research article evolutionary game-based secrecy rate...

Research ArticleEvolutionary Game-Based Secrecy Rate Adaptation inWireless Sensor Networks

Guanxiong Jiang,1 Shigen Shen,1,2 Keli Hu,1 Longjun Huang,1,3

Hongjie Li,2 and Risheng Han2

1Department of Computer Science and Engineering, Shaoxing University, Shaoxing 312000, China2College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing 314001, China3College of Computer Science and Technology, Zhejiang University of Technology, Hangzhou 310014, China

Correspondence should be addressed to Guanxiong Jiang; [email protected]

Received 22 December 2014; Revised 26 February 2015; Accepted 4 March 2015

Academic Editor: Chuan-Ming Liu

Copyright © 2015 Guanxiong Jiang et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Physical layer security, whose aim is to maximize the secrecy rate of a source while keeping eavesdroppers ignorant of datatransmitted, is extremely suitable forWireless Sensor Networks (WSNs).We therefore, by developing the classical wire-tap channel,construct an approach to compute the secrecy rate between a sensor node and its responsible cluster head in the clustered WSNs.A noncooperative secrecy rate game towards WSNs is formulated to solve contradictions between maximizing the secrecy rateof a sensor node and minimizing power consumed for data transmission. Using evolutionary game theory, we set up a selectiondynamics upon which a power level can be adaptively selected by a sensor node. Thus, the objective of secrecy rate adaptation formaximizing the fitness ofmember sensor nodes is achieved.We also prove the game is stable; that is, there exist evolutionarily stablestrategies (ESSs) that explain which strategies will be selected by a sensor node in the end. Moreover, a corresponding algorithmof secrecy rate adaptation is given. Numerical experiments show our proposed approach can adaptively adjust the secrecy rate of asensor node, which provides a novel way to guarantee the confidentiality of WSNs.

1. Introduction

Providing security for Wireless Sensor Networks (WSNs)is challenging, due to the characteristics of wireless com-munications, limited resources of sensor nodes, dense andenormous networks, as well as the unattended environmentswhere sensor nodes are prone to physical attacks. A largenumber of security approaches [1] such as cryptography,attack detection, and secure routing have been proposed todefend various threats and vulnerabilities inWSNs. Differentto these traditional methods, physical layer (PHY) security[2, 3] using the physical properties of the radio channel tohelp provide secure wireless communications is currentlyattracting considerable attention. This approach is extremelysuitable for WSNs since it achieves security through takingadvantage of the fundamental ability of the physical layer, notadding extra components.

The main idea of PHY security is to maximize therate of reliable information from a source to a destination,while keeping eavesdroppers ignorant of data transmitted.This rate is referred to as secrecy rate and its maximumreliable value for a channel is known as secrecy capacity.Wyner [4] in his pioneering work introduced a wire-tapchannel and showed that perfect secrecy could be attainedeven without the application of shared secrets. Since itsinception, secrecy capacity has been fundamentally devel-oped in other channels including Gaussian wire-tap channel[5], broadcast channel [6], relay channel [7], multiple-accesschannel [8], interference channel [9], multiple-input single-output (MISO) channel [10], multiple-input multiple-output(MIMO) channel [11], and feedback channel [12].

Along with the foundation of secrecy capacity in vari-ous channels, a recent trend of long-standing interest hasemerged in how to improve PHY security under the same

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2015, Article ID 975454, 13 pageshttp://dx.doi.org/10.1155/2015/975454

2 International Journal of Distributed Sensor Networks

surroundings. These include methods such as cooperationrelays [13], joint relay and jammer selection [14], chan-nel frequency selectivity [15], and game theory [16–26].In practice, during the process of transmitting data, thetransmission power is a key factor that influences the secrecyrate of a source. From a noncooperative perspective, eachsensor node, due to its selfishness, attempts to increaseits transmission power for maximizing its secrecy rate inresponse to actions of the other sensor nodes. In this manner,it will however cause more and more interference to theother nodes and consume its own energy extraordinarily.These interactions among sensor nodes naturally result inapplications of game theory.

As an efficient method to solve problems of optimalstrategies, game theory provides a rich set of mathematicaltools for investigating the multiperson strategic decision-making, which has been broadly employed inWSNs security[27–33]. It is generally assumed that players in the classicalgame theory have complete information about the gameand about behaviors of opponents and must act completelyrationally. Distinguishable to these assumptions, evolution-ary game theory (EGT) to be borrowed in this paper imaginesthat conditioned players who are randomly drawn from alarge population play the game repeatedly. Moreover, overtime players have the opportunity to optimize their individualutilities by reacting to simple observations from their oppo-nents. This nature satisfies the requirement for the currentlarge-scale wireless networks with characteristics of self-organization, self-configuration, and self-optimization.Thus,many researchers are currently engaged in developing EGTbased schemes [34, 35]. Typical examples exist in evaluatingan incentive protocol [36], adjusting multiple-access controland power control [37], maximizing TCP throughout [38],learning for an optimal strategy [39], coevolving rationalstrategies [40], implementing network selection [41], sensingthe spectrum collaboratively [42], selecting dynamic service[43], enforcing cooperation among network nodes [44],promoting the selfish nodes to cooperatewith each other [45],and so forth.

Up to now, there are some papers on combining secrecycapacity with game theory in different environments. Thework in [19] formulates a noncooperative game between asource and a jammer relay for achieving the optimal secrecyrate of the source. Also, a noncooperative game betweenwireless users and a malicious node is introduced in [20],where the wireless users, through choosing a relay station,are to maximize a utility function considering their mutualinterference and security of the chosen path whereas themalicious node is to reduce the overall network’s secrecycapacity. From the perspective of information theory, thework in [22] provides a mathematical formulation for thenoncooperative zero-sumgame existed in the two-userMISOGaussian interference channel with confidential messages. In[23] the authors model a two-person zero-sum gamewith theergodicMIMO secrecy rate as the payoff function, in order toexamine the trade-offs for the legitimate transmitter and theadversary. Since cooperation, on the contrary, is capable ofimproving PHY security, the work in [21] introduces friendlyjammers to interfere with the eavesdropper to increase the

secrecy rate of a source. A Stackelberg game is investigatedto reflect interactions between the source that pays jammersfor their interference and jammers who charge the sourcewith a certain price for the jamming. Different from [21], aStackelberg game [18] between primary users and secondaryusers in cognitive radio networks is modeled to maximizetransmission rates of secondary users. Consequently, primaryusers’ secrