int. j. autonomous and adaptive communications systems...

Int. J. Autonomous and Adaptive Communications Systems, Vol. 10, No. 1, 2017 23

Copyright © 2017 Inderscience Enterprises Ltd.

A cooperation guarantee mechanism based on altruistic punishment for nodes in distributed environments

Jiaqi Liu* School of Software, Central South University, Changsha 410083, Hunan Province, China Email: [email protected] *Corresponding author

Guojun Wang School of Information Science and Engineering, Central South University, Changsha 410083, Hunan Province, China Email: [email protected]

Zhigang Chen School of Software, Central South University, Changsha 410083, Hunan Province, China Email: [email protected]

Hui Liu Department of Computer Science, Missouri State University, Springfield, MO 65897, USA Email: [email protected] Abstract: Aiming at non-cooperation behaviours in highly distributed environments such as P2P systems, researchers have presented many algorithms and mechanisms based on traditional economic theories. The hypothesis of rational man is the basic assumption in traditional economic theories. The hypothesis means that any man is a rational man who is selfish and desires to maximise his needs or desires. However, researches on behaviour economics have proved by many demonstrations that the assumption of rational man is defective. Research achievements in demotics and other fields find that humans have a predisposition to punish those who violate group-beneficial norms, even when this imposes a fitness cost on the punisher. Based on the altruistic punishment, we present a whole new mechanism considering both rational nodes and irrational nodes in distributed environments reference from rational man and irrational man in social area without central controller. A mechanism based on altruistic punishment is proposed to improve the cooperation degree. Theory analysis and simulation results prove the ration and validity of this paper.

24 J. Liu et al.

Keywords: distributed environments; cooperation; altruistic punishment; ration; behaviour economics.

Reference to this paper should be made as follows: Liu, J., Wang, G., Chen, Z. and Liu, H. (2017) ‘A cooperation guarantee mechanism based on altruistic punishment for nodes in distributed environments’, Int. J. Autonomous and Adaptive Communications Systems, Vol. 10, No. 1, pp.23–37.

Biographical notes: Jiaqi Liu is a Postdoctoral scholar in School of software at Central South University, China. She received her BSc in Computer Science and Technology, MSc and PhD in Computer Application Technology at Central South University, China, in 2005, 2008, 2012, respectively. Her current research interests are network and distributed computing.

Guojun Wang is a Professor of Computer Science at the Central South University, China. He received his BSc in Geophysics, MSc in Computer Science, and PhD in Computer Science at Central South University, China, in 1992, 1996, 2002, respectively. His research interests include network and information security, internet of things, transparent computing, and cloud computing, internet of things, and cloud computing.

Zhigang Chen is a Professor of Computer Science at the Central South University, China. He received his PhD in Computer Science from Central South University, China in 1998. His current research focuses on wireless networks and opportunistic network.

Hui Liu received her BSc and MSc in Computer Science from Central South University, China in 1998 and 2001, respectively. She received her PhD degree from the Department of Computer Science at Georgia State University in 2005 and is an Associated Professor in Missouri State University. Her research interests include wireless networks, mobile computing and parallel algorithms.

1 Introduction

It is a hot spot that how to guarantee the cooperation among nodes in a highly dynamic, heterogeneous and distributed environments such as P2P systems. Incentive mechanisms and game theory are widely used to improve the cooperation. However, those mechanisms or theories are all based on the hypothesis of rational man. They assume that nodes are rational nodes including either full rational nodes (e.g., assumptions in incentive mechanism and game theory) or limited rational nodes (e.g., assumptions in evolutionary game theory). Since developments of behaviour economics, a lot of researches proved that the assumption of rational man is defective (Gintis, 2000; Fehr and Rockenbach, 2003; Sánchez and Guesta, 2005). Human behaviour could be affected by many factors including fairness preference, altruism, reciprocity, and so on (Hetzer, 2011). Humans have a predisposition to punish those who violate group-beneficial norms, even when this imposes a fitness cost on the punisher (Dominique et al., 2004). This paper fully considers those irrational behaviours of nodes, and then, presents a whole new cooperation guarantee mechanism based on altruistic punishment. Theory analysis and simulation results prove the ration and validity of the mechanism.

