A Caching Algorithm using Evolutionary Game Theory in a File-Sharing System

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1 A Caching Algoithm using Evolutionay Game Theoy in a File-Shaing System Masahio Sasabe Cybemedia Cente Osaka Univesity 1-32 Machikaneyama-cho, Toyonaka-shi Osaka 56-43, Japan m-sasabe@cmc.osaka-u.ac.jp Naoki Wakamiya, Masayuki Muata Gad. School of Infomation Science and Technology Osaka Univesity 1-5 Yamadaoka, Suita-shi Osaka , Japan {wakamiya, muata}@ist.osaka-u.ac.jp Abstact In a P2P file-shaing system, a node finds and etieves its desied file. If multiple nodes cache the same file to povide othes, we can achieve a file-shaing system with low latency and high file availability. Howeve, a node has to spend costs, e.g., pocessing load o stoage capacity, on caching of a file. Consequently, a node may selfishly behave and hesitate to cache a file. In such a case, thee is a possibility that unpopula files disappea fom the system. In this pape, we aim to accomplish effective caching in the whole system that emeges fom autonomous and selfish node behavio. We fist discuss elationship between selfish node behavio and system dynamics accoding to evolutionay game theoy. As a esult, we show that a file-shaing system can be obust to file disappeaance depending on a cost and demand model fo caching even if nodes behave selfishly. 1. Intoduction In a Pee-to-Pee (P2P) file-shaing system, a node exchanges infomation and files with othe nodes on a logical netwok that the nodes constuct by establishing a logical link between two nodes. Each node caches its oiginal o etieved files into a local stoage to shae them with othe nodes. If multiple nodes coopeatively cache an identical file into thei stoages, a node can find it with highe pobability and etieve it moe quickly. This leads to enhance pefomance, availability, and eliability of the system [6, 11]. Howeve, caching of a file incus costs such as pocessing load and stoage capacity. A node consisting of the fileshaing system selfishly behaves because it is pimitively a use teminal. As a esult, some nodes may not be coopeative to cache files [3]. In such a situation, thee is a poblem that a node cannot use a file due to disappeaance caused by its unpopulaity. It is difficult fo a system administato to monito and manage all nodes so that the system achieves effective caching. Thus, it is desiable that selfish and autonomous node behavio leads to effective caching in the whole system. Game theoy can eveal how the selfish node behavio affects dynamics of the whole system [5, 1]. In Ref. [5], they assume that a node behaves to minimize its own costs fo using a file. The costs ae stoage capacity consumed by caching and latency to etieve a file fom its coesponding file holde, i.e., povide. Then, they pove that Nash equilibia, namely stable states of a system, exist in such a situation by using game theoy. Futhemoe, they show an optimum state of the system is equal to one of the Nash equilibia by intoducing a payment model in which a node can obtain payments to cache a file fom othe nodes that etieve the cache file. On the othe hand, in Ref. [1], they discuss coopeative node behavio in a file-shaing system unde the famewok of Multi-Peson Pisone s Dilemma. If thee is no incentive fo caching a file, all nodes become fee-ides in a Nash equilibium. Though analyses and simulation evaluations, they pesent that a node intends to contibute to caching if it can obtain payments o eputations fom othe nodes in compensation fo caching a file. Although these studies eveal how selfish node behavio affects the system pefomance in a Nash equilibium, they insist that incentive mechanisms, e.g., payments o eputations fom othe nodes, ae essential to achieve coopeative caching. Howeve, such incentive mechanisms ae not necessaily applied to a file-shaing system. Fo example, thee is a file-shaing system that hides a povide fom nodes equesting the coesponding file so as to impove the anonymity among nodes. In such a system, it is difficult fo a povide to obtain payments fom the equesting nodes. In this pape, we focus on a moe geneal file-shaing system without such incentive mechanisms. We aim to accomplish effective caching in the whole system that emeges fom selfish and autonomous node behavio taking into account its demand and cost fo caching. Fo this pupose,

