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Modeling Networked Systems Using the Topologically Distributed Bounded Rationality Framework DHARSHANA KASTHURIRATHNA, MAHENDRA PIRAVEENAN, AND SHAHADAT UDDIN Complex Systems Research Group, Faculty of Engineering and IT, University of Sydney, Sydney, 2006, Australia

Received 1 December 2015; revised 18 March 2016; accepted 28 March 2016

In networked systems research, game theory is increasingly used to model a number of scenarios where distributed decision making takes place in a competitive environment. These scenarios include peer-to-peer network formation and routing, computer security level allocation, and TCP congestion control. It has been shown, however, that such modeling has met with limited success in capturing the real-world behavior of computing systems. One of the main reasons for this drawback is that, whereas classical game theory assumes perfect rationality of players, real world entities in such settings have limited information, and cognitive ability which hinders their decision making. Meanwhile, new bounded rationality models have been proposed in networked game theory which take into account the topology of the network. In this article, we demonstrate that game-theoretic modeling of computing systems would be much more accurate if a topologically distributed bounded rationality model is used. In particular, we consider (a) link formation on peer-to-peer overlay networks (b) assigning security levels to computers in computer networks (c) routing in peer-to-peer overlay networks, and show that in each of these scenarios, the accuracy of the modeling improves very significantly when topological models of bounded rationality are applied in the modeling process. Our results indicate that it is possible to use game theory to model competitive scenarios in networked systems in a C 2016 Wiley Periodicals, Inc. way that closely reflects real world behavior, topology, and dynamics of such systems. V Complexity 21: 123–137, 2016 Key Words: networked systems; peer-to-peer networks; network routing; computer security; game theory; TDBR model

1. INTRODUCTION

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Correspondence to: Mahendra Piraveenan; [email protected]

Q 2016 Wiley Periodicals, Inc., Vol. 21 No. S2 DOI 10.1002/cplx.21789 Published online 7 May 2016 in Wiley Online Library (wileyonlinelibrary.com)

E-mail:

raditionally, it was assumed in information technology that all users of a computer network would be willing to co-operate for the greater good of the community at large [1,2]. However, phenomena such as freeriding in peer-to-peer networks [3], selfish link formation

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in overlay networks [4], or unfairness in ad hoc networks [5] challenge this traditional wisdom. Therefore, recent efforts in modeling such systems have focused on considering the participants in such systems as selfish and competing entities [1,2,4,6–8], and relying on incentive mechanisms to induce the selfish entities to co-operate. Game theory is ideally suited to model the dynamics of such competitive entities, and as such, many recent studies have focused on game theoretic modeling of networked systems [1,2,4,6]. It has, however, been found that system modeling using game theory often does not precisely match the real world dynamics, topology, and decision making patterns in such systems. For example, if a game theoretic model were to be used to capture security level settings of computers against a distributed denial of service (DDoS) attack, as done in [1], the resulting Nash equilibrium suggests that all computers must have an identical level of security setting. However, in real world, security settings tend to follow a heterogeneous distribution within a computer network [9,10]. Similarly, the current game theoretic modeling seems not to result in scale-free networks, which are by now a well-established phenomenon cutting across domains [11–15]. For example, peer-to-peer overlay networks constructed using existing game-theoretic overlay network construction models [4] are not particularly scalefree, while real world overlay networks, such as Gnutella, are shown to be scale-free [16]. Furthermore, even when game theoretic models do produce scale-free networks, the corresponding scale-free exponents are too low compared to the scale-free exponents of real-world peer-topeer overlay networks, as we show later in the article. Similarly, in peer-to-peer network routing, the existing game theoretic models suggest that without any incentive mechanism in place, the pure Nash equilibrium will result in the ‘‘tragedy of the commons’’—a situation where users do not share any files and refuse to route any data to minimize the costs they incur [1]. It is suggested that incentive mechanisms and reputation systems need to be used to facilitate cooperation in a network of self-interested players, and some deployed peer-to-peer systems such as KaZaA and BitTorrent indeed rely on such incentive mechanisms. However, in many real-world peer-to-peer networks, collaboration seems to be sustained as a strategy even without any particular incentive or reputation mechanisms in place [1,17]. These and many other examples (such as TCP congestion control) mentioned in literature suggest that classical game theory does not adequately model the dynamics and topological evolution of such networked computer systems. In discussing the reasons for this inadequacy, researchers have been quick to point out that the ‘‘absolute rationality’’ assumption of classical game theory is to blame [1]. Indeed, one of the key underlying assumptions in Nash

