A Note on Competitive Diffusion Through Social Networks

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1 A Note on Compettve Dffuson Through Socal Networks Noga Alon Mchal Feldman Arel D. Procacca Moshe Tennenholtz Abstract We ntroduce a game-theoretc model of dffuson of technologes, advertsements, or nfluence through a socal network. The novelty n our model s that the players are nterested partes outsde the network. We study the relaton between the dameter of the network and the exstence of pure Nash equlbra n the game. In partcular, we show that f the dameter s at most two then an equlbrum exsts and can be found n polynomal tme, whereas f the dameter s greater than two then an equlbrum s not guaranteed to exst. Introducton Socal networks such as Facebook and Twtter are modern focal ponts of human nteracton. The pursut of nsghts nto the nature of ths nteracton calls for a game-theoretc analyss. Indeed, a number of papers (see, e.g., [5]) nvestgate varatons on the followng settng. The socal network s represented by an undrected graph, where the vertces are users and edges connect users who are n a socal relatonshp. Suppose, for example, that there are several competng applcatons, e.g., voce over IP systems, that are not nteroperable. The users play a coordnaton game, where f two neghbors adopt the same system they get some reward that s based on the nherent qualty of the system. The goal s to study the dffuson of technologes through the socal network. The pont of vew here s completely decentralzed, and the players n the game are the users of the socal network. We propose a dfferent, global pont of vew regardng the ncentves that govern the dffuson process. Suppose we have several frms that would lke to advertse competng products va vral marketng. Each frm ntally targets a small subset of users, n the hope that the rumor about ts product would spread throughout the network. However, a user that adopts one product s reluctant to adopt another, hence the campagn of one frm negatvely affects the success of another frm s campagn. To the best of our knowledge our model s the frst game-theoretc model to deal wth Mcrosoft Israel R&D Center, 3 Shenkar Street, Herzelya 46725, Israel, and Schools of Mathematcs and Computer Scence, Tel Avv Unversty, Tel Avv, 69978, Israel, Emal: nogaa@tau.ac.l. Research supported n part by a USA Israel BSF grant, by a grant from the Israel Scence Foundaton, by an ERC advanced grant and by the Hermann Mnkowsk Mnerva Center for Geometry at Tel Avv Unversty. Mcrosoft Israel R&D Center, 3 Shenkar Street, Herzelya 46725, Israel, and School of Busness Admnstraton and Center for the Study of Ratonalty, The Hebrew Unversty of Jerusalem, Jerusalem 9904, Israel. Emal: mfeldman@huj.ac.l School of Engneerng and Appled Scences, Harvard Unversty, 33 Oxford Street, Cambrdge, MA 0238, Unted States. Emal: arelpro@seas.harvard.edu. Ths work was done whle the author was at Mcrosoft Israel R&D Center. Mcrosoft Israel R&D Center, 3 Shenkar Street, Herzelya 46725, Israel, and Technon, IIT, Hafa 32000, Israel. Emal: moshet@mcrosoft.com

