A A Game Theoretic Model for the Formation of Navigable Small-World Networks the Tradeoff between Distance and Reciprocity

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1 A A Game Theoretic Model for the Formation of Navigable Small-World Networ the Tradeoff between Ditance and Reciprocity ZHI YANG, Peing Univerity WEI CHEN, Microoft Reearch Kleinberg propoed a family of mall-world networ to explain the navigability of large-cale real-world ocial networ. However, the underlying mechanim that drive real networ to be navigable i not yet well undertood. In thi paper, we preent a game theoretic model for the formation of navigable mall world networ. We model the networ formation a a game called the Ditance-Reciprocity Balanced DRB) game in which people ee for both high reciprocity and long-ditance relationhip. We how that the game ha only two Nah equilibria: One i the navigable mall-world networ, and the other i the random networ in which each node connect with each other node with equal probability, and any other networ tate can reach the navigable mall world via a equence of bet-repone move of node. We further how that the navigable mall world equilibrium i very table a) no colluion of any ize would benefit from deviating from it; and b) after an arbitrary deviation of a large random et of node, the networ would return to the navigable mall world a oon a every node tae one bet-repone tep. In contrat, for the random networ, a mall group colluion or random perturbation i guaranteed to bring the networ out of the random-networ equilibrium and move to the navigable networ a oon a every node tae one betrepone tep. Moreover, we how that navigable mall world equilibrium ha much better ocial welfare than the random networ, and provide the price-of-anarchy and price-of-tability reult of the game. Our empirical evaluation further demontrate that the ytem alway converge to the navigable networ even when limited or no information about other player trategie i available, and the DRB game imulated on real-world networ lead to navigability characteritic that i very cloe to that of the real networ, even though the real-world networ have non-uniform population ditribution different from the Kleinberg mall-world model. Our theoretical and empirical analye provide important new inight on the connection between ditance, reciprocity and navigability in ocial networ. Categorie and Subject Decriptor: G.. [Dicrete Mathematic]: Graph Theory Networ problem Additional Key Word and Phrae: Small-world networ, game theory, navigability, reciprocity INTRODUCTION In 967, Milgram publihed hi wor on the now famou mall-world experiment [Milgram 967]: he aed tet ubject to forward a letter to their friend in order for the letter to reach a peron not nown to the initiator of the letter. He found that on average it too only ix hop to connect two people in U.S., which i often attributed a the ource of the popular term ix-degree of eparation. Thi eminal wor inpired A preliminary verion of thi wor appeared in Proceeding of the 4th International World Wide Web Conference WWW 05). The current verion contain a number of new reult comparing to the preliminary verion, uch a heterogeneou utility function, analytical reult on the bet-repone dynamic, non-exitence of non-uniform equilibria, price of anarchy and price of tability, and empirical evaluation on real dataet. The wor wa partly done when the firt author wa a viiting reearcher at Microoft Reearch Aia. Thi wor wa upported by the National Baic Reearch Program of China Grant No. 04CB340400). Author addree: Zhi Yang, Computer Science Department, Peing Univerity, Beijing, China; e- mail:yangzhi@pu.edu.cn. Wei Chen, Microoft Reearch, Beijing, China; weic@microoft.com. Permiion to mae digital or hard copie of all or part of thi wor for peronal or claroom ue i granted without fee provided that copie are not made or ditributed for profit or commercial advantage and that copie bear thi notice and the full citation on the firt page. Copyright for component of thi wor owned by other than ACM mut be honored. Abtracting with credit i permitted. To copy otherwie, or republih, to pot on erver or to reditribute to lit, require prior pecific permiion and/or a fee. Requet permiion from permiion@acm.org. c YYYY ACM /YYYY/0-ARTA $5.00 DOI:

2 A: Zhi Yang and Wei Chen. numerou tudie on the mall-world phenomenon and mall-world model, which lat till the preent day of information age. In [Watt and Strogatz 998] Watt and Strogatz invetigated a number of realworld networ uch a film actor networ and power grid, and howed that many networ have both low diameter and high clutering meaning two neighbor of a node are liely to be neighbor of each other), which i different from randomly wired networ. They thu propoed a mall-world model in which node are firt placed on a ring or a grid with local connection, and then ome connection are randomly rewired to connect to long-range contact in the networ. The local and long-range connection can alo be viewed a trong tie and wea tie repectively in ocial relationhip originally propoed by Granovetter [Granovetter 973; Granovetter 974]. Kleinberg notice an important dicrepancy between the mall-world model of Watt and Strogatz and the original Milgram experiment: the latter how not only that the average ditance between node in the networ are mall, but alo that a decentralized routing algorithm uing only local information can contruct hort path. Here, we call a routing algorithm decentralized in that given a ource node u and a detination node v, the algorithm attempt to come up with a path u = x 0, x, x,..., x m = v, only uing the acquaintance relationhip of thee m intermediate node x 0, x, x,..., x m. By contrat, a centralied algorithm e.g., Dijtra algorithm) require the node to now full networ i.e., the acquaintance relationhip among all people in the world) to find an optimal route, but obviouly they cannot now thi in real networ. To addre thi iue, Kleinberg adjuted the Watt-Strogatz model o that the longrange connection are elected not uniformly at random among all node but inverely proportional to a power of the grid ditance between the two end