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equilibrium is that players are fully rational [18], and there would not be any cognitive or temporal limitation to their decision making ability. However, most real-world strategic decision making scenarios involve players with nonoptimal or bounded rationality, prompting them to deviate their behavior from that of Nash equilibrium [19]. The possible limitations, such as the amount of information at hand, cognitive capacity, and the computational time available, may force a self-interested autonomous player or agent to have bounded rationality and, therefore, to make nonoptimal decisions [20,21]. Therefore, acknowledging that Nash equilibria might be too restrictive in modeling real world behavior, studies have attempted to use competitive equilibria of more general forms [1], or used equilibria resulting from learning behavior [22]. Nevertheless, these studies do not make a direct attempt to model bounded rationality of players as a heterogeneous distribution and then to use it to predict system dynamics and evolution. Meanwhile, recent work in the field of networked game theory has proposed a topologically distributed bounded rationality (TDBR) model [23], which attempts to quantify the distribution of rationality of players who are part of a heterogeneous network. As explained later in detail, this model is based on studies made in psychology and cognitive science, which suggest that the rationality of individuals may be correlated to the level of their social interactions [24–27]. Moreover, the model assumes that the level of information available about the environment and the cognitive capacity may affect the rationality of an autonomous agent, which may be reflected by the social placement of that agent. In this article, we attempt to use this TDBR model (which we abbreviate as the TDBR model for convenience, though the proposing paper did not use this abbreviation), and the quantal response equilibrium (QRE) [18], which is a more generalized form of Nash equilibrium, to model the behavior and topological evolution (where applicable) of networked systems. In particular, we undertake three case studies: (a) link formation in peer-to-peer overlay networks (b) assigning security levels to computers in computer networks (c) routing in peer-to-peer overlay networks. In each case, we compare our modeling approach with more ‘‘classical’’ game theoretic modeling approaches prescribed in the literature, and show that our approach results in a better fit to the real world behavior of these systems. Therefore, the contribution of this paper is to show that several networked systems are better modeled using the TDBR framework compared to traditional game theoretic models. The rest of the article is organized as follows. The next section provides the background for this study: it is divided into several subsections, beginning with an overview of classical game theory, including the concepts of Nash and Quantal Response Equilibria. It then describes

Q 2016 Wiley Periodicals, Inc. DOI 10.1002/cplx

the traditional game theoretic models used to simulate each of the case studies mentioned above. Following this, it introduces the TDBR model, which we utilize in this work. The following section is the results section, which is divided into three subsections corresponding to the three case studies we undertake. In each subsection, the results of our modeling are compared to the classical models described in the background section. The final section includes our conclusions, discussion, and pointers to future research.

2. BACKGROUND 2.1. Strategic Games Game theory studies strategic decision making among self-interested entities [28,29]. Game theory originated as a branch of micro-economics [28]. However, due to the prevalence of strategic decision making scenarios in many disciplines of study, it has been widely adopted to myriad disciplines such as evolutionary biology, computer science, political science, and even quantum mechanics [30–34]. Different games are used to define a plethora of abstract decision making scenarios. Game theory provides the necessary conceptual background to model complex socioecological systems that involve multiple self-interested entities and decision making scenarios [31,32,35,36]. Also, it is often used as a predictive tool to optimize the decision making in complex decision making environments [37]. In recent years, networked game theory, or the study of games played on (typically heterogeneous) networks, has emerged as an important branch of game theory [38–45]. Perhaps the next frontier to emerge in this regard will be the adoption of networks of networks (or multilayered networks) [46] into networked game theory, and the resultant analysis of evolutionary games on such multilayered networks [47].

2.1.1. Nash Equilibrium Nash equilibrium is one of the pivotal concepts in game theory [48]. Nash equilibrium predicts that in a strategic decision making environment, there exists an ‘‘equilibrium’’ from which no player would benefit by deviating. That is, the payoff of each player would be optimum at this equilibrium. Nash equilibrium is defined for both pure and mixed strategies [49]. A pure strategy is a strategy that is adopted in a binary fashion. If a mixed strategy is applied, each strategy available to a player is selected according to a probability distribution. In fact, pure strategies can be thought of as special cases of mixed strategies, where each strategy is selected with a probability 1 or 0.

2.1.2. Bounded Rationality It has been observed that in experimental settings, players deviate substantially from the predictions given in Nash equilibrium [19]. One key reason for this deviation is

Q 2016 Wiley Periodicals, Inc. DOI 10.1002/cplx

the nonperfect, or bounded rationality that they possess. Nash equilibrium assumes that the players are perfectly rational [18]. In other words, they always adopt the strategy that maximizes their utility, where rationality is identified as the tendency to maximize one’s own utility. However, in the real-world, the players may not be perfectly rational and could possess ‘‘bounded-rationality.’’ The reasons for this bounded rationality may be the lack of information available about the strategies adopted by the opponents and their respective payoffs, limitations in cognitive capacity of the player or the limitation of computational time available to make the strategic decision [20,23]. As all these limitations exist in various proportions in real-world game settings, it is expected that the players in such settings would have bounded rationality.

2.1.3. Quantal Response Equilibrium Due to the existence of bounded rationality, players may make nonoptimal decisions. Thus, players could be modeled as noisy players to account for bounded rationality. The QRE [18] presents an analogous way to model games with noisy players. Probabilistic choice models such as logit and probit, used in the calculation of QRE, have the inherent feature where the deviations of optimal decisions are negatively correlated with the associated costs. In other words, players are likely to select better choices than worse choices, although there is no guarantee that they will always select the best possible choice. Formally put, a quantal response function maps the vector of expected payoffs from available choices into a vector of choice probabilities that is monotone with the expected payoffs [23]. Such a function as a strategy adoption rule was used perhaps for the first time in the work of Blume [50]. Considering a general payoff matrix similar to that given in Figure 1, we can use the quantal response logit function given in Eq. (1) to derive the equilibrium of a player with a nonperfect rationality. The logit function given in Eq. (1) is often used to derive the equilibrium probabilities at QRE [18,23,51]. i

i

eki E ðsj ;PÞ Pji 5 Xj i i e ki E ðsk ;PÞ k51

(1)

where Pji denotes probability of player i selecting the strategy j. The expected utility player i receives by choosing strategy j is given by E i ðsji ; PÞ, assuming that other players play according to the probability distribution P. As noted in [23], this distribution is also sometimes denoted P2i to emphasize that entries ‘‘belonging’’ to player i should be counted out when the other players are considered collectively. The symbol j denotes the total number of strategies that player i can choose from [23].

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FIGURE 1

The payoff matrix of a generic two-player game, (Reproduced from Ref. 23, with permission [The figure was drawn by the author]) (Fig. 1).

Thus, QRE can be regarded as a mixed strategy equilibrium, where the choice probabilities give the equilibrium probabilities of a player with a particular value of bounded rationality. In the logit QRE function given in Eq. (1), ki is known as the rationality parameter of player i. Hence, by increasing it, it is possible to vary player i’s ability to respond to the opponent’s strategy distribution and the payoffs obtained under each strategy. Simply put, by varying the rationality parameter it is possible to vary player i’s rationality. The rationality parameter ki has a range from zero to infinity, and as ki ! 1, the equilibrium probabilities approach those given by the Nash equilibrium and indicate a perfectly rational player, while ki ! 0 indicates a player who is completely irrational or makes random decisions [18,23].

2.2. Game Theoretic Modeling of Networked Computing Systems Several recent studies in the field of networked computing systems have modeled the dynamics of these systems as games between self-interested players. Examples include TCP congestion control, computer security level allocation, peer-to-peer routing, peer-to-peer overlay network formation, and peer-to-peer file sharing patterns [1,2,4,6]. In this article, we consider three particular case studies, and demonstrate how the TDBR framework could be used to model systems with increased accuracy in these cases. Therefore, we describe these three cases, and the traditional approaches to model these scenarios as games, below.

2.2.1. Peer-to-Peer Overlay Network Formation

based on their topological positioning. Free-running or free-loading behavior is prevalent in such networks where certain players act in an extortionate manner exploiting others while certain players would contribute toward the common good [6,52]. In a non-cooperative game-theoretic perspective, free-runners could be thought as ‘‘rational’’ players while contributors could be regarded as ‘‘irrational’’ players, under the assumption that rationality is based on the tendency to maximize one’s own utility. Such heterogeneity of rationality is more prevalent when the network is formed by human players, and could even be relevant to autonomous agents due to the limitation of information that they might have about the overall network [53]. An overlay network construction model has been proposed by Chun et al. [4] that defines a network construction game. In this particular network construction model, each node is assumed to be running the link state protocol, where each node would periodically add and drop links. The total cost of a node being part of the network is a function of cost paid for maintaining links and the distances from the node in concern to the other nodes in the overlay network. The strategy adopted by a node would be the subset of other nodes in the network that the node chooses to connect to. The construction model uses Eq. (2) to calculate the cost incurred by a particular node by being part of the overlay network [4]. Ci ðsÞ5a

X jNBi

tj 1

n21 X

dG½s ði; jÞ

(2)

j50

Here, NBi is the set of neighbors of node i, tj is the cost incurred to connect to node j and dG½s ði; jÞ is the distance from node i to node j in the overlay network G[s]. The distance between two nodes is calculated by measuring the shortest path between them in the overlay and then adding the distances of the intermediate links along the underlying base network. Here, a can be regarded as the relation between the cost of establishing a link and the change in distance to other nodes caused by the addition of that link [4]. Also, the cost can be regarded as the inverse of payoff obtained by a node. The higher the cost, the less would be the payoff. The following algorithms (reproduced from [4]) depict the link addition and link dropping protocols used in this network formation model.

Algorithm 1: Link Addition for node i, reproduced from [4]

Peer-to-peer overlay routing networks are formed by self-interested players who want to share resources among themselves. These players may be human or software agents and they have limited visibility of the network

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Q 2016 Wiley Periodicals, Inc. DOI 10.1002/cplx

Algorithm 2: Link Dropping for node i, reproduced from [4]

The resulting network that evolves based on the assumption of perfect rationality could be regarded as the Nash equilibrium solution [4] under the network construction game.