A cooperation guarantee mechanism based on altruistic punishment 25

2 Related work

In dynamic distributed environments especially in mobile environments such as peer-to-peer (P2P) networks, ad hoc, wireless sensor networks and so on, the cooperation among nodes is very important for the performance of the environment. Currently, there have been many researches on nodes’ cooperation in distributed environments (Wei et al., 2011; Li and Shen, 2011). Based on the assumption that users will selfishly act to maximise their own rewards, Golle et al. (2001) consider the free-rider problem that arises in P2P file sharing networks, and constructs a formal game theoretic model of the system, then analyses equilibria of user strategies under several novel payment mechanisms. It supports and extends upon our theoretical predictions with experimental results from a multi-agent reinforcement learning model. Fehr and Gachter (2002) generalise from the traditional symmetric evolutionary prisoners dilemma (EPD) to the asymmetric transactions of P2P applications, maps out the design space of EPD-based incentive techniques, and simulates a subset of these techniques. Yang et al. (2007) used a trust-management system to give priority and premium services to users with high reputation, while Fernandes et al. (2004) introduce economical incentives to motivate users to provide the feedback needed to setup the trust infrastructure. Such incentives are given in terms of rewards that can be spent to get a discount on future service requests, while there is no way to cash them. Zhang et al. (2007) present a collaborative system merging trust and economical incentive mechanisms. Peers providing the same type of service are grouped into a club, within which the price for the service is first determined by the club on the basis of classical competition models, and then adjusted on the basis of the trust of the club on the peer requesting the service. Bogliolo et al. (2012) propose a general framework which combines full-fledged trust-management mechanisms and virtual currency systems to build a comprehensive incentive mechanism specifically tailored to user-centric networks. Xu et al. (2013) propose a scheme which is based on reputation in opportunistic networks. The nodes whose reputation value is higher than the threshold get in the vicinity of each other, and they exchange messages so that their reputation values will increase, if any party refuses to forward messages then its reputation value will decrease. Wu et al. (2014) present a reward-based incentive mechanism to stimulate the peers to contribute their local storage resources in a P2P-VoD system. In particular, the content provider proposes a price for each video.

Above all, though there are many good incentive mechanisms or game theory for nodes cooperation now, most of those researches regard the assumption as the basic that users will selfishly act to maximise their own rewards. Differing from current cooperation guarantee researches, the paper does not assume the rational users but it admits the irrational behaviour such as altruistic punishment. Theory analysis and simulations results show that the altruistic punishment-based mechanism can induce nodes to form the cooperation preference.

3 The analysis of rationality

This paper uses some research achievements in demotics and behaviour economics for reference. It is rational because there are some similar characteristics among distributed environments and those academic subjects as follows:

26 J. Liu et al.

1 each node in a distributed environment has its own benefit, limited energy and resources

2 the degree of contributing or consuming resource by each node is affected by its own and others’ behaviours

3 the nodes decide the patterns of behaviour they will display by themselves

4 nodes are heterogeneous and dynamic

5 the feasibility of a distributed environment is strongly based on the premise of cooperative behaviours.

From above similarities, we can draw a conclusion that the useful mechanisms in demotics and corresponding fields will also be useful in some degree for those similar problems in distributed environments.

The cooperation is a very important and necessary problem in many research fields. It is reasonable that the altruistic punishment can be used to guarantee the cooperation in distributed environments for following reasons.

1 A lot of researches in economics show that a group’s cooperation degree will be improved if members have the ability of punishment (Fehr and Gachter, 2002). For example, the utility of punishment in repeated interaction is more than that in unrepeated interaction. The latter means each interaction begins between different and strange entities. In distributed environments, there is much repeated interaction.

2 The multilateral trading (i.e., multilateral interaction) in human society is very obvious. Many researches (Gintis, 2000; Fehr and Rockenbach, 2003; Sánchez and Guesta, 2005) have proved that the punishment is an effective mechanism for preventing selfish entities from non-cooperation in the public good game. Though the incentive mechanism is also useful, its effect is less than the punishment (Fehr and Gachter, 2002). Thus, in distributed environments, the cooperation mechanism based on actively punishing is better than that based on being passively incented, although the punishment is costly for them and yields no material gain.

3 It is decided by the inherence of distributed environments where are highly dynamic. It is found that as a representative incentive mechanism, reputation mechanisms can get a good result only when the group size is very small and nodes’ migration is little (Damiani et al., 2002). So it is a reasonable inference that reputation mechanisms are not good enough for distributed environments. Moreover, many approaches show that if there are punishments, the cooperation will be maintained in high degree even though in unrepeated interaction or at the last round of repeated interaction (Fehr and Gachter, 2002).