point of the connection [Kleinberg 00]. More pecifically, Kleinberg modeled a ocial networ a compoed of n node on a -dimenional grid, with each node having local contact to other node in it immediate geographic neighborhood. Each node u alo etablihe a number of long-range contact, and a long-range lin from u to v i etablihed with probability proportional to d M u, v) r, where d M u, v) i the grid ditance between u and v, and r 0 i the model parameter indicating how liely node prefer to connect to remote node, which we call connection preference in the paper. The Watt-Strogatz model correpond to the cae of r = 0, and a r increae, node are more liely to connect to other node in their vicinity. Kleinberg modeled Milgram experiment a decentralized greedy routing in uch networ, in which each node only forward meage to one of it neighbor with coordinate cloet to the target node. He howed that when r =, greedy routing can be done efficiently in Olog n) time in expectation, but for any r, it require Ωn c ) time for ome contant c depending on r, exponentially wore than the cae of r =. Therefore, the mall world at the critical value of r = i meant to model the real-world navigable networ validated by Milgram and other experiment, and we call it the navigable mall-world networ. After Kleinberg theoretical analyi, a number of empirical tudie have been conducted to verify if real networ indeed have connection preference cloe to the critical value that allow efficient greedy routing [Liben-Nowell et al. 005; Adamic and Adar 005; Cho et al. 0; Goldenberg and Levy 009; Schaller and Latan 995]. Since real population i not evenly ditributed geographically a in the Kleinberg model, Liben-Nowell et al. [Liben-Nowell et al. 005] propoed to ue the fractional dimenion D, defined a the bet value to fit {w : d M u, w) d M u, v)} = c d M u, v) D, averaged over all u and v. They howed that when the connection preference r = D, the networ i navigable. They then tudied a networ of 495,836 LiveJournal uer in the continental United State who lit their hometown, and find that D 0.8 while r =., reaonably cloe to D. We apply the ame approach to a ten million

3 A Game Theoretic Model for the Formation of Navigable Small-World Networ A:3 CDF Renren data Linear fit Ditance d m) Fig.. The fraction of node within ditance d in Renren. Lin probability Pd) Friendhip probability v. ditance in Ren- Fig.. ren Pd) d Ditance d m) node Renren networ [Jiang et al. 00; Yang et al. 0], one of the larget online ocial networ in China. We map the hometown lited in uer profile to longitude, latitude) coordinate. The reolution of our geographic data i limited to the level of town and citie and thu we cannot get the exact ditance of node within 0m. We found that D Figure ) and r 0.9 Figure ) in the Renren networ. Other - tudie [Adamic and Adar 005; Cho et al. 0; Goldenberg and Levy 009; Schaller and Latan 995] alo reported connection preference r to be cloe to in other online ocial networ including Gowalla, Brightite and Faceboo). Even though they did not report the fractional dimenion, from both the LiveJournal data in [Liben-Nowell et al. 005] and our Renren data, it i reaonable to believe that the fractional dimenion i alo cloe to. Therefore, empirical evidence all ugget that the real-world ocial networ indeed have connection preference cloe to the critical value and the networ i navigable. A natural quetion to a next i how navigable networ naturally emerge? What are the force that mae the connection preference become cloe to the critical value? A Kleinberg pointed out in hi urvey paper [Kleinberg 006] when taling about the above triing coincidence between theoretical prediction and empirical obervation, it ugget that there may be deeper phenomena yet to be dicovered here. There are everal tudie trying to explain the emergence of navigable mall-world networ [Mathia and Gopal 00; Hu et al. 0; Clauet and Moore 003; Sandberg and Clare 006; Chaintreau et al. 008], motly by modeling certain underlying node or lin dynamic ee additional related wor below for more detail). In thi paper, we tacle the problem in a novel way uing a game-theoretic approach, which i reaonable in modeling individual behavior in ocial networ without central coordination. One ey inight we have i that connection preference r i not a global preference but individual own preference ome prefer to connect to more faraway node while other prefer to connect to nearby node. Therefore, we etablih mall-world formation game where individual node u trategy i it own connection preference r u Section ). Thi game formulation i different from mot exiting networ formation game where individual trategie are creating actual lin in the networ c.f. [Tardo and Wexler 007]). It allow u to directly explore the entire parameter pace of connection preference and anwer the quetion on why node end up chooing a particular parameter etting leading to the navigable mall world. In term of payoff function, we firt conider minimizing greedy routing ditance to other node a the payoff, ince it directly correpond to the goal of navigable networ. However, Gulyá et al. [Gulyá et al. 0] prove that with thi payoff the navigable networ cannot emerge a a equilibrium for the one-dimenional cae. Our empirical analyi alo indicate that node will converge to random networ r u = 0, u) rather than navigable networ for higher dimenion. Our empirical analyi further how