2.2.2. Protection Against Security Threats in Networks Setting the security level of a particular node in a computer network is a compromise between the amount of resources that can be allocated for providing security and the level of security desired [54]. Moreover, there could be a compromise between computational and data transfer time and the security level provided [54]. As these are often conflicting interests, game theory is effectively used to model the desired security levels at each node within a computer network. We particularly focus on an example, where protection against security threats have been analyzed in a game theoretic setting within the context of DDoS attacks [1]. A DDoS attack begins with an attacker looking for vulnerable machines to get hold of and subsequently using them to launch a larger-scale attack. The vulnerability of nodes to such an invasion would depend on the security level set at each node. Even though the attack itself is often automated, the setting up of security is often a decision that is made with human cognition and experience, which nevertheless is a decision made with bounded rationality. Christin et al. [1] model a network of size N, where each user i is vulnerable against the initial stage of a DDoS attack. The level of computer security adopted by each user is denoted by si. The nodes that would be compromised would be those with the lowest security level minðsi Þ5smin . An assumption is made that the cost born by each user i to implement his security policy is a monotonically increasing function of si. All compromised users would incur a fixed cost of P  smin . It is assumed that users ‘‘probe’’ the security levels of other users and adjust their security levels accordingly. While this representation of a security model against the initial stage of a DDoS attack is fairly simple, it can be used to accurately model the DDoS attacks that have been

Q 2016 Wiley Periodicals, Inc. DOI 10.1002/cplx

performed in real networks [1]. The network size does not play a significant role in this model, unlike in real-world attacks. Thus, it would be more appropriate for networks of reasonably small size, such as corporate and university networks. The Nash equilibrium derived for this game occurs when all users within the network choose an identical security level si 5 P. The proof of this can be given as follows, as described by [1]. Suppose users 1; ::; k( 1  k  N ) apply a security level of si  smin for all ii; . . . ; N . Therefore, each user i1; ::; k is compromised, thus their utility would be ui 52P. Now suppose user i in 1,. . .., k increases their security level to si 5smin 1h where h  0. Then, user i’s utility would be 2smin 2h. As the original distribution of security levels form a Nash equilibrium, any change in strategy should decrease the utility of player i for any h  0, which results in the inequality, 2smin 2h  2P

(3)

This can be reduced to smin  P by continuity. However, it was originally hypothesized that smin  P. For both these inequalities to hold smin should be equal to P. As for any i, smin  si  P, it follows that si 5 P for all si. Thus, the Nash equilibrium occurs when si 5 P for all i users. Therefore, for this particular network security game, the Nash equilibrium occurs when all users have identical security level of P and thus the identical utility of-P. A more detailed explanation of this proof has been presented by Christin et al. [1]. Although the Nash Equilibrium of this particular game model predicts that all users within a network would have an identical security level, real-world network security systems operate otherwise [9,10]. Security levels tend to follow a heterogeneous distribution within a network of users.

2.2.3. Routing in a Peer-to-Peer Overlay Network In a peer-to-peer resource sharing network, each node relies on other nodes to forward its requests, and in turn is expected to forward the requests sent by other nodes [6]. However, the self-interested nodes may refuse to forward requests to conserve local bandwidth. It is suggested that incentive mechanisms and reputations systems can be used to facilitate cooperation as a robust and subgame-perfect equilibrium in a network of selfinterested players [6]. However, in real-world peer-topeer networks, collaboration seems to take place even without any particular incentive or reputation mechanism in place [17]. A variation of the random matching game has been used to model peer-to-peer routing in the literature [6]. In each round, players are randomly matched, and then each

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pair plays a single-round of the prisoner’s dilemma game [55]. As prisoner’s dilemma game has been widely used to model the behavior of self-interested agents, this model could be used to model peer-to-peer networks that consists of self-interested players. When each request is propagated, the intermediate nodes are considered to play a single instance of the prisoner’s dilemma game with the original node that generated the request. As the model is explained in [6], once a request originates from the node s, the request is forwarded to the immediate next node in the routing path. Then a single instance of the prisoner’s dilemma game is played between those two nodes. If the second node cooperates, that is if it forwards a request, it would get a payoff of 22 while it would get a payoff of 0 if it defects and ignores the request. Node s would get a payoff of 0 in both instances. If the request is forwarded, then the game would be repeated with s and the next node in the routing path until the destination node is reached. Thus, if each routing node is at Nash equilibrium it would always choose to defect as it gives a higher payoff of 0. This would create the scenario where there is the ‘‘tragedy of the commons,’’ where the collaborative environment cannot function at all, due to all players behaving in a self-interested manner [56]. However, this is contrary to observed peer-to-peer routing networks where cooperation occurs even without incentives or a reputation system in place [1,17].

2.3. The Topologically Distributed Bounded Rationality Model As mentioned earlier, the QRE is a generalized version of Nash equilibrium. Yet, a criticism can be made against the QRE that it uses the rationality parameter as an arbitrarily set parameter [51]. Quite often, it is used as a model to fit empirical results by varying the rationality parameter [57]. Social cognitive theories [24] and social brain hypothesis [25,58] suggests that there is a strong correlation between the cognitive capacity of a player and the amount of social interaction that the player may have. Based on this argument, a TDBR model has been recently proposed [23]. In this TDBR model, the rationality parameter k is defined as a function of social interactions [23], as shown in Eq. (4). In a more general context, cumulative weights of a node’s connections to its neighbors can be used to quantify the social interactions of that node. If the weights of the links are all identical, the degree of the node can be used as a relative measure of its social interactions. The rationality parameter is, therefore, calculated as: ki 5r:f