4 The altruistic punishment-based cooperation guarantee mechanism

4.1 Basic assumptions

There are three types of nodes called altruistic punisher (AP), cooperation node (CN), non-cooperation node (NCN), respectively. The AP unconditionally cooperates with others and would like to cost itself to punish NCNs. The CN only cooperates with others


unconditionally. The NCN is the traditional so-called rational node who will selfishly act to maximise its own rewards. The NCN will not cooperate with others in some probability until its income of non-cooperation will be less than its cost of punishing. The paper assumes that the distributed environment has characteristics as following:

1 Nodes in the environment are divided into many different groups. The size of each group is little enough to guarantee each node in the same group to know any other node’s situation and communication with.

2 The figure of central authorities or servers is lacking, that is, leaving the decision to cooperate or not under the sole responsibility of the participant, without any outside influence or incentive.

3 In each group, the correlation degree between any two nodes is low so that each node is strange to others.

4 Sharing is the main characteristic differing from other environments. In each group, the benefit will be shared jointly by nodes in the same group either it is produced by solo node or it is produced by nodes’ cooperation.

5 The punishment means that the punished node must leave the environment.

4.2 Definitions of parameters

φi The benefit of one interaction of the node i. When nodes work cooperatively in a group, each node produces an amount b at cost c (all benefits and costs are in fitness units). It is assumed the output of one group is shared equally by nodes in the group, so if all group members work, each member has a net income φi = b – c.

s The node will cost s if it is punished.

fa The proportion of AP in one group.

fc $the proportion of CN in one group.

ρs Suppose a NCN shirks (that is, does not cooperate) at a fraction ρs in the system, so the average rate of shirking is given by ρ = (1 – fa – fc)ρs, ρs ∈ [0, 1].

μ The fitness value of group output, μ = n(1 – ρ)b, where n is the size of the group. Since output is shared equally, each node receives (1 – ρ)b. Resulting from a NCN’s shirking, the loss of the group is bρs.

λ(θ) the fitness cost of cooperating function, where θ ≡ 1 – ρs. It is increasing and convex in its argument (i.e., , 0,λ λ′ ′′ > λ(1) = c, λ(0) = 0). If we assume that benefits from cooperation are always more than costs from cooperation to the cooperators, we can get the inequation as θb > λ(θ) > when ρs ∈ [0, 1). Thus, at every level of effort θ, the cooperation helps the group more than it hurts the cooperator. λ(θ) does not include the cost for punishing NCNs. One node costs c for cooperation, while it costs 0 for non-cooperation.

28 J. Liu et al.

4.3 The altruistic punishment mechanism

The altruistic punishment-based cooperation guarantee mechanism (APCGM) works as follows. The fitness cost to an AP for punishing a shirker is cp > 0. It is obvious that not all shirking nodes will be punished at rate 100% in a real environment. Thus, it is assumed that a node shirking at rate ρs will be punished with probability faρs. Punishment means being ostracised from the group. NCNs, given their individual assessment s of the cost of being ostracised, and with the knowledge that there is a fraction fa of APs in their group, choose a level of shirking, ρs, to maximise expected fitness. Writing the cost for obtaining expected fitness, ω(ρs), as the cost of cooperation plus the expected cost of being ostracised, plus the node’s share in the loss of output associated with others’ shirking, we have:

( ) ( )1 /s s a s sω ρ λ ρ sf ρ ρ b n= − + + (1)

Then, NCNs select a non-cooperation probability *sρ to minimise the ω(ρs). Assuming an

interior solution, this is given by

( ) ( )* *1 / 0s s aω ρ λ ρ sf b n′ ′= − + + = (2)

Requiring the shirker to equate the marginal fitness benefits from non-cooperation (the first and third terms) with the marginal costs of greater shirking, namely the increased likelihood of bearing the fitness cost of ostracism (the second term). Since ( ) 0,λ θ′′ > there is at most one solution to this equation, and it is a minimum. This first-order condition (2) shows that NCNs who inherit a large s (i.e., who believe the cost of ostracism is very high) will shirk less, as also will those in groups with a larger fraction of APs.