2 we use evolutionay game theoy to examine how local inteaction between selfish nodes affects the dynamics of the whole system [7]. In a society of oganisms, vaious individuals influence each othe. Evolutionay game theoy oiginally ties to figue out a mechanism in which optimum behavio comes down to offsping in evolutionay pocess of oganisms by using game theoy. Suppose that individual behavio defined by genes is a stategy of game theoy and the numbe of offsping following cetain behavio is payoffs acquied by the coesponding stategy. In such a case, vaious individuals ae in stategically mutual dependence elation of game theoy. Thus, by using game theoy, we can explain the phenomenon that supeio behavio speads ove a society of oganisms though inheitance fom ancestos to offsping. Fo example, many eseaches focus on how individual behavio that seems to be selfish and uncoopeative affects the emegence of coopeative phenomena in the whole society [7, 9, 12]. We expect that evolutionay game theoy can also eveal how much selfish node behavio based on local inteaction has impact on caching condition in the whole system. In this pape, we deive the elationship between node behavio and the numbe of cache files in a stable state of the whole system by using eplicato dynamics and agent-based dynamics of evolutionay game theoy. Then, we examine effective caching in the whole system that emeges fom selfish node behavio. Replicato dynamics and agent-based dynamics ae models to deive a stategy distibution in an evolutionay stable state. Replicato dynamics is a mathematical model in which a stategy inceases when it can yield moe payoffs than aveage payoffs of all stategies. Note that eplicato dynamics can be applicable when the numbe of individuals composed of the society is elatively lage and the netwok among the individuals is mean-field like. As a esult, we can obtain the system chaacteistics in the case that a node selfishly behaves in accodance with global infomation on caching condition in a full mesh netwok. On the othe hand, agent-based dynamics models a phenomenon that a supeio stategy speads ove the netwok in a hop-by-hop manne. In agent-based dynamics, an individual plays a game once with all neighboing individuals, detemines supeioity of its own stategy based on the game esults, and finally decides the next stategy. Consequently, we can find out the system chaacteistics in the case that a node selfishly behaves based on its local infomation in vaious netwok topologies. In this pape, we fist make modeling a file-shaing system as a game taking into account cost and demand fo caching. Then, we theoetically deive the numbe of cache files in the whole system by eplicato dynamics when a node can obtain global infomation. Futhemoe, though simulation expeiments based on agent-based dynamics, we Table 1. payoff matix (geneal) playe 2 coopeato defecto playe 1 coopeato (R, R) (S, T ) defecto (T, S) (P, P ) evaluate the numbe of cache files and distance fom a node to a povide in the case that a node behaves accoding to its local infomation. Finally, we descibe an appopiate caching model to achieve effective caching fom the analyses and simulation esults. The est of the pape is oganized as follows. In section 2, we descibe models of caching games with which we deal in this pape. Then, we evaluate the system chaacteistics by eplicato dynamics in section 3 and by agentbased dynamics in section 4. Fom the esults, we examine whethe the models ae suited to effective caching. Finally, section 5 gives conclusions and futue wok. 2. Caching Game In evolutionay game theoy, a game between two abitay playes is defined as a pay-off matix as shown in Tab. 1. A defecto exploiting a coopeato gets T and the exploited coopeato eceives S. Both playes eceives R (P ) when they coopeate (defect) each othe. Pisone s Dilemma Game (T > R > P > S) and SnowDift Game (T > R > S > P ) ae examples of well-known games. In this pape, we fist model caching in a file-shaing system as a caching game between two neighboing nodes. If thee is no capacity limit on nodes that esults in no competition of stoage capacity among diffeent files, we can assume that the caching is composed of multiple sepaated caching games each of which deals with a single file. In what follows, we define caching games fo a single file fo sake of simplicity. Fo the single file caching, each node has two stategies: caching o no caching. Since a node satisfies its demand to the file by leveaging the file, we egad the benefit obtained by the use of the file as its demand b. On the othe hand, a node has to spend a cost on caching a file. In this pape, we investigate two kinds of costs: pocessing load c l caused by self o othe node s access to a file and stoage capacity c s consumed by caching a file. As futue wok, we plan to conside dynamic costs, e.g., netwok bandwidth o delay equied to etieve a file fom a povide. We do not conside the incentive mechanisms fo caching [5, 1] because we assume a geneal file-shaing system in this pape. In the case that the cost is the pocessing load, the paametes of payoff matix become R = b c l, T = b, S = b 2c l, P = (Tab. 2). Note that b 2c l is geate than. Since the elationship among paametes of Tab. 2 sat-

3 Table 2. payoff matix (caching game, cost: load) playe 2 caching no caching playe 1 caching (b c l, b c l ) (b 2c l, b) no caching (b, b 2c l ) (,) Table 3. payoff matix (caching game, cost: stoage) playe playe 1 2 caching no caching caching (b c s, b c s ) (b c s, b) no caching (b, b c s ) (,) isfies T > R > S > P, Tab. 2 is the same as SnowDift Game. On the contay, we obtain R = b c l, T = b, S = b c l, P = in the case that the cost is the stoage capacity (Tab. 3). Note that b c s is geate than. Tab. 3 holds T > R = S > P and does not coespond to any kind of existing games. In the succeeding sections, we exploe how the numbe of nodes caching a file, that is the numbe of nodes taking the stategy caching, changes in accodance with the paametes of payoff matix and examine the ealization of a file-shaing system with high file availability. 3. Theoetical Analysis by Replicato Dynamics In this section, we theoetically deive the elationship between the paamete settings in Tabs. 2 and 3 and the numbe of nodes caching a file in a stable state of a fileshaing system by eplicato dynamics [7]. If the payoff matix can enfoce moe nodes to take the stategy caching, we can enhance the file availability of the file-shaing system composed of nodes that selfishly behave based on the payoff matix Replicato Dynamics The basic concept of eplicato dynamics is that the gowth ate of nodes taking a stategy is popotional to the payoff acquied by the stategy. Thus, the stategy that yields moe payoff than aveage payoff of the whole system inceases, and vice vesa. Note that eplicato dynamics assumes that an node plays a game with all othe nodes. In what follows, we explain the deivation pocess of eplicato dynamics in a geneal game descibed as Tab. 1. Suppose that the atio of coopeatos at a cetain time is x and that of defectos is 1 x. The aveage payoff of coopeatos becomes On the contay, that of defectos is Rx + S(1 x). (1) T x + P (1 x). (2) Consequently, the aveage payoff of the whole nodes can be deived as follows: x(rx + S(1 x)) + (1 x)(t x + P (1 x)). (3) The payoff diffeence between the aveage payoff of coopeatos and that of the whole nodes is expessed as {Rx + S(1 x)} {x(rx + S(1 x)) + (1 x)(t x + P (1 x))} = {(R + P T S)x + S P }(1 x). (4) Finally, eplicato dynamics ẋ that indicates the tansition of x is defined as follows: ẋ = {(R + P T S)x + S P }(1 x)x. (5) 3.2. The Numbe of Cache Files in a Stable State The equilibia of x that satisfy ẋ = ae, 1, and R+P T S. Next, we descibe the stability of thei equilibia. In Eq. (5), (1 x)x is constantly ove zeo fo x 1. The emaining pat (R + P T S)x + S P becomes positive fo x < R+P T S and negative fo x > R+P T S because of the definition of games in section 2. As a esult, x appoaches to R+P T S