X n

 wij

(4)

j51

where ki is the rationality of node i. The quantity r denotes a ‘‘network rationality constant’’ that is a property of the network. The higher this constant, the more sensi-

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tive the nodes’ rationality to the amount of social interactions that they have. The weight of the link which connects node i with each neighbor j is denoted by wij, whereas n is the number of neighbors of node i. Therefore, a node may exhibit random decision making behavior if the network rationality constant is zero or when the node degree is zero. By contrast, a node may make decisions as predicted by the Nash equilibrium when network rationality constant r ! 1, or when the node degree is very high. Thus, this model captures the two extremes of rationality and the discrete levels of rationality in between those two extremes. In the case studies considered, we use unweighted networks (or networks with identical link weight of unity). This TDBR model is comparatively more realistic because it captures the heterogeneity of rationality levels of individual nodes, and makes the very reasonable assumption that such rationality is proportional to the interaction of the node with other nodes. Used along with the QRE the TDBR framework thus allows realistic modeling of competition in networked computer systems, as we show in the next section. It is mentioned in [23] that either a convex, concave, or linear function f can be used in Eq. (4), and the examples pffiffiffiffiffiffiffi used there are f ðxÞ5x2 ; f ðxÞ5 ðxÞ and f(x) 5 x. In the results presented in this study, we choose to use the convex function f ðxÞ5x2 , though we have verified that our results shown below are not qualitatively affected by the choice of f. Similarly, unless otherwise stated, the network rationality constant is set to r 5 0.01 in the results presented below. Again, we have tested our results for a range of r values from 104 to 10214 , and we can report that our results do not qualitatively change in most cases, and for this reason we have chosen to consistently report the results with the network rationality constant set to r 5 0.01. It will be seen however that there is one instance where the choice of r does matter, and we have reported the results for a range of r values in this case.

3. RESULTS In this section, we apply the TDBR model, along with the QRE defined by the Eq. (1), to the three game models that we have chosen to study as set out in the background section. We describe our results in three subsections corresponding to these game models.

3.1. Peer-to-Peer Network Formation We described the overlay network construction model [4] which is used to simulate peer-to-peer network formation in the background (Algorithm 1 and Algorithm 2). Yet, this model does not take into account the bounded heterogeneous rationality distribution of nodes into account. Therefore, we modified the link-generating algorithms to incorporate the TDBR model, as shown in Algorithms 3

Q 2016 Wiley Periodicals, Inc. DOI 10.1002/cplx

FIGURE 2

The distribution of rationality parameters against node degree during the network construction game. Please note that all nodes with the same degree have the same rationality parameter, according to Eq. (4), and thus the plot can be interpreted as either a scatter plot or a plot of average k values for each degree. The hubs typically display higher values, and thus will be more ‘‘rational’’ in making decisions.

and 4. In the modified Algorithm 3, to add a link, the probability pa is calculated using the Eq. (5). The probability of dropping a link used in Algorithm 4 is calculated using Eq. (6). These probabilities would depend on the rationality of each node, thus the hubs would have to make more ‘‘rational’’ decisions in adding and dropping links, compared to leaf nodes. In Figure 2, the rationality parameters of nodes used in this network construction game are shown against node degree [(as a scatter-plot: however, note that all nodes which have the same degree will have the same k, according to Eq. (4)], and as expected, it can be seen that the hubs have higher rationality parameters. Note that the opponent of each node, that is the node at the receiving end, is assumed to be in an always-connected state, and the decision to create or drop a link would be purely made by the active node of the link in that particular instance.

Algorithm 3: Link Addition for node i under bounded rationality and QRE.

Q 2016 Wiley Periodicals, Inc. DOI 10.1002/cplx

Algorithm 4: Link Dropping for node i under bounded rationality and QRE.

pa 5

e ki :P e ki :P 1e ki :0

(5)

Here, pa is the probability of adding a new link, ki is the rationality parameter of node i, and P is the payoff of creating the link, which is the difference between Costnew and Costold. The node would thus obtain a payoff of P if the link is created and a payoff of zero if not. This simple subgame with an incorporation of a rationality parameter could be used to model a node with nonoptimal bounded rationality. pd 5

eki :P eki :P 1e ki :0

(6)

Similarly, in Eq. (6), pd is the probability of dropping a new link, ki is bounded rationality of node i, and P is the payoff of dropping the link, which is the difference between Costnew and MinCost. Thus, P could be regarded as the expected payoff of dropping the link in concern, in comparison to the payoff 0 of keeping the link. Using the two sets of algorithms given in this section and the background section, and the corresponding equations, we could generate peer-to-peer overlay networks that are constructed using (a) perfectly rational nodes (Nash equilibrium) in the classical way (b) boundedly rational nodes whose rationality levels are topologically distributed, using the TDBR framework. We did so and compared the topological properties, particularly the scale-free exponent and the R-squared correlation to the power-law degree distribution between the two sets of networks. Real-world peer-to-peer overlay networks such as Gnutella have been observed to show scale-free topologies [16]. By comparing the topologies of each of the generated networks, we could compare each network construction game in its ability to generate networks that are topologically similar to the realworld overlay networks.