If there is no cost, the fitness value of each node is (1 – ρ)b. Since there are different costs according to different behaviours (i.e., altruistic punishment, cooperation and non-cooperation), there will be different expected benefit of each group member. The benefit of a CN is,

(1 )cη ρ b c= − − (3)

For a NCN, the benefit is equal to the node’s fitness minus the cost of cooperation and the expected cost of ostracism, that is,

( )(1 ) 1s s a sη ρ b λ ρ sf ρ= − − − − (4)

Each AP chooses a random node to monitor; this node is a NCN with probability (1 – fc – fa) and this node shirks with probability ρs. Thus, this gives the benefit of the AP is as follows:

( )(1 ) 1a p c a sη ρ b c c f f ρ= − − − − − (5)

We can gain more insight into the dynamics of this mechanism by choosing a specific function λ (1 – ρ) satisfying the conditions λ(1) = c, λ(0) = 0, (1 ) 0λ ρ′ − < and

(1 ) 0.λ ρ′′ − > Extensive simulations suggest that the exact form of this function is unimportant, and the simplest function satisfying these conditions is


2(1 ) (1 )λ ρ c ρ− = − (6)

Using (6) and (2), it is easy to get the non-cooperation probability of NCNs according to the function

( )max

max

2 /120

aa a

s a

a a

f sn b c b nf fρ f cn s

f f

+ −⎧ − ≤ =⎪= ⎨⎪ >⎩

(7)

where the maxaf is the most proportion of APs in one group. Figure 1 shows the dynamic

change of behavioural types. This diagram is based on s > 2c, that is, the cost of being ostracised is greater than twice the cost of cooperating for one period.

Figure 1 Within group dynamics with behaviour types change

In this figure, each point in the simplex is a distribution of behavioural types. The vertex S refers to the all-NCN composition (fa = fc = 0), vertex A refers to the all-AP composition (fa = 1) and C refers to the all-CN case (fc = 1). For the parameter values

illustrated in the figure, there is some fraction of APs max 2 /a

c b nfs

−= (the line BD) at

and above which NCNs do not shirk at all, but below which they shirk at a strictly positive rate. Thus, in the triangle RBD all three types have equal payoffs, equal to b – c. Along the whole CA segment, APs and CNs do equally well, since there is no NCN to be punished (the payoffs are again b – c). Along the CS segment, APs are absent (i.e., fa = 0), so NCNs get more benefits than CNs (since λ(1) = c while λ(0) = 0). On the segment DS, NCNs get more benefits than APs. It can be clearly shown in equation (8).

( )1 /s a a pη η f c b n− ≥ − + (8)

30 J. Liu et al.

where the non-cooperation can optimise NCNs. So when σs ∈ (0, 1), we know from (2) that an increase in fa holding the frequency of NCNs constant, must entail a decline in σs. This, along with equation (7), means that lowering the fraction of NCNs increases the payoff to APs relative to NCNs in the area DBCS. Moreover, CNs always have higher payoffs than APs in the interior of this area. We conclude that the only asymptotically stable equilibrium of the system is the all-selfish point S, and its basin of attraction consists of all interior points below the line BD in Figure 1.

Clearly, then, if cooperation is to be sustained, it must be because CNs, who undermine the cooperative equilibrium by driving out APs, must themselves be harmed by shirking NCNs when APs are rare. Figure 1 illustrates exactly this process, with NCNs proliferating at the expense of both CNs and APs once the frequency of APs falls below a threshold. We shall show by simulation that this is indeed the case for a wide range of plausible parameter values.

In the paper, the initial APs is the CNs who have the high contribution level. The setting results from achievements of Fehr and Gachter (2002). Fehr and Gachter (2002) found that most (more than 70%) acts of punishment were executed by above-average contributors in public good game. And hence, the bigger the contribution difference between the contributors and the defectors is, the more probability that the contributor on their own initiative to punish the defectors is. That is, the CN with the high contribution level could be mutated to be a punisher. The contribution level (gp/GP) in the paper is defined as the ratio of the ability which one node contributing (gp) to the node’s total ability (GP). It is clear that the contribution level is a relative value rather than an absolute value. For example, the node A’s total ability (GPA) can be valued as 10 units, and the node B’s total ability (GPB) is 20 units. If the node A contributes 80% ability while the node B contributes 70% ability, the contribution level of the node A is higher than that of the node B (80% > 70%). The absolute value (10*80%) < (20*70%), that is gpA < gpB, but it is shown by the percentage that the node A is more likely to contribute its ability (i.e., 80%) than the node B (i.e., 70%).