4 A:4 Zhi Yang and Wei Chen. that if we adjut the payoff with a cot proportional to the grid ditance of remote connection, the equilibria are enitive to the cot factor. The above unucceful attempt ugget that beide the goal of hortening ditance to remote node, ome other natural objective may be in play. Reciprocity i regarded a a baic mechanim that create table ocial relationhip in a peron life [Gouldner 960]. A number of prior wor [Java et al. 007; Liben-Nowell et al. 005; Milove et al. 007] alo ugget that people ee reciprocal relationhip in online ocial networ. Therefore, we propoe a payoff function that i the product of average ditance of node to their long-range contact and the probability of forming reciprocal relationhip with long-range contact. We call thi game the ditance-reciprocity balanced DRB) game. In practice, increaing relationhip ditance capture that individual attempt to create ocial bridge by lining to ditant people, which can help them earch for and obtain new reource. Meanwhile, increaing reciprocity capture that individual loo at ocial bond by lining to people lie them, which could help them preerve or maintain reource. Therefore, the DRB game i natural ince it capture ource of bridging and bonding ocial capital in building ocial integration and olidarity [Gittell and Vidal 998]. We further allow heterogeneou utility function in that different uer may weigh the tradeoff between ditance and reciprocity in different way. Even though the payoff function for the DRB game i very imple, our analyi demontrate that it i extremely effective in producing navigable mall-world networ a the equilibrium tructure. In theoretical analyi Section 3), we firt how that navigable mall world r u =, u) and random mall world r u = 0, u) are the only two Nah equilibria of the DRB game, depite the flexible and heterogeneou utility function. Moreover, for any trategy profile that i not the random networ, it can alway reach the navigable mall world through a cacade of nearby node adopting trategy in a bet-repone dynamic. In term of the tability of NE, we prove that the navigable mall world i a trong Nah equilibrium, which mean that it tolerate colluion of any ize trying to gain better payoff. Moreover, it alo tolerate arbitrary deviation without the objective of increaing anyone payoff) of large group of random deviator, ince the ytem i guaranteed to return bac to the navigable NE a oon a every node tae one bet-repone tep. In contrat, random mall world can be moved away from it equilibrium tate by either a random perturbation of one node or a colluion of two nearby node, and when a mall random et of node perturb to different trategie, we prove that the ytem i guaranteed to converge to the navigable mall world a oon a every node tae one bet-repone tep. Our theoretical analyi provide trong upport that the navigable mall-world NE i the unique and table equilibrium that would naturally emerge in the DRB game. We further examine the global function of ocial welfare i.e., the total payoff of all node) and how elfih behavior of uer affect the ocial welfare. Interetingly, we find that the global optimum can be reached by a fraction of node acrificing their ditance payoff to focu on reciprocity by electing a trategy greater than ) o that their neighbor could elect trategy to reach a high balanced payoff of both ditance and reciprocity. Thi ituation remind u ocial relationhip generated by different ocial tatu e.g. employee-employer relationhip) or by tight bond with mutual undertanding and upport uch a marriage). Next we compare the ocial welfare of navigable and random mall-world networ with the global optimum through the - tandard price of anarchy PoA) and price of tability PoS) metric, which i the ratio of ocial welfare between the global optimum and the wort or the bet) Nah equilibrium, repectively. We how that navigable networ ha the better ocial welfare, and being only logarithmically wore than the global optimum.

5 A Game Theoretic Model for the Formation of Navigable Small-World Networ A:5 To complement our theoretical analyi, we conduct empirical evaluation to cover more realitic game cenario not covered by our theoretical analyi Section 5). We firt tet random perturbation cae and how that arbitrary initial profile alway converge to the navigable equilibrium in a few tep, while a very mall random perturbation le than theoretical prediction) of the random mall world caue it to quicly converge bac to the navigable equilibrium. Next, we imulate more realitic cenario where node have limited or no information about other node trategie. We how that if they only learn their friend trategie with ome noie), the ytem till converge cloe to the navigable equilibrium in a mall number of tep. Further, even when the node ha no information about other player trategie and can only ue it obtained payoff a feedbac to earch for the bet trategy, the ytem till move cloe to the navigable equilibrium within a few hundred tep in the grid). Finally we imulate the DRB game on Renren and LiveJournal networ, which have non-uniform population ditribution different from Kleinberg grid-baed mall-world model. Our imulation reult how that in both networ, the game quicly converge to an equilibrium where connection preference of uer are cloe to the empirical one. In ummary, our contribution are the following: a) we propoe the mall-world formation game and deign a balanced ditance-reciprocity payoff function to explain the navigability of real ocial networ; b) we conduct comprehenive theoretical and empirical analyi to demontrate that navigable mall world i the unique robut equilibrium that would naturally emerge from the game under both random perturbation and trategic colluion; and c) our game reveal a new inight between ditance, reciprocity and navigability in ocial networ, which may help future reearch in uncovering deeper phenomena in navigable ocial networ. To our bet nowledge, thi i the firt game theoretic tudy on the emergence of navigable mall-world networ, and the firt tudy that lining relationhip reciprocity with networ navigability. Additional related wor. We provide additional detail of prior wor on explaining the emergence of navigable mall-world networ, and other related tudie not covered in the introduction. Some tudie try to explain navigability by auming that node form lin to optimize for a particular property. Mathia et al. [Mathia and Gopal 00] aume that uer try to mae trade-off between wiring and connectivity. Hu et al. [Hu et al. 0] aume that people try to maximize the entropy under a contraint on the total ditance of their long-range contact. Thee wor rely on imulation to tudy the networ dynamic. Moreover, the navigability of a networ i enitive to the weight of wiring cot or the ditance contraint, and it i unliely that navigable networ a defined by Kleinberg [Kleinberg 00] would naturally emerge. Another type of wor propoe