2 (a) Tme. (b) Tme 2. (c) Tme 3, the process termnates. Fgure : An llustraton of the dffuson process, wth N = {, 2}. the ncentves of nterested partes outsde the socal network. Note that some prevous papers dd consder the problem of choosng an nfluental set of users as an optmzaton problem (see, e.g., [6]), but not n a compettve game-theoretc settng. Other papers, whch deal wth Vorono games on graphs, provde a game-theoretc study of a faclty locaton problem that does not nvolve a dffuson process, where rather each vertex s assgned to the closest agent and the utlty of an agent s the number of vertces assgned to t (see, e.g., [3, 7]). The model. Let G = V, E be an undrected graph. Furthermore, let N = {,..., n} be the set of agents (the nterested partes). The dffuson process unfolds as follows. There are n + 2 colors: a color for each agent N, as well as two addtonal colors: whte and gray. Intally, at tme, some of the vertces are colored n the colors of N, whle the others are whte. At tme t + each whte vertex that has neghbors colored n color, but does not have neghbors colored n color j for any j N \ {}, s colored n color. A whte vertex that has two neghbors colored by two dstnct colors, j N s colored gray. In other words, we assume that f two agents compete for a user at the same tme they cancel out and the user s removed from the game. The process contnues untl t reaches a fxed pont, that s, all the remanng whte vertces are unreachable due to gray vertces. See Fgure for an llustraton of the dffuson process. A game Γ = G, N s nduced by a graph G, representng the underlyng socal network, and the set of agents N. The strategy space of each agent s the set of vertces V n the graph, that s, each agent selects a sngle node that s colored n color at tme. Note that f two or more agents select the same vertex at tme then that vertex becomes gray. A strategy profle s a vector x = x,..., x n V n, where x V s the ntal vertex selected by agent. We also denote x = x,..., x, x +,..., x n. Gven a strategy profle x V n, the utlty of agent N, denoted U (x), s the number of nodes that are colored n color when the dffuson process termnates. For nstance, n the example gven n Fgure the utlty of each of the agents s two. A strategy profle x s a (pure strategy) Nash equlbrum of the game Γ f an agent cannot beneft from unlaterally devatng to a dfferent strategy,.e., for every N and x V t holds that U (x, x ) U (x). Our results. Gven a graph G and u, v V, let d(u, v) be the length of the shortest path between u and v (n terms of the number of edges). The dameter of the graph, denoted D(G), s the maxmum dstance between a par of vertces, that s, D(G) = max u,v V d(u, v). Our nvestgaton focuses on the relaton between the dameter of the graph and the exstence of Nash equlbra n the nduced dffuson game. Indeed, f we can fnd a Nash equlbrum then we can often predct the behavor of the agents and the outcome of ths compettve dffuson process, or, alternatvely, advse the agents how to play. Our frst theorem s the followng. 2

3 Theorem 2.. Every game Γ = G, N where D(G) 2 admts a Nash equlbrum. Furthermore, an equlbrum can be found n polynomal tme. Note that a random graph on n labeled vertces where each edge appears wth probablty p, usually denoted G(n, p), has dameter at most two wth hgh probablty whenever p (c ln n)/n for c > 2 (see, e.g., [2] for more detals about the dameter of random graphs). In partcular (by takng p = /2) almost all graphs over n vertces have dameter at most two. Fnally, socal networks typcally have a very small dameter. Therefore, t can be argued that assumng a dameter of two s not very restrctve. It s now natural to ask whether the exstence of Nash equlbra can also be guaranteed for dameters larger than two. It s not too dffcult to construct a graph wth dameter four that does not admt an equlbrum. Our second theorem gves a negatve answer even wth respect to dameter three. Theorem 2.2. Let N = {, 2}. There exsts a graph G wth D(G) = 3 such that the game Γ = G, N does not admt a Nash equlbrum. The constructon n the proof of Theorem 2.2 can easly be extended to a larger number of agents or to any (fnte or nfnte) dameter greater than three. Dscusson. In order to facltate the game-theoretc analyss we consder a very smple model of dffuson. In partcular, conflcts are determnstcally resolved by ntroducng gray vertces, and each agent ntally selects just one vertex. Rcher (probablstc) models of dffuson through a socal network exst n the lterature, e.g., [6, 4]. On the other hand, the assumpton of dscrete tme steps s qute common. Theorem 2. mples that wth hgh probablty a random graph (even a relatvely sparse one) nduces a game that admts a Nash equlbrum. However, socal networks are normally not completely random, but rather often exhbt structure. Ideally one would be able to extend our result by showng that under a convncng random graph model of socal networks (see, e.g., [, 9]) the nduced game admts a Nash equlbrum wth hgh probablty. 2 Proofs We begn by provng Theorem 2.; we subsequently dscuss some mplcatons of the proof. Theorem 2.. Every game Γ = G, N where D(G) 2 admts a Nash equlbrum. Furthermore, an equlbrum can be found n polynomal tme. Proof. If D(G) then the graph s a clque and the theorem follows trvally. Therefore, we may assume that D(G) = 2. Gven a profle x V N, let P (x) = {, j : d(x, x j ) = }, that s, the number of pars wth dstance one from each other. Furthermore, denote the neghborhood of vertex u V by N u = {v : d(u, v) }, and let N(x) = n = N x. Consder the potental functon Φ(x) = N(x) n + P (x). It s suffcent to show that for every x V n, N, and x V, U (x, x ) > U (x) Φ(x, x ) > Φ(x). () 3

4 Indeed, gven Equaton () t clearly holds that any strategy profle x V n that maxmzes Φ(x) must be a Nash equlbrum. Moreover, n order to fnd one such profle we may start from some preference profle, and n each step attempt to fnd a proftable devaton for one of the agents. We termnate f there s no such devaton (whch, by defnton, means that we have found a Nash equlbrum). Ths algorthm termnates after a polynomal number of steps snce Φ(x) s bounded from above by n V + n 2 for every x, and by Equaton () every proftable devaton by an agent ncreases the value of the potental functon by at least one. We turn to provng Equaton (). If the dameter of the graph s two then vertces can only be colored by an agent N at tme or 2. Specfcally, the vertces colored by agent are roughly the vertces n the neghborhood of x that are not neghbors of x j for some j N \ {} (snce these vertces are ether gray or colored by j). Formally, defne A = {x : j N \ {} s.t. d(x, x j ) = }. Assumng that x x j for all j, the utlty of agent under the strategy profle x V n s U (x) = N x j (N x N xj ) + χ A (x), where χ A s the ndcator functon that returns f x A and 0 otherwse. The rghtmost term s requred snce even f x s a neghbor of some x j, t s stll colored by agent at tme, but s nevertheless ncluded n the mddle term. Now, suppose U (x, x ) > U (x). It follows that N x (N x N xj ) + χ A (x, x ) > N x x N xj ) + χ A (x). (2) j j (N Snce χ A s a Boolean functon, ths mples that N x (N x N xj ) N x x N xj ). (3) j j (N We dstngush between two cases. If Equaton (3) holds as a strct nequalty then N xj N x = N xj + N x (N x N xj ) > N xj + N x x N xj ) j j j j j (N = j N xj N x, whch mples that N(x, x ) N(x) +. In addton, a devaton of a sngle agent can decrease the number of adjacent pars of agents by at most n,.e., P (x, x ) > P (x) n. We conclude that Φ(x, x ) = N(x, x ) n+p (x, x ) N(x) n+n+p (x, x ) > N(x) n+n+p (x) n = Φ(x). Otherwse, Equaton (3) holds as an equalty, and hence N(x, x ) = N(x). It then follows from Equaton (2) that χ A (x, x ) > χ A (x). That s, agent has no neghbors among x under x but has at least one neghbor under x. Thus the number of neghbors of agent ncreases and 4

5 the number of neghbors of agents j N \ {} does not decrease,.e., P (x, x ) > P (x). We conclude that Φ(x, x ) = N(x, x ) n + P (x, x ) = N(x) n + P (x, x ) > N(x) n + P (x) = Φ(x). Ths establshes Equaton (), and hence completes the proof of the theorem. What the proof of Theorem 2. essentally shows s that when the dameter of the graph s two the dffuson game s a potental game [8]; specfcally, a functon that satsfes () s known as a generalzed ordnal potental functon. Potental games have the property that better response dynamcs converge to a Nash equlbrum; n other words, f at every stage the agents smply behave myopcally, that s, some agent devates to a more proftable strategy, then they wll eventually reach an equlbrum. We are now ready to prove our second theorem. Theorem 2.2. Let N = {, 2}. There exsts a graph G wth D(G) = 3 such that the game Γ = G, N does not admt a Nash equlbrum. Proof. We frst gve our constructon, then establsh that t has dameter three and that t does not admt a Nash equlbrum. The constructon. Let G = V, E be defned as follows. The vertces of the graph are V = {v,..., v 6 } C C 2 C 3, where for =, 2, 3, C = C C 5. Each C j contans ten vertces, that s, V = 56. The edges of the graph are defned as follows. Each C, for =, 2, 3, s a clque. There s an edge v, u for every u C C 2 C 3 C 2 C 22 C 23 ; an edge v 2, u for every u C C 4 C 5 C 2 C 24 C 25 ; an edge v 3, u for every u C C 2 C 4 C 3 C 32 C 33 ; an edge v 4, u for every u C C 3 C 5 C 3 C 34 C 35 ; an edge v 5, u for every u C 2 C 22 C 24 C 3 C 32 C 34 ; an edge v 6, u for every u C 2 C 23 C 25 C 3 C 33 C 35. An llustraton of the graph G s gven as Fgure 2. We refer to the vertces v,..., v 6 as hubs; we say that v and v 2 are parallel hubs, and so are v 3 and v 4, v 5 and v 6. If the hub v s connected by an edge to some of the vertces of clque C j, we say that v s adjacent to C j ; for nstance, v and C are adjacent, whereas v and C 3 are not. The constructon possesses the followng mportant propertes:. Let v and v j be two parallel hubs that are adjacent to a clque C k. Then (N v \ N vj ) C k contans exactly two of the sets C kl, l =,..., Let v and v j be two nonparallel hubs that are adjacent to a clque C k. Then (N v \ N vj ) C k contans exactly one of the sets C kl, l =,..., 5. Note that the constructon s essentally symmetrc wth respect to the hubs. G has dameter 3. Usng Fgure 2, t s easy to verfy that G has dameter 3. For example, a path from v to u C 32 s gven by v w v 3 u, where w C. A path from u C 3 to w C 24 s gven by u v x w, where x C 2. G does not admt a Nash equlbrum. We consder strategy profles x, x 2 V 2 for the two agents. The symmetres of our constructon allow us to restrct our attenton to sx cases. 5