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We used scale-free networks of 500 nodes as the underlying base network. As the Internet has been observed to be scale-free in nature [11,59], using scale-free networks as the underlying networks is justifiable. Next, an overlay network was generated randomly over the physical base network. Then, the link-state algorithms were applied (1 and 2 as a pair and 3 and 4 as another pair). The resulting networks were compared for their scale-free properties. As previously mentioned, parameter a used in the cost function Eq. (2) is an indication of the relationship between the cost of creating a link and the change in distance to other nodes that occurs due to the creation of that link [4]. For instance, if a  1, that means that it is always beneficial to create a link than having to traverse at least two nodes to reach a non-neighboring node. If a is significantly large, a link addition would only happen if it substantially reduces the distances to the other nodes in the overlay network. By varying a, we were able to construct different overlay topologies. We compared networks generated using three different a values, 0.6, 1.5, and 10. This enabled us to observe how the perfectly rational and boundedly rational nodes would construct overlay networks under varying cost-benefit ratios of creating links. Figure 3 depicts the comparison of networks formed as a result of nodes operating at Nash equilibrium and TDBR induced QRE, respectively. The two parameters compared are the scale-free exponent and the R-squared correlation to the respective power-law degree distribution. The scalefree exponent is observed to be between 2 and 3 in most real-world scale-free networks, including peer-to-peer overlay networks [11,16]. The R-squared correlation indicates the level of fitness of the network topology to a perfectly scale-free network with a power-law degree distribution. As evident from Figure 3, the networks that result from TDBR 1 QRE demonstrate scale-free topological features, reminiscent to real-world peer-to-peer overlay networks [11,16]. The higher R-squared correlation in TDBR 1 QRE based networks indicate that they fit better with a powerlaw degree distribution. The relatively higher scale-free exponent is also similar to real-world peer-to-peer overlay networks. It has been observed that for very large a values, the resulting Nash equilibrium network may show scalefree characteristics [4], and this is to some extent confirmed by Figure 3, where the R-squared correlation of the network is higher for a 5 10. However, it is still much lower than the R-squared correlation of the TDBR 1 QRE network. In any case, in real-world peer-to-peer networks, it is unlikely that the cost of creating a link would be substantially large compared to the reduction of distance to the other nodes [2]. To verify that the results discussed above are not in any way dependent on the size of the system, we conducted similar experiments with varying system sizes, but did not detect any qualitative difference in results. For example, in Figure 4 we show the results for a system of N 5 3000

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nodes in the base network, where the results are extremely similar. If anything, the difference between Nash and TDBR modeling is now starker, because even for very large values of a (e.g., a 5 10), the Nash equilibrium network now shows rather low scale-free R-squared correlation. Thus, our results may suggest that the TDBR framework, along with the generalized QRE could be used to model network formation among peers more accurately, compared to classical models in literature based on Nash equilibria where players are assumed to have perfect rationality.

3.2. Protection Against Security Threats in Networks As our second case study, we discussed game theoretic modeling of assigning security levels to computers to protect them from DDoS attacks, which is introduced as a ‘‘security game’’ by Christin et al [1]. We saw that the Nash equilibrium in this security game occurs when all computer nodes have identical security settings, and thus identical Utility—U, yet in real world it has been observed that the security settings are heterogeneously distributed [9,10]. To incorporate the TDBR framework into this game setting, we extend this game where each user would ‘‘decide’’ whether to apply a security level or not. Accordingly, the ‘‘opponent’’ of each potential player would be the network itself. If a user does not get any security level applied, we assume si 5 0 and thus the payoff ui would be 0. The players who engage in the security game would probe others and adjust their security levels accordingly, and again the Nash equilibrium is when all players have identical security settings and thus identical utility—U. Now suppose that each user makes this decision based on a rationality level ki, as per the TDBR framework. Thus, following from Eq. (1), the probability of a player being part of the security game is: pi 5

e ki :U e ki :U 1e ki :0

(7)

where ki is computed from Eq. (4). When the user is not part of the network (node degree of user is zero), ki 50 for that user and pi can be computed as 0.5. Thus, he still has the probability of 0.5 of participating in the network game and his expected utility, therefore, is 2U=2. Conversely, when the node degree increases from zero, the probability of participation would tend toward 1, with an expected utility of—U. Thus, this model suggests that the users have a utility of ui ½2U=2; 2UÞ and thus the security level si is distributed in the range si ½U=2; UÞ. Figure 5(a) shows security level distributions of some simulated computer networks, using the TDBR 1 QRE modeling, and it is clear that these distributions are heterogeneous and not uniform, matching the real world phenomena. Furthermore, Figure 5(b) plots the same results as security levels against node degree, and from this plot it is clear that the hubs are assigned higher security levels. Therefore, with the

Q 2016 Wiley Periodicals, Inc. DOI 10.1002/cplx

FIGURE 3

The evolution of the scale-free exponent and the R-squared correlation of the Nash equilibrium network and the QRE network with TDBR. The network size is N 5 500 nodes. The a value was set to 0.6, 1.5, and 10, respectively.