node/lin dynamic that converge to navigable mallworld networ. Clauet and Moore [Clauet and Moore 003] propoe a rewiring dynamic modeling a Web urfer uch that if the urfer doe not find what he want in a few tep of greedy earch, he would rewire her long-range contact to the current end node of the greedy earch. They ue imulation to demontrate that a networ cloe to Kleinberg navigable mall world will emerge after long enough rewiring round. Sandberg and Clare [Sandberg and Clare 006] propoe another rewiring dynamic where with an independent probability of p each node on a greedy earch path would rewire their long-range contact to the earch target, and provide a partial analyi and imulation howing that the dynamic converge to a networ cloe to the navigable mall world. Chaintreau et al. [Chaintreau et al. 008] ue a move-and-forget mobility model, in which a toen tarting from each node conduct a random wal move) and may alo go bac to the tarting point forget), and ue the ditribution of

6 A:6 Zhi Yang and Wei Chen. the toen on the grid a the ditribution of the long-range contact of the tarting node. They provide theoretical analyi howing that there exit a critical forgetting value for which the move-and-forget model provide navigability. However, the underlying mechanim driving the critical value to be choen in practice remain unclear. The approach taen by thee tudie can be viewed a orthogonal and complementary to our approach: they aim at uing natural dynamic rewiring or mobility dynamic) to explain navigable mall world, while we focu on directly exploring the entire parameter pace of connection preference of node and ue game theoretic approach to how, both theoretically and empirically, that the node would naturally chooe their connection preference to form the navigable mall world. The connection preference can be conidered a higher level deciion-maing variable for individual that puhe them to mae long-range connection over time. In particular, electing connection preference capture the people proce of cognitively creating behavioral plan i.e., intenion) on how to ditribute the finite time and effort among node of different ditance. Once the preference i elected, the player would engage in activitie uch a rewiring or mobility dynamic to create long-range contact with the correponding connection intenion. Moreover, all the prior tudie only how that they converge approximately to the navigable mall world, while in our game the navigable mall world i preciely the only robut equilibrium. Finally, none of thee wor introduce reciprocity in their model and we are the firt to lin reciprocity with navigability of the mall world. Some tudie ue hyperbolic metric pace or graph to try to explain navigability in mall-world networ e.g. [Boguñá et al. 009; Papadopoulo et al. 00; Kriouov et al. 00; Kriouov et al. 009; Chen et al. 03; Gulyá et al. 05]). However, they do not explain why connection preference in real networ are around the critical value and how navigable networ naturally emerge. In particular, Chen et al. [Chen et al. 03] how that the navigable mall world in Kleinberg model doe not have good hyperbolicity. Mot recently, Gulyá et al. [Gulyá et al. 05] propoe a game where each player trie to minimize the number of lin in order to be able to greedily route to all other node. The equilibrium of the game i a cale-free networ whoe degree ditribution follow a power law. However, thi game i not intended and doe not explain the emergence of navigable mall-world networ validated by Milgram and other experiment, where greedy routing can be done efficiently in Olog n) time in expectation, and relationhip reciprocity i not included in any apect of the game. SMALL-WORLD FORMATION GAMES In thi ection, we firt preent the game formulation baed on Kleinberg mall-world model, and we then tudy the payoff function which i ey to undertanding the underlying mechanim that give rie to navigable mall world networ.. Game Formulation baed on Kleinberg Small-World Model Small-world model. Let V = {i, j) : i, j [n] = {,,..., n}} be the et of n node forming an n n grid. For convenience, we conider the grid with wrap-around edge connecting the node on the two oppoite ide, maing it a toru. For any two node u = i u, j u ) and v = i v, j v ) on thi wrap-around grid, the grid ditance or Manhattan ditance between u and v i defined a d M u, v) = min{ i v i u, n i v i u } + min{ j v j u, n j v j u }. The Kleinberg mall-world model ha two univeral contant p, q, uch that a) each node ha undirected edge connecting to all other node within lattice ditance p, called it local contact, and b) each node ha q random directed edge connecting to poibly faraway node in the grid called it long-range contact, drawn from the

7 A Game Theoretic Model for the Formation of Navigable Small-World Networ A:7 following ditribution. Each node u ha a connection preference parameter r u 0, uch that the i-th long-range edge from u ha endpoint v with probability proportional to /d M u, v) ru, that i, with probability p u v, r u ) = d M u, v) ru /cr u ), where cr u ) = v u d M u, v) ru i the normalization contant. Let r be the vector of r u value on all node. We ue r to denote r u =, u V. The above model can be eaily extended to dimenional grid with wraparound) for any =,, 3,..., where each long range contact i till etablihed with probability proportional to /d M u, v) ru. We ue Kn,, p, q, r) to refer to the cla of Kleinberg random graph with parameter n,, p, q, and r. Small-world formation game. A game i decribed by a ytem of player, trategie and payoff. Connection preference r u in Kleinberg model reflect u intention in etablihing long-range contact: When r u = 0, u chooe it long-range contact uniformly among all node in the grid; a r u increae, the long-range contact of u become increaingly clutered in it vicinity on the grid. Our inight i to treat connection preference a node trategy in a game etting and tudy the game behavior. More pecifically, we model thi via a non-cooperative game among node in the networ. Firt, we aume that each r u i taen from a dicrete et Σ = {0, γ, γ, 3γ,..., }, where γ repreent the granularity of connection