asily allows for th cas whn companis can targt customrs, but th targtd customr adopts th tchnology only with a crtain probability. W bliv our rsults could also b xtndd to includ mor gnral cass, for xampl by having diffrnt accptanc probabilitis for th two tchnologis, or by allowing th rat at which influnc travls in th graph diffr for th two tchnologis. From a gam-thortic prspctiv, th qustion w study is that of finding a bst rspons to th first playr s mov in a Stacklbrg gam. A natural nxt stp would b to study th optimal bhavior of th first playr, givn that sh knows that th scond playr will us our approximat bst rspons, and ultimatly to study th Nash quilibria of this Stacklbrg gam. W hav shown that A can obtain a significant portion of a markt dspit choosing consumrs scond. In our xprimnts, th huristic that chooss high dgr nods was th bst stratgy for th first playr among th thr stratgis w compard. If B dosn t choos thir initial st in this way, A can in fact outprform B with a rlativly small budgt. It would b intrsting to find a provably good huristic for th first playr. Othr intrsting gams that could b considrd using our modls ar th simultanous vrsion of this gam, and th gam whr th two playrs tak turns in targting nods. Lastly, using our first modl with dg probabilitis qual to, ths problms can also b sn in th contxt of comptitiv facility location [, 4] on a ntwork, but w ar not awar of any prvious rsults for comptitiv location gams on a ntwork. 7. ACKNOWLEDGMENTS W ar gratful to Eric Fridman, Jon Klinbrg, David Williamson, and an anonymous rfr for thir hlpful commnts on an arlir vrsion of this papr. W also thank David Shmoys, Christin Shomakr, and David Williamson for financial support. 359

10 Ratio of Inital St Siz to Grdy Initial St Siz Ndd Grdy A High Dgr A Cntral A Siz of Activ St for A Grdy B High Dgr B Cntral B k = I A Figur 7: Th bnfit of knowing I B,(Wavpropagation modl: high-dgr I B) 8. REFERENCES [] H.-K. Ahn, S.-W. Chng, O. Chong, M. Golin, and R. van Oostrum. Comptitiv facility location: th Voronoi gam. Thort. Comput. Sci., 30(-3): , 004. [] W. B. Arthur. Compting tchnologis, incrasing rturns, and lock-in by historical vnts. Economic Journal, 99(394):6 3, March 989. [3] P. A. David. Tchnical Choic, Innovation and Economic Growth. Cambridg Univrsity Prss, 975. [4] G. Dobson and U. S. Karmarkar. Comptitiv location on a ntwork. Opr. Rs., 35(4): , 987. [5] H. Eislt and G. Laport. Comptitiv spatial modls. Europan Journal of Oprational Rsarch, 39:3 4, 989. [6] J. Farrll and G. Salonr. Installd bas and compatibility: Innovation, product prannouncmnts, and prdation. Amrican Economic Rviw, 76: , 986. [7] D. Fudnbrg and J. Tirol. Gam Thory. MIT Prss, 99. [8] J. Goldnbrg, B. Libai, and E. Mullr. Talk of th ntwork: A complx systms look at th undrlying procss of word-of-mouth. Markting Lttrs, (3): 3, 00. [9] S. Hill, F. Provost, and C. Volinsky. Ntwork-basd markting: Idntifying likly adoptrs via consumr ntworks. Journal of Computational and Graphical Statistics, ():56 76, 006. [0] M. O. Jackson and L. Yariv. Diffusion on social ntworks. Économi publiqu, 6():69 8, 006. [] M. L. Katz and C. Shapiro. Systms comptition and ntwork ffcts. Journal of Economic Prspctivs, 8():93 5, 994. [] D. Kmp, J. Klinbrg, and É. Tardos. Maximizing th sprad of influnc through a social ntwork. In 9th ACM SIGKDD Intrnational Confrnc on Knowldg Discovry and Data Mining, pags 46 57, 003. [3] D. Kmp, J. Klinbrg, and É. Tardos. Influntial nods in a diffusion modl for social ntworks. In 3nd Figur 8: A s gain in adding an arly adoptr (Wav propagation modl: grdy I A) Intrnational Colloquium on Automata, Languags and Programming (ICALP), pags 7 38, 005. [4] S. Khullr, A. Moss, and J. S. Naor. Th budgtd maximum covrag problm. Information Procssing Lttrs, 70():39 45, 999. [5] J. Lskovc, L. A. Adamic, and B. A. Hubrman. Th dynamics of viral markting. In ACM Confrnc on Elctronic Commrc, 006. [6] J. Lskovc, A. Singh, and J. Klinbrg. Pattrns of influnc in a rcommndation ntwork. In Proc. Pacific-Asia Confrnc on Knowldg Discovry and Data Mining (PAKDD), 006. [7] D. Lópz-Pintado. Contagion and coordination in random ntworks. Tchnical rport, Columbia, Sptmbr 9, 005. [8] G. L. Nmhausr, L. A. Wolsy, and M. L. Fishr. An analysis of approximations for maximizing submodular st functions-. Mathmatical Programming, 4:65 94, 978. [9] M. E. J. Nwman. Th structur of scintific collaboration ntworks. Proc. National Acadmy of Scincs USA, 98(): , 00. [0] U. of M-A. Knowldg discovry lab proximity databass. Accssd April 7, 006. [] M. Richardson and P. Domingos. Mining knowldg-sharing sits for viral markting. In Eighth Intl. Conf. on Knowldg Discovry and Data Mining, pags 6 70, 00. [] M. Sviridnko. A not on maximizing a submodular st function subjct to a knapsack constraint. Oprations Rsarch Lttrs, 3():4 43, 004. [3] M. Tomochi, H. Murata, and M. Kono. A consumr-basd modl of comptitiv diffusion: th multiplicativ ffcts of global and local ntwork xtrnalitis. Journal of Evolutionary Economics, 5:73 95, 005. [4] T. W. Valnt. Ntwork Modls of th Diffusion of Innovations. Quantitativ Mthods in Communication Subsris. Hampton Prss, Nw York, NY,