TDBR 1 QRE model, we could argue that the heterogeneity of security levels in real-world networks could be accounted for. Moreover, it is commonly found that the servers that are highly connected get assigned higher level security compared to individual workstations that may not have high level of security [60]. This is also the case in Figure 5(b), where there is clear correlation between node degree and the security levels allocated.

Q 2016 Wiley Periodicals, Inc. DOI 10.1002/cplx

3.3. Routing in a Peer-to-Peer Overlay Network We have described a random matching game which has been used in literature to model peer-to-peer routing [6]. We saw that if each routing node is at Nash equilibrium it would always choose to defect and not forward the data packets, and unless there are incentives for forwarding the collaborative system would collapse [56]. While existing papers have used incentive mechanisms to make the nodes co-operate [6], in

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FIGURE 4

The evolution of the scale-free exponent and the R-squared correlation of the Nash equilibrium network and the QRE network with TDBR. The network size is N 5 3000 nodes. The a value was again set to 0.6, 1.5, and 10, respectively.

real world peer-to-peer routing often works even without incentives or a reputation system in place [1,17]. If we modify this game by introducing the TDBR 1 QRE framework, the non-zero probability of an intermediate node deciding to forward a request would be given by, pf 5

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e 22:kn e 22:kn 1e 0:kn

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(8)

where pf is the probability of forwarding the request and kn is the rationality of node n. With this probability distribution, a purely nonrational node with kn 50 would forward a request with 0.5 probability while a fully rational node, where kn ! 1, would never cater for another’s request. Therefore, this model suggests that the probability of forwarding a request is distributed in a probability distribution of (0:1/2], where hubs have a higher tendency of not forwarding incoming requests, compared to leaf nodes.

Q 2016 Wiley Periodicals, Inc. DOI 10.1002/cplx

FIGURE 5

The distribution of security setting levels against (a) node ID (b) against node degree (averaged), in a computer network (modeled as a scale-free network) of size N 5 1000 nodes and M 5 2000 links, simulating the ‘‘security game,’’ using the Nash equilibria and TDBR-QRE framework. Note that the TDBR - QRE distribution is heterogeneous, and not uniform as Nash equilibria precits.

To further analyze the dynamics of the modified game, we devised the following experiment. We simulated the random matching routing game on a scale-free network (of size N 5 6000 nodes and M 5 20,000 links) by generating broadcast requests from each node. Each node would send requests to all other nodes within the network along the shortest paths. We assume that nodes have the necessary routing information [61] contained in them, to forward the requests. We measured the fraction of such requests generated from all the nodes in the network, that would successfully make it to the destination. We varied the network rationality constant r to see how the routing game responds to the overall rationality level of the network. Then, we compared the results obtained with those of a structured lattice network and a real-world Gnutella

peer-to-peer overlay network, both with comparable average degrees. The Figure 6(a,b) show the comparison between the TDBR 1 QRE framework, and the Nash equilibria model, for a synthesized scale-free network and the Gnutella peer-to-peer network topology (which is also scale-free), respectively. No incentive mechanism was used. We may see from these results that for both topologies, the TDBR 1 QRE framework shows that a considerable fraction of requests are successful, as it happens in real world. Only when the network rationality constant is relatively high does the system ‘‘break down,’’ obviously because such high rationality implies a high level of selfishness. Conversely, the Nash-equilibrium based solution suggests that only a negligible fraction of requests will be

FIGURE 6

The variation of the fraction of overlay requests that are successfully routed against the network rationality constant. The results of TDBR 1 QRE and Nash equilibria are compared for a (a) synthesized scale-free network of size N 5 6000 nodes and M 5 20,000 links (b) Gnutella peer-to-peer overlay network (size N 5 6000 nodes and M 5 20,000 links).

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FIGURE 7

The scale-free topology goes through a sudden transition in the fraction of requests allowed, when the network rationality constant r increases beyond a particular point (logðrÞ522). Both structured and nonstructured topologies are applied in peer-to-peer overlay networks [62]. Therefore, these results suggest that if the rationality of agents are topologically distributed, nonstructured topologies such as scale-free topologies may be more robust against free-riding behavior. This observation does not strengthen or weaken the case for using TDBR 1 QRE but nevertheless is an important result.