preference and i in the form of /g for ome poitive integer g. Uing dicrete trategy et avoid nuance in continuou trategy pace and i alo reaonable in practice ince people are unliely to mae infiniteimal change. Next, we model the mall-world networ formation a a game Γ = Σ, π u ) u V, where V i the et of node player) in the grid, connection preference r u Σ i the trategy of a player u, and π u : S R i the payoff function of u, with S = Σ Σ... Σ. An element r = r, r,..., r n ) S i called a trategy profile. Let C = V \ denote the et of all coalition. For each coalition C C, let C = V \C, and if C = {u}, we denote C by u. We alo denote by S C the et of trategie of player in coalition C, and r C the partial trategy profile of r for node in C. Objective. Greedy routing on the mall-world networ from a ource node u to a target node v i a decentralized algorithm tarting at node u, and at each tep if routing reache a node w, then w elect one node from it local and long-range contact that i cloet to v in grid ditance a the next tep in the routing path, until it reache v. In [Kleinberg 00], Kleinberg how that given a two-dimenional grid, when r, the expected number of greedy routing tep called delivery time) i Olog n), but when r, it i Ωn c ) for ome contant c related to. More generally, for any dimenional grid, it i hown that r i the critical value allowing efficient greedy routing. Hence, we call Kleinberg mall world with r the navigable mall world. Interetingly, empirical evidence have demontrated that the real-world networ i navigable with the connection preference cloe to the critical value [Liben-Nowell et al. 005; Cho et al. 0; Adamic and Adar 005; Goldenberg and Levy 009; Lambiotte et al. 008; Illenberger et al. 03]. We aim to explain thi triing coincidence from the perpective of individual incentive. In particular, our objective i to tudy intuitively appealing payoff function π u and find one that individual effort to get thi payoff lead fairly quicly to the emergence of navigable mall-world networ.. Routing-baed Payoff A navigable mall world achieve bet greedy routing efficiency, it i natural to conider the payoff function a the expected delivery time to the target in greedy routing. Given the trategy profile r S, let t uv r u, r u ) be the expected delivery time from

8 A:8 Zhi Yang and Wei Chen. Expected delivery time r u r u Strategy of player u Bet trategy of a player u r u r u Cot factor Fig. 3. The expected delivery time for a player u with different trategie. Fig. 4. The bet repone of a player u given different cot factor. ource u to target v via greedy routing. The payoff function i given by: π u r u, r u ) = t uv r u, r u ). ) v u We tae a negation on the um of expected delivery time becaue node prefer horter delivery time. Although the above payoff function i intuitive and imple, it ha ome eriou iue. Prior wor [Gulyá et al. 0] ha already proved that, with the length of greedy path a the payoff, player u bet repone i to lin uniformly i.e., r u = 0) for the one-dimenional cae. For higher dimenion, Figure 3 how the expected delivery time for a ingle node u at a grid, where each node generate q = 0 lin. We ee that when other node fixed their trategy e.g., r u ), the bet trategy of a ingle node u i 0. More tet on different initial condition reach the ame reult that the ytem will converge to the random mall-world networ. The intuitive reaon i that to reach other node quicly, it i better for a node to evenly pread it long-range contact from the individual propective or equivalently, eeing the longrange contact of the larget ditance on average given r u 0). Thi i inconitent with empirical evidence that real-world networ are navigable one, where lin are much more liely to connect neighbor node than ditant node. In practice, creating and maintaining long-range lin have higher cot, o one may adapt the above payoff function by adding the grid ditance of long-range contact a a cot term in the payoff function: π u r u, r u ) = t uv r u, r u ) λ p u v, r u )d M u, v), ) v u where λ i a factor controlling the long range cot and p u v, r u ) = d M u, v) ru /cr u ) i the probability that u tae v a a long-range contact under the trategy of r u. A larger λ mean uer are more concerned with ditance cot. Figure 4 how that the bet trategy of a uer u i ignificantly influenced by the cot factor. Similar reult i alo hown in [Gulyá et al. 0]. Thu, it i unclear if the navigable mall-world networ can naturally emerge from thi type of game. In the above payoff function, we ue the expected delivery time to meaure the routing efficiency to an arbitrary node. It i alo poible to give more complex payoff function by conidering the ditribution function of delivery time, uch a the percentage of node that can be delivered within a given number of tep. However, given that individual can have different trategie, it i very difficult to obtain the explicit form of delivery time t uv r u, r u ) in term of uer trategie r. Due to thi diadvantage of the delivery time-baed game, there i no theoretical guarantee that the networ formation would converge to the deired navigable mall world. v u

9 A Game Theoretic Model for the Formation of Navigable Small-World Networ A:9.3 Ditance-Reciprocity Balanced Payoff The previou ection demontrate that eeing hort routing ditance alone cannot explain the emergence of navigable mall world, and thu people in the ocial networ mut have ome other objective to achieve. Reciprocity i regarded a a baic mechanim that create table ocial relationhip in the real world [Gouldner 960]. Several empirical tudie [Java et al. 007; Liben-Nowell et al. 005; Milove et al. 007] alo how that high reciprocity i alo a typical feature preent in real mall-world networ uch a Flicr, YouTube, LiveJournal, Orut and Twitter). Therefore, we conider the payoff of a uer u a the following balanced objective between ditance and reciprocity: π u r u, r u ) = p u v, r u )d M u, v) v u αu p u v, r u )p v u, r v ), 3) where v u p uv, r u )d M u, v) i the mean grid ditance of u long-range contact, v u p uv, r u )p v u, r v ) i the mean probability for u to form bi-directional lin with it long-range contact, i.e., reciprocity, and α u α u > 0) i a contant exponent with repect to node u, capturing how that uer weigh the relative importance of ditance and reciprocity. Note here the tradeoff exponent α u could be heterogeneou among player, modeling uer having different weight on the balance between the ditance and reciprocity tradeoff. So our utility function i very flexible and actually repreent a large cla of tradeoff function. We refer the mall-world formation game with payoff function in Eq.3) the Ditance-Reciprocity Balanced DRB) game. The payoff function in Eq.3) reflect two natural objective uer in a ocial networ want to achieve: firt, they want to connect to remote node, which may give them divere information a in the famou the trength of wea tie argument by Granovetter [Granovetter 973]; econd, they want to etablih reciprocal relationhip which are more table in the long term. However, thee two objective can be in conflict for a node u when other prefer lining in their vicinity i.e., other node v chooing poitive exponent r v ). In thi cae, faraway long-range contact are le liely to create reciprocal lin. Therefore, node u hould obtain the maximum payoff when it achieve a balance between the two objective. We ue the imple product of ditance and reciprocity objective to model thi tradeoff, and allow different node to have different emphai on ditance-reciprocity tradeoff with their own exponent. One may alo conider the addition of the ditance term and the reciprocity term to model the tradeoff, but ince the two quantitie have different unit of cale ditance cale from to On) while reciprocity i a probability between 0 and, we believe the multiplicative formulation mae more ene. We remar that the reciprocity term v u p uv, r u )p v u, r v ) doe not conider reciprocity formed by fixed local contact. Effectively, we diregard local contact and treat p = 0 in the mall world etting Kn,, p, q, r). Thi treatment mae our analyi more treamlined and only focued on long-range contact, and it alo mae intuitive ene: the local contact are paively given baed on geographic location, while longrange contact are actively etablihed by node baed on their connection preference, and thu reciprocity baed on long-range contact could mae more ene. For example, your neighbor in the ame apartment building are your local contact by phyical location, but it doe not mean that they are your friend, and you till need to intentionally etablih friendhip baed on your preference) among your neighbor, and thu reciprocity only by phyical location doe not mean much but reciprocity baed on actively etablihed relationhip doe mean a lot for an individual. v u

10 A:0 Zhi Yang and Wei Chen. Exiting networ formation game typically ue pure-lin-baed trategy and lead to motly trivial equilibria uch a clique or tar. Different from prior game, we ue the lin probability function a the trategie, which can be viewed a a mix trategy on pure lin, but with retricted ditribution. Here, we focu on the power-law ditribution aumed in Kleinberg mall-world model, which i alo upported from finding in everal real complex networ, uch a human travel networ [Gonzlez et al. 008; Zhao et al. 05], communication networ [Kring et al. 009], trade networ [Bhattacharya et al. 008] and other ocial networ [Liben-Nowell et al. 005; Cho et al. 0]. In practice, the trategy capture certain behavioral preference of player related to connection. One concrete example i mobility preference in human travel networ, where the lin ditribution can be undertood a the trip ditance ditribution by taing the grid location of a node a it home. In thi networ, the trategy r u capture the mobility preference of individual u, large r u reult in large poibility of hort ditance travel. Our game mean that each uer adjut it mobility preference to the heterogeneou preference of other for a better payoff, uch a obtaining non-redundant information via long-ditance viit and ocial upport enforced by mutual viit. 3 PROPERTIES OF THE DRB GAME In thi ection, we conduct theoretical analyi to dicover the propertie of the DRB game. We begin by conidering the problem of the exitence of equilibria in the game, and if the anwer i ye, whether there exit multiple equilibria. In Section 3., we prove that DRB game ha only two Nah equilibria r and r 0, correponding to the navigable and random mall-world networ, repectively. Given multiple Nah equilibria, we further invetigate if the navigable mall world poee further propertie maing it the liely choice in practice. Thi i the ta of the next two ection. One way to olve the problem of multiple equilibria i to conider a more appealing equilibrium concept trong Nah equilibrium SNE). While in a NE no player can improve it payoff by unilateral deviation, in a SNE there i no coalition of player that can improve their payoff by collective deviation. In Section 3., we how that the navigable mall-world equilibrium i a SNE in the game, which i much more table than the random mall-world equilibrium. Another way to approach the problem i to tudy the convergence to equilibrium under the bet repone dynamic. Thi dynamic could help to elect among multiple equilibria of the game. In Section 3.3, we how that the navigable mall-world equilibrium i reachable via bet repone dynamic from any tate not in the other equilibrium. We alo prove that the navigable mall-world NE can alo tolerate large perturbation of player under bet repone dynamic, wherea the random mallworld NE i extremely untable under perturbation. We finally give a decription how the navigable mall-world networ i formed by ummarizing our reult in Section Equilibrium Exitence Nah equilibrium NE) for the trategic game Γ = Σ, π u ) u V i a trategy profile r S uch that each player trategy ru u V ) i a bet repone to the other player trategie u, where the bet repone i defined a follow: Definition 3. Bet repone). Player u trategy r u Σ i a bet repone to the trategy profile r u S u if π u r u, r u ) π u r u, r u ), r u Σ \ {r u},

11 A Game Theoretic Model for the Formation of Navigable Small-World Networ A: Moreover, if above i actually > for all r u r u, then u i the unique bet repone to u. We denote thi unique bet repone a B u u ). Strategy profile r i a trict Nah equilibrium if for every player u V, r u i the unique bet repone to r u. We firt how that the navigable mall-world networ i a Nah Equlibrium of the DRB game. To do o, we focu on a local region centering around a node w preferring local connection, and we have the following important lemma. LEMMA 3.. In the -dimenional DRB game, for any contant δ, there exit n 0 N may depend on δ), for any n n 0, for any non-zero trategy profile r 0, if a node w atifie r w or r w = max v V r v, then for any node u within δ grid ditance of w i.e. d M u, w) δ), u ha the unique bet repone of r u =. PROOF SKETCH). The intuition i a follow. When a node w atifying r w or r w = max v V r v, it prefer it long-range contact to be in it vicinity. For a nearby node u with d M u, v) δ, the cae of r u = provide the bet balance between good grid ditance to long-range contact and high reciprocity even jut counting the reciprocity received from w). In other cae, the node u obtain either too low reciprocity or too hort average grid ditance to long-range contact. In the cae of r u <, the node u could increae the average grid ditance to long-range contact by an factor of Oln n), but the reciprocity can be reduced by a factor of Ωn γ ), a compared with thoe provided by r u =. Thu, the ratio of payoff for r u < to payoff for r u = i at mot Oln αu n/n γ ), which i maller than one given ufficient large n. Similarly, in the cae of r u >, the node u could increae the reciprocity by an factor of Oln n), but the average grid ditance to long-range contact can be reduced by a factor of Ωn γ ), a compared with thoe provided by r u =. Thu, the ratio of payoff for r u > to payoff for r u = i alo at mot Oln n/n αuγ ), which i alo maller than one given ufficient large n. The detailed proof i included in Appendix B. The above lemma how that given a non-zero profile, we can find a local region where the bet repone of every node i. Thi lemma i intrumental to everal analytical reult, including the following theorem. THEOREM 3.. For the DRB game in a -dimenional grid, the following i true for ufficiently large n: For every node u V, every trategy profile r, and every Σ, if r u, then u ha a unique bet repone to r u : { if > 0, B u r u ) = 0 if = 0. PROOF SKETCH). For the cae of > 0, given the trategy profile of r u, for every node u, each of it nearet neighbor w i.e., d M u, w) = ) atifie r w = max v V r v. Thu by Lemma 3., node u unique bet repone to r u i r u =. When = 0, all other node lin uniformly. In thi cae, the reciprocity for node u become a contant independent of it trategy r u. Thu, r u hould be elected to maximize average ditance of u long-range contact, which lead to r u = 0. The detailed proof of thi cae i included in Appendix C. Theorem 3. how that when all other node ue the ame nonzero trategy, it i trictly better for u to ue trategy ; when all other node uniformly ue the 0 trategy, it i trictly better for u to alo ue 0 trategy. When etting = and = 0, we have: Technically, a tatement being true for ufficiently large n mean that there exit a contant n 0 N that may only depend on model contant uch a, γ and α u, uch that for all n n 0 the tatement i true in the grid with parameter n.

12 A: Zhi Yang and Wei Chen. COROLLARY 3.. For the DRB game in the -dimenional grid, the navigable mallworld networ r ) and the random mall-world networ r 0) are the two trict Nah equilibria for ufficiently large n, and there are no other uniform Nah equilibria. We next examine if there exit any non-uniform equilibrium. THEOREM 3.. In the -dimenional DRB game, there i no non-uniform Nah e- quilibrium for ufficiently large n. PROOF. Given any non-uniform trategy profile r, let V = {v r v }. If V, we can find a pair of grid neighbor u, w) with r u and r w. If V =, we can find a pair of grid neighbor u, w) with r u and r w = max v V r v. In either cae, we now the node u could obtain better payoff by unilaterally deviating to the trategy r u = by Lemma 3.. Therefore, non-uniform trategy profile r i not a Nah equilibrium. Combining the above theorem with Corollary 3., we ee that DRB game ha only two Nah equilibria r and r 0, correponding to the navigable and random mallworld networ, repectively. 3. Equilibrium Stability under Colluion While in an NE no player can improve it payoff by unilateral deviation, ome of the player may benefit ometime ubtantially) from forming alliance/coalition with other player. So we tudy a more general t-strong Nah equilibrium t-sne) to tudy the reilience to coalition. Definition 3. t-strong Nah equilibrium). For a number t {,,..., V }, a trategy profile r S i a t-trong Nah equilibrium if for all C C with C t, there doe not exit any r C S C uch that u C, π u r C, r C) π u r ), u C, π u r C, r C) > π u r ). When t = V, we imply call r the trong Nah equilibrium SNE). Note that -SNE fall bac to NE. We firt how the important reult that the navigable mall-world networ i able to tolerate colluion of any group of player, i.e., r i a V -SNE or imply SNE. THEOREM 3.3. For the DRB game in the -dimenional grid, the navigable mallworld networ r ) i a trong Nah equilibrium for ufficiently large n. PROOF SKETCH). We prove a lightly tronger reult any node u in any trategy profile r with r u ha trictly wore payoff than it payoff in the navigable mall world. Intuitively, when u deviate to 0 r u <, it lo on reciprocity would outweigh it gain on lin ditance; when u deviate to r u >, it lo on lin ditance i too much to compenate any poible gain on reciprocity. The detailed proof i in Appendix D. The above theorem how that the navigable mall-world equilibrium i not only immune to unilateral deviation, but alo to deviation by coalition of any ize, and in particular it i Pareto-optimal, uch that no player can improve her payoff without decreaing the payoff of omeone ele. After howing that the navigable mall-world i robut to colluion of any ize, we now how that random mall world equilibrium i not table even under the colluion of a pair of node. THEOREM 3.4. For the DRB game in a -dimenional grid, the random mall-world NE r 0 i not a -trong Nah equilibrium for ufficiently large n.