independently of the initial value of x. Thus, the stable equilibium is P S x = R + P T S. (6) Eq. (6) indicates that the atio of coopeatos at a stable state depends on the paametes of the payoff matix. Based on Refs. [9, 12, 8], we fist define as the cost-tobenefit atio of mutual coopeation. means a isk that a node should take when it behaves as a coopeato. anges (,1]. The smalle value of indicates that coopeatos incease. In this pape, is detemined by b and c l o by b and c s. Theefoe, can be egaded as the atio of demand to cost fo caching. If the cost is the pocessing load, fom Eq. (6) and Tab. 2, the atio x of coopeatos at a stable state becomes x = b 2c l. (7) b c l On the othe hand, the cost-to-benefit atio of mutual coopeation is as follows: c l =. (8) b c l

4 Finally, the elationship between x and satisfies x = 1. (9) We can also deive the elationship between x and in the case whee the cost is the stoage capacity as follows: 1 x = 1 +. (1) Fom Eqs. (9) and (1), the atio x of coopeatos deteioates with the incease of the, namely the decease of the demand b to the file, independently of the cost models. In this pape, we aim to achieve a file-shaing system with high file-availability that is obust to file disappeaance. If the cost is the pocessing load, Eq. (9) denotes that a file disappeas fom the system when is close to 1 that means the demand to the file is low. On the othe hand, the cost model of the stoage capacity enhances the file availability by peventing x falling in even if is Simulation-based Analysis by Agent-based Dynamics Agent-based dynamics [7] can eveal how much caching based on local inteaction among neighboing nodes has impact on the numbe of cache files in the whole system. Moeove, we can evaluate seach latency fo a file since we can obtain the stategies of all nodes, namely the locations of cache files Agent-based Dynamics In agent-based dynamics, a node detemines its stategy by compaing its own payoff with that of a neighboing node. A node initially selects a stategy at andom. The initial atio of coopeatos and that of defectos ae fifty-fifty in Refs. [8, 9, 12]. Once a node i detemines its stategy, it plays a game once with all neighboing nodes. This is one geneation. At the end of the geneation, the node i calculates aveage A i of payoffs acquied, then detemines the stategy of the next geneation as follows. Step 1: Selection of a neighboing node fo compaison of payoffs The node i andomly chooses a node j fom neighboing nodes. Step 2: Decision of the next stategy based on the compaison of aveage payoffs If A j > A i is satisfied, the node i imitates the stategy of node j with the following pobability: P A (i, j) = A j A i T P. (11) Othewise, it does not change its stategy. The node i tends to imitate the stategy of a node that obtained Table 4. maximum hop count between two abitay nodes m = 2 m = 4 scale-fee netwok andom netwok moe payoffs than it. In addition, P A inceases in popotion to the payoff diffeence. In an actual system, we should conside the oveheads incued by playing games and exchanging payoff with neighboing nodes. We expect that these pocesses can be ealized by applying the keep-alive messaging as in Gnutella with slight modifications Simulation Expeiments Though seveal simulation expeiments, we evaluate how node behavio based on the payoff matix affects file availability and seach latency of the whole system in scalefee and andom netwoks. We evaluate the file availability by the atio x of nodes taking the stategy caching. In the case of single-file caching, the numbe of cache files is the same as the poduct of x and the numbe of nodes. On the othe hand, we define the seach latency as the aveage hop count between a node to its closest povide including itself. Note that we altenatively use the maximum hop count between two abitay nodes when a file disappeas fom the system Simulation Model. We used NetLogo [2] in ou simulation expeiments. Based on Refs. [8, 9, 12], we set simulation configuations as follows. We geneated scale-fee and andom netwoks of 1 nodes by using the topology geneato BRITE [1]. The scale-fee netwok was based on Baabási-Albet (BA) model [4] and the andom netwok followed waxman algoithm [13] with α =.15 and β =.2. We also set the numbe m of connections that a newly paticipating node established to 2 and 4. Table 4 epesents the aveage of maximum hop count between two abitay nodes in twenty netwoks. We set the caching costs c l and c s to 1, espectively. Thus, independently of the cost models, b = 1+ was deived. We configued that the initial atio of stategy caching and that of stategy no caching wee fifty-fifty. To investigate the system chaacteistics in a stable state, we show esults when 1 geneations passed. The following esults indicate the aveage of twenty simulations. We abbeviate eplicato dynamics to RD, agent-based dynamics to AD, scale-fee to SF, and andom to RND in the following figues Impact of Payoff Matix. Figue 1(a) illustates that the elationship between and the atio x of nodes taking

5 1 4.5 x AD (load).2 AD (stoage) RD (load) RD (stoage) hop count to the closest povide AD (load) AD (stoage) (a) faction of coopeatos (b) hop count to the closest povide Figue 1. payoff matix vs. file availability and seach latency (scale-fee: m=4) the stategy caching that is deived by agent-based dynamics in a scale-fee netwok with m = 4. We discove that x deteioates with the incease of, namely the decease of demand, egadless of the cost models. In addition, the cost model of pocessing load causes the disappeaance of files with lage. On the othe hand, the file availability is enhanced by using the cost model of stoage capacity even if is 1. Next, Fig. 1(a) also shows x deived by eplicato dynamics. If the cost is the pocessing load, x of agent-based dynamics is lowe than that of eplicato dynamics excluding the case of smalle. On the contay, the cost model of stoage capacity enhances the file availability as x of agent-based dynamics achieves almost the same as that of eplicato dynamics. Figue 1(b) depicts that the seach latency inceases with the gowth of, independently of the cost models. This is because the numbe of povides deceases as shown in Fig. 1(a). In the case that the cost is the stoage capacity, x is highe than.4 even fo a low-demand file. Consequently, a node can educe the seach latency by finding out a close povide Impact of Netwok Stuctue. Figue 2 illustates x and the seach latency in scale-fee netwoks with m = 2, 4. Since the aveage degee is 2m, lage m makes a netwok dense. Figue 2(a) shows that smalle m pomotes to incease high-demand files, independently of the cost models. This is due to the effect of high degee nodes. A high degee node tends to acquie moe payoffs than othe nodes and be chosen fo compaison of payoffs by its neighboing nodes. As a esult, the stategy of a high degee node is likely to spead ove the netwok. The impact of high degee nodes is acceleated in a netwok with small m whee low degee nodes fequently exist. On the othe hand, Fig. 2(b) pesents that the seach latency of m = 2 is lowe than that of m = 4 if is smalle than.3 in the cost model of pocessing load and.5 in the cost model of stoage capacity. This is because files with smalle ae moe cached in the case of m = 2 as shown in Fig. 2(a). Since the incease of esults in the decease of povides, the seach latency of m = 4 becomes supeio to that of m = 2. Note that thee is slight diffeence between m = 2 and m = 4 in the cost model of stoage capacity. Figue 2 also shows that x and the seach latency in the andom netwok with m = 4. We find that x in the scalefee netwok is mainly lage than that in the andom netwok as shown in Fig. 2(a). Contay to ou expectation, Fig. 2(b) pesents that the seach latency in the scale-fee netwok is not so supeio to that in the andom netwok despite of its lowe diamete of the netwok (Tab. 4). This is because files tend to be cached at egula nodes athe than high degee nodes in the scale-fee netwok. In othe wods, we can alleviate load concentation on high degee nodes while suppessing the seach latency. Note that the diffeence of degee distibution does not so much affect to the seach latency if the cost is the stoage capacity. In summay, we can accomplish a file-shaing system with high file-availability and low seach-latency by using the cost model of stoage capacity independently of the netwok stuctues. 5. Conclusions In this pape, we evealed the elationship between node behavio and effective caching in the whole system by evolutionay game theoy so as to accomplish a file-shaing system with high file availability and low seach latency even if nodes behaved selfishly and autonomously. We fist made modeling a file-shaing system as two kinds of caching games between two neighboing nodes. Then, we