4. CONCLUSIONS

The variation of the fraction of overlay requests that are successfully routed against the network rationality constant. The results of scale-free and lattice topologies are compared with that of a Gnutella network. All networks have size N 5 6000 nodes and M 5 20,000 links.

successfully routed, no matter what the network rationality constant r is (indeed, r is not needed in the Nash equilibria calculation, so the main point here is that the fraction is nearly zero). As mentioned below, incentive mechanisms are needed in this modeling process to simulate a working file sharing system. Therefore, it is obvious that the TDBR 1 QRE framework is much more realistic. Interestingly, we also see that the topology of the realworld Gnutella network is better than the synthetic scalefree network in successfully routing requests, implying that it has evolved topological attributes other than scalefreeness which help further in successful routing. As an aside, the Figure 7 depicts the results obtained by comparing the lattice and scale-free networks on their ability to facilitate peer-to-peer routing under varying network rationality constraints. The network rationality constant is shown in logarithmic scale. As the figure shows, the scale-free topology facilitates a considerable fraction of requests under the TDBR framework and over a wide range of network rationality constants. Conversely, lattice topology allows significantly less fraction of messages to be routed, due to its homogeneous degree distribution and the higher average path length. Most real-world peerto-peer overlay networks show scale-free topology [16]. Thus, this result may indicate that by being distributed in a scale-free topology, overlay networks obtain the ability to sustain message routing, provided that the nodes have heterogeneous and topologically distributed rationality levels. The real-world Gnutella network too, which shows scale-free characteristics, facilitates message forwarding over a significant range of network rationality constants.

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In networked systems field, game theory has often been used to model, and predict the dynamics of, systems where cognitive selfish entities compete to maximize their own utility. These entities include humans who construct and use networked systems (such as peer-to-peer networks) as well as entities with artificial intelligence. A traditional drawback of this modeling process has been that players were assumed to be fully rational. However, in reality, players are constrained by many factors, including lack of information, limited computing time, and limited inherent cognitive capacity, which prevent them from displaying fully ‘‘rational’’ behavior [23]. Even though generalized equilibria and incentive schemes have been used to model behavior of players which cannot be explained by classical game theory, an explicit modeling effort which accounts for the distributed heterogeneous rationality of players has not been made in the networked systems domain. Such modeling is particularly vital in systems which naturally display heterogeneous topology, such as peer-to-peer networks, as it could be argued that heterogeneous topology must naturally result in heterogeneous rationality distribution. In such systems, the highly connected nodes have relatively more opportunities to observe other nodes and gather information from other nodes, and thus be more rational. Meanwhile, inspired by observations made in the fields of psychology, learning theory, and brain science, a TDBR model has been proposed recently. In this article, therefore, we apply this TDBR model, to predict the behavior of players based on the more generalized QRE. Our choice of this model is backed up by intuitive understanding of how the systems work in many examples we choose. In particular, we choose three examples where considerable work has been done in networked systems research to model systems with game theoretic concepts, yet the models thus obtained have not exactly matched the topological evolution and behavioral dynamics of the systems concerned. The first case we study is peer-to-peer network formation, where the existing network construction game models do not result in scale-free networks, yet many

Q 2016 Wiley Periodicals, Inc. DOI 10.1002/cplx

peer-to-peer networks in real world are scale-free. We show that by modifying this game using the TDBR-QRE framework, the model could be made to generate scalefree networks which match well with the topology of real world networks such as Gnutella. The second case we considered was setting the security levels of computers against a DDoS attack, and the classical models suggested the equilibria occurs when all computers are set the same level of security, yet this does not happen in real world. Our TDBR-QRE resulted in equilibria which were heterogeneous in terms of security settings, and servers in particular received higher security settings, as happens in real world. As the third example we studied peer-to-peer network routing, where existing literature suggests that incentive mechanisms are needed to make players co-operate. The TDBR-QRE framework was able to achieve cooperation among players without having incentive mechanisms, and this matches with the fact that several real world peer-to-peer networks function without actual incentive mechanisms for individuals. These three cases confirmed that the new TDBR-QRE framework can be successfully used in modeling the behavior of intelligent entities in networked systems research. As a peripheral yet important observation, we also found that non-structural topologies, such as scale-free networks, might be more robust to free-riding behavior in peer-to-peer networks, thus being able to function without collapsing even with many free-riders. Therefore, this might be a reason why peer-to-peer overlay networks ‘‘evolve’’ to be scale-free. The observations demonstrate that in modeling competitive behavior in networked systems using games, it is important to explicitly take into account the bounded rationality of players, which is a physical reality. Future work in this regard may involve using more accurate methods to quantify the amount of interaction of a player

with the rest of the network, or using custom functions to map this amount of interaction to the level of rationality of players. The first can be achieved using information measures, such as transfer entropy, instead of simply node degree as we have done in this article, to quantify the amount of interaction a player has with other players. The second could be achieved by gathering data of player responses in particular scenarios in real-world networked systems, and use this data to ‘‘fit’’ a suitable function f, then use the fitted function f in predicting the future behavior of nodes in that system. Nevertheless, the simple TDBR model used in this paper (along with the quantal response equilibria used in predicting player behavior) already gives considerably better results compared to existing game theoretic models in matching real-world dynamics and topology of the networked systems studied.

DISCLOSURE The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

AUTHOR CONTRIBUTIONS D.K and M.P conceived the study and designed the experiments. D.K wrote the software and carried out the simulation experiments. D.K., M.P. and S.U. wrote the article.

ACKNOWLEDGMENTS The authors thank Albert Zomaya, Sanjay Chawla, and Anatasios Viglas from the School of Information Technologies, University of Sydney, and Mikhail Prokopenko from Complex Systems Research Group, School of Civil Engineering, University of Sydney for inspiring discussions.

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