13 A Game Theoretic Model for the Formation of Navigable Small-World Networ A:3 PROOF SKETCH). If a pair of grid neighbor collude to deviate their trategie to, they could gain much benefit in term of reciprocity, a compared with the lo of relationhip ditance. A a reult, they would both get better payoff than their payoff in r 0. The detailed proof i in Appendix E. 3.3 Convergence under Bet Repone Dynamic For our game, we finally tudy it bet repone dynamic to invetigate it propertie of convergence to Nah equilibria. Bet repone dynamic are typically pecified in term of aynchronou tep: in each aynchronou tep, one player move from it current trategy to it bet repone to the current trategy profile, and thu the entire trategy profile move one tep accordingly. To facilitate the tudy of convergence peed, we alo loo into ynchronou tep for the bet repone dynamic: in each ynchronou tep, every player move from it current trategy to it bet repone to the current trategy profile, and collectively we count thi a one ynchronou tep. With the concept of bet-repone dynamic, we firt how that for any non-zero profile, we can find a node that trigger a cacade of adopting trategy from neighbor to neighbor of neighbor, and o on, ultimately leading to the navigable mall world equilibrium. THEOREM 3.5. In the -dimenional DRB game, for ufficiently large n, the navigable mall-world equilibrium r i reachable via bet repone dynamic from any non-zero trategy profile r 0. Moreover, if all node move ynchronouly in the bet repone dynamic, then it tae at mot n/ ynchronou tep for any non-zero trategy profile to converge to the navigable mall-world equilibrium r. PROOF. Let V w j) = {v d M v, w) j}. Given a non-zero profile r, we can find a node w atifying r w or r w = max v V r v. Given a contant δ δ ), Lemma 3. implie that for ufficiently large n, for every u V w δ), in one aynchronou tep u will et r u =. Then conider u neighbor V u δ), in one aynchronou tep each of them will alo et their trategy to. Following thi cacade it i clear that there exit a tep equence uch that the non-zero profile r will reach the navigable mall world r. We now conider that all node move ynchronouly. Again we firt find a node w atifying r w or r w = max v V r v. By Lemma 3. all node in V w δ/) δ/ ) move to trategy in the firt ynchronou tep. Conider the econd ynchronou tep. Even though we are not ure if node w adopt trategy in the firt ynchronou tep, we now that w adopt in the econd ynchronou tep ince w ha neighbor adopting after the firt ynchronou tep. Moreover, for all node in V w δ/), their mutual grid ditance i at mot δ, and thu Lemma 3. applie to thee node in the econd ynchronou tep and they all tay at trategy. Finally for their grid neighbor within grid ditance δ/, eentially node in V w δ) \ V w δ/), they will alo adopt trategy in the econd ynchronou tep. Repeating the above procedure, all node that have adopted will eep while their grid neighbor will alo adopt. Since the longet grid ditance among node in the -dimenion grid i n/, after at mot n/ ynchronou tep, all node adopt. The proof of the above theorem provide valuable inight into the calability of the game. Notice that a -dimenion grid contain a total of V = n player, o the above theorem tate that, for any non-zero trategy profile, the convergence time to navigable NE i at mot O V ) ynchronou tep if player move ynchronouly in the bet repone dynamic. Alo, any player u involved in the cacade of adopting r u = can mae thi bet deciion locally according to the trategie of the player in hi neighborhood. Thu our game i calable with the number of player.

14 A:4 Zhi Yang and Wei Chen. Next, we would lie to ee if the navigable equilibrium can alo tolerate perturbation of player under bet repone dynamic, where the perturbation could be arbitrary and there i no guarantee that perturbed player are better off. From Theorem 3.5, we now that a long a not all node deviate to zero, there exit a bet repone dynamic equence for the ytem to go bac to the navigable mall world, and if all node move ynchronouly, the ytem reache the navigable mall world in at mot n/ ynchronou tep. We now give a further reult on the tability of navigable mall-world in tolerating perturbation of random player: we how that even if each individual independently perturb to an arbitrary trategy with a fairly large probability, the ytem move bac to the navigable mall world in jut one ynchronou tep, and even if player move aynchronouly, it i guaranteed that the ytem move bac to the navigable mall world after each node tae at leat one aynchronou tep. THEOREM 3.6. Conider the navigable mall-world equilibrium r for the DRB game in a -dimenional grid > ). Suppoe that with probability p u each node u V independently perturb r u to an arbitrary trategy r u Σ, and with probability p u r u = r u. Let α min = min u V α u, then for any contant ε with 0 < ε < min{, α min }γ/4, there exit n 0 N depending only on, γ, and ε), for all n n 0, if p u n ε, with probability at leat /n, the perturbed trategy profile r move bac to the navigable mall world r ) in one ynchronou tep, or a oon a every node tae at leat one aynchronou tep in the bet repone dynamic. PROOF SKETCH). The independently elected deviation node et atifie that with high probability, for any node u, at ufficiently many ditance level from u there are enough fraction of non-deviating node. We then how that u obtain higher order payoff jut from thee non-deviating node than any poible payoff he could get from any poible deviation. The detailed proof i in Appendix F. Notice that the bound of n ε i cloe to when n i ufficiently large, meaning that the navigable equilibrium tolerate arbitrary deviation from a large number of random node. For the random mall-world networ, which i hown to be the other NE, Theorem 3.5 already implie that even one deviating player could poibly drive the ytem out of the random mall-world equilibrium and lead it toward the navigable mallworld equilibrium. However, converging to navigable mall world i not guaranteed in thi cae. In the following, we how a tronger convergence reult: if each individual u deviate from r u = 0 independently with even a mall probability, then the ytem could witch to the navigable mall world in jut one ynchronou tep, or after each node tae at leat one aynchronou tep, and the convergence to the navigable mall world i guaranteed in thi cae. THEOREM 3.7. For the DRB game in a -dimenional grid > ) with the initial trategy profile r 0 and a finite perturbed trategy et S Σ with at leat one nonzero entry 0 < max S β), for any contant ε with 0 < ε < γ/, there exit n 0 N depending only on, γ, and ε), for all n n 0, if for any u V, with independent probability of p n )ε +β, ru S \ {0} after the perturbation, then with probability at leat /n, the networ converge to the navigable mall world in one ynchronou tep, or a oon a every node tae at leat one aynchronou tep in the bet repone dynamic. PROOF SKETCH). We conider the gain of a node u when electing r u = eparately from each group of node with the ame trategy after the perturbation, and then apply the reult in Theorem 3.. The full proof i in Appendix G.