6 1 4.5 x load, SF (m=2) load, SF (m=4) load, RND (m=4).2 stoage, SF (m=2) stoage, SF (m=4) stoage, RND (m=4) hop count to the closest povide load, SF (m=2) load, SF (m=4) load, RND (m=4) stoage, SF (m=2) stoage, SF (m=4) stoage, RND (m=4) (a) faction of coopeatos (b) hop count to the closest povide Figue 2. impact of netwok stuctue (m=2 vs. m=4, scale-fee vs. andom) showed the basic chaacteistics of the models by analytically deiving the numbe of cache files in the system based on eplicato dynamics. Futhemoe, though simulation expeiments based on agent-based dynamics, we evaluated how much local node behavio had impact on the system pefomance. Simulation esults showed that modeling the cost fo caching as the stoage capacity made a file-shaing system obust to fie disappeaance independently of the netwok stuctues even if nodes behave selfishly. As futue eseach wok, we should examine a caching game that takes into account combination of multiple costs including dynamic costs, e.g., delay and bandwidth. Futhemoe, we plan to analyze the dynamics of infomation netwoks othe than the file-shaing system and popose contol mechanisms based on the analysis. Fo example, infomation tavels along a chain of intemediate nodes towad its destination in the following infomation systems: infomation distibution on an ovelay netwok, e.g., application level multicast, and infomation diffusion and gatheing on a senso netwok. Since fowading infomation takes costs, such as netwok bandwidth and electicity consumption, an effective fowading mechanism taking into account the significance of infomation is needed. Evolutionay game theoy can eveal node behavio suitable fo such effective infomation tansfe. Acknowledgement This eseach was suppoted by a Gant-in-Aid fo Young Scientists (B) and New Infomation Technologies fo Building a Netwoked Symbiosis Envionment of The 21st Centuy Cente of Excellence Pogam of the Ministy of Education, Cultue, Spots, Science and Technology in Japan. Refeences [1] BRITE. available at bite/. [2] NetLogo. available at edu/netlogo/. [3] E. Ada and B. A. Hubeman. Fee Riding on Gnutella. Technical Repot Xeox PARC, 2. [4] A.-L. Baabási and R. Albet. Emegence of Scaling in Random Netwoks. Science, 286, [5] B.-G. Chun, K. Chaudhui, H. Wee, M. Baeno, C. H. Papadimitiou, and J. Kubiatowicz. Selfish Caching in Distibuted Systems: A Game-Theoetic Analysis. In Poceedings of PODC 24, pages 21 3, Aveio, July 24. [6] K. P. Gummadi, R. J. Dunn, S. Saoiu, S. D. Gibble, H. M. Levy, and J. Zahojan. Measuement, Modeling, and Analysis of a Pee-to-Pee File-Shaing Wokload. In Poceedings of SOSP 23, pages , New Yok, Oct. 23. [7] S. Gyogy and F. Gabo. Evolutionay Games on Gaphs. AXiv Condensed Matte e-pints, July 26. [8] C. Hauet and M. Doebeli. Spatial Stuctue Often Inhibits The Evolution of Coopeation in The Snowdift Game. Natue, 428, 24. [9] J. M. Pacheco and F. C. Santos. Netwok Dependence of the Dilemmas of Coopeation. In Poceedings of CNET 24, pages 9 1, Aveio, Aug. 24. [1] K. Ranganathan, M. Ripeanu, A. Sain, and I. Foste. Incentive Mechanisms fo Lage Collaboative Resouce Shaing. In Poceedings of CCGRID 24, pages 1 8, Washington, DC, 24. [11] A. Rowston and P. Duschel. Stoage Management and Caching in PAST, a Lage-Scale, Pesistent Pee-to-Pee Stoage Utility. In Poceedings of SOSP 21, pages , New Yok, Oct. 21. [12] F. C. Santos and J. M. Pacheco. A New Route to the Evolution of Coopeation. Jounal of Evolutionay Biology, 19(3): , 26. [13] B. M. Waxman. Routing of Multipoint Connections. IEEE Jounal on Selected Aeas in Communications, 6(9): , Dec