6 v v 2 C C 2 C 3 C 4 C 5 C 2 C 22 C 23 C 24 C 25 v 3 v 4 v 5 v 6 C 3 C 32 C 33 C 34 C 35 Fgure 2: The constructon of the proof of Theorem 2.2. The clques C, C 2, C 3 are outlned by dashed ellpses, and the edges nsde the clques are not shown. An edge between v and C jk mples that v s connected to all the vertces u C jk. Case : x = v, x 2 C (hub and adjacent clque). Agent colors some of the vertces of C 2 and some hubs, that s, U (x, x 2 ) < 60. By devatng to x = v 5, agent colors C 2, C 22, C 24, C 3, C 32, C 34,.e., U (x, x 2) 60. Case 2: x = v, x 2 C 3 (hub and nonadjacent clque). Agent 2 colors the vertces of C 3 and some hubs, hence U 2 (x, x 2 ) < 60. By devatng to x 2 C, agent 2 colors C 4, C 5, and C 3, thus U 2 (x, x 2 ) 70. Case 3: x = v, x 2 = v 3 (nonparallel hubs). Agent colors C 3, C 2, and some hubs, therefore U (x, x 2 ) < 70. By devatng to x C, agent colors C 3, C 5, C 2, so U (x, x 2) 70. Case 4: x = v, x 2 = v 2 (parallel hubs). Agent colors C 2, C 3, C 22, C 23, and some hubs (v 3,..., v 6 are gray and C 3 remans whte), hence U (x, x 2 ) < 50. By devatng to x C 3, agent can guarantee a utlty of at least 50 (snce t colors C 3 ). Case 5: x C, x 2 C 3 (dfferent clques). If x / C, x 2 / C 3, then agent can beneft by devatng to x C, snce then t colors both v and v 2 at tme 2 (rather than just one of them), and colors twenty vertces of C 2 at tme 3 (rather than ten). Hence we can assume wthout loss of generalty that x C. In that case, agent 2 colors C 3 and some hubs, therefore U 2 (x, x 2 ) < 60. By devatng to x 2 = v 5, agent 2 colors at least C 2, C 22, C 24, C 3, C 32, C 34, hence U 2 (x, x 2 ) 60. Case 6: x C, x 2 C (same clque). Snce C \ {x, x 2 } s gray, there are at most 08 vertces that are not gray, therefore t must hold that ether U (x, x 2 ) < 60 or U 2 (x, x 2 ) < 60. By devatng to v 5 an agent can guarantee a utlty of at least 60. References [] A. L. Barabás and R. Albert. Emergence of scalng n random networks. Scence, 286:509 52, 999. [2] B. Bollobás. Random Graphs. Cambrdge Unversty Press, 2nd edton, 200. [3] C. Dürr and N. K. Thang. Nash equlbra n Vorono games on graphs. In Proceedngs of the 5th Annual European Symposum on Algorthms (ESA), pages 7 28,

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