When a CN’s benefit is less than b – c, there are NCNs in the system because the non-cooperation behaviours minish the benefit. Then the CN decides to whether cost itself to punish NCNs or not. The former choice will make the CN to be mutated to an AP. When the proportion of NCNs in the system is less than a threshold value, APs will punish those NCNs. The more the APs are, the more the probabilities of NCNs being punished are. Given the pattern of punishment, the investment behaviour of NCNs seems quite rational. To avoid punishment, NCNs are going to cooperate by a relatively high probability. Thus, the total cooperation level of the whole system will be improved so as to increase the system’s income. From equation (7), it is clear that NCNs will cooperate

by 100% when the proportion of APs satisfies 2 / .ac b nf

s−> Thus, the system is a full

cooperation system. Figure 2 shows the group average payoffs as a function of the distribution of types

within a group. The curved lines are iso-group-average-fitness loci, showing clearly that average group fitness increases as we move away from the unique stable equilibrium S.


Figure 2 Average payoffs and group composition

5 Simulations and evaluations

In the paper, the interaction is defined as the process that all nodes have completed one time to choose a type of behaviour. The experiments are based on a mobile P2P-like environment with Java programme language. To simulate the mobile characteristic, the paper sets that there are the proportion α of new nodes to all nodes in the same group are independent of and permitted to join the group in the interaction, at the same time, there are fraction β of group members to leave the group. To avoid the too big or too small group size, each group’s size will be changed in every interaction. There will be an upper limit nmax and a lower limit nmin of the group size. The node number will not be more than nmax. If the node number is less than nmin, the group will be dismissed. Nodes in that group will be regarded as new nodes to rejoin other groups. Table 1 Baseline parameters

Parameters Values Detailed Descriptions b 0.2 Income per node, no shirking c 0.1 Cost of cooperating, no shirking cp 0.1 Cost of punishing

β 0.2 Emigration rate

α 0.1 Immigration rate n 15 Initial group size m 30/60/120 Number of groups nmin 3 Minimum group size nmax 20 Maximum group size s [0, 1] Initially seeded expected cost of ostracism Uniformly distributed on this interval

0sf 0.9 The initial proportion of NCNs

32 J. Liu et al.

For our simulation, we setup various group number including 30 groups, 60 groups and 120 groups respectively, each group starting out with 15 members, the nmin is 3, and the nmax is 20. We set the initial frequency of CNs at 10%, and the initial frequency of NCNs at 90%. We assigned each NCN a cost s of being ostracised, using a random number drawn from a uniform distribution on the interval [0, 1]. We experiment 100 times respectively for each group numbers (i.e., 30/60/120) to take the 1,000 times interaction each time. The detailed parameter set is shown in Table 1.

Simulations are used to prove the validity of the APGCM, that is, to show if it can decrease the number of NCNs. Thus, the paper concentrates on several factors including the proportion of each type of nodes, the non-cooperation level and the proportion of ostracised nodes who are readmitted to a group. We assume an ostracised node works alone for a period of time before being readmitted to a group. Above factors are simulated and evaluated as follows respectively.

5.1 The proportion of various nodes

After every interaction, each node chooses a type of behaviour in next interaction. The proportion of each type nodes will be counted. Figure 3 shows the change of the proportion of each type nodes with the increase of the number of the interaction. To be convenient for read, the paper only shows data in 300 times interaction.

From Figure 3, the proportion of NCNs observably decreased in the first 50 times interaction, while the proportion of CNs and APs distinctly increased. From the 50 times to the 250 times interaction (as shown by dash line), the proportions of various type nodes change slowly. In the system with N = 450, the proportion of each type nodes will tend to be stable that 40% of nodes are NCNs, 32% of nodes are CNs and 28% of nodes are APs. Figure 3 only shows two network scales of 450 and 1,800 nodes. It is clear that the network scale little affects the proportion of nodes’ types. Whatever the network scale is, the trend and the last value of each type’s proportion are respectively similar.

5.2 The non-cooperation level

The term ‘the non-cooperation level’ is used to evaluated the cooperation performance of system. It is decided by the non-cooperation probability of system. Figure 4 shows the trend of non-cooperation probabilities on three different network scales of 450, 900 and 1,800 nodes.

From Figure 4, the non-cooperation probability of each network scale tends to be stable after about 150 times interaction, as shown by dash line. When the network scale is 450 nodes (i.e., N = 450), the non-cooperation probability is less than 10%. The non-cooperation probability is near 10% for 900 nodes scale and is less than 12% for 1,800 nodes scale. Since the network scale increases, the non-cooperation probability increases little. But whatever the network scale is, the trend of the non-cooperation probability decreases quickly at the first interaction and is going to be stable at last. As the figure shows, simulation results agree well with analytical results. The altruistic punishment prevents NCNs from non-cooperation behaviours, that is, decreases the non-cooperation level so as to improve the system’s cooperation.


Figure 3 The proportion of various type nodes, (a) N = 450 (b) N = 1,800 (see online version for colours)

(a)

(b)

34 J. Liu et al.

Figure 4 The non-cooperation probability of system (see online version for colours)

5.3 The system’s incomes

Figure 5 shows the system’s incomes for various network scales. When there are 450 nodes in the system, the income tends to be stable after about 70 times interaction (as shown by the left dash line). The numbers of interaction times are both about 120, as shown by the right dash line, when the network scale is 900 and 1,800, respectively.

Figure 5 The change of the system’s income (see online version for colours)

The system’s income is affected by factors such as proportions of nodes’ types, the average non-cooperation probability and so on. When those factors are stable, the system’s income will be stable.


5.4 The proportion of punished nodes

In Figure 6, it is shown the proportion of punished nodes changes since the interaction time increases.

Figure 6 The proportion of punished nodes (see online version for colours)

At first, with the increase of interaction time, the proportion of punished nodes increases quickly. That is because there are many NCNs (i.e., 90%) with high non-cooperation probabilities at beginning. Thus, there will be many nodes to be punished. After about 50 times interaction, the proportion of punished nodes slowly increases until after about 300 times interaction when the proportion tends to be stable. That is, there is no node to be punished at that time. The stabilisation of the proportion does not mean that there is no NCN in the system. From Figure 3, we know that there are always many NCNs in the system. The stabilisation in Figure 6 means that the proportion of NCNs’ non-cooperation behaviours lies in an acceptable range, which the APs think that it is not necessary to cost more to punish NCNs. Thus, the number of punished nodes will not be increased though there are still many NCNs in the system.

5.5 The cooperation proportion of rejoined nodes

Those ostracised nodes will be readmitted to the system after working alone for a period of time. The cooperation proportion of rejoined nodes is shown in Figure 7.

In the normal course of events, the rejoined nodes will tend to cooperate since they have painful experiences of being ostracised. It is assumed that working alone leads to too few incomes to live. If there are many incomes when working alone, a node would not be eager to join a system which has rigorous rules. Figure 7 proves above analysis. In Figure 7, the more interaction times are, the more ostracised nodes rejoin. After about 50 times interaction, all ostracised nodes rejoin the system as CNs. That is, the altruistic punishment improves the cooperation by ostracising NCNs.

36 J. Liu et al.

Figure 7 The cooperation proportion of rejoined nodes (see online version for colours)

At the beginning, there are too many NCNs so that several CNs who have high contribution level change from CNs to APs voluntarily. Thus, the number of APs in Figure 3 increases, and by ostracising NCNs, many NCNs are transformed to be CNs as shown in Figure 7. As shown in Figure 3, the number of NCNs decreases while the number of CNs increases even though some primary CNs are changed to be APs. The more APs are, the less the non-cooperation probability of the whole system is. As shown in Figure 4, the non-cooperation probability quickly decreases and is near 10% after about 50 times interaction. From above simulation results, the APCGM is useful.

6 Conclusions

Differing from current researches on cooperation guaranteeing, the paper introduces approaches in demotics and other fields about entities’ behaviours and presents a whole new mechanism considering both rational nodes and irrational nodes in distributed environments. Theory analysis and simulations results have proved the ration and validity of this mechanism. This paper concentrates on the theoretic analysis of the validity of the cooperation guaranteeing mechanism. The evaluation is used to validate if the mechanism can agree well with analytical results. Thus, a realistic behaviour exhibited in P2P systems such as BitTorrent is not considered in this paper instead of the theoretic analysis as described in Section 3. Simulations of the realistic behaviour in real P2P systems will be the future work.

Acknowledgements

This research is supported by the National Natural Science Foundation of China under Grant No. 61272149 and No. 61309001, by Doctoral Program Foundation of Institutions of Higher Education of China under Grant No. 20100162120015 and No. 20130162120080, and by the postdoctoral Foundation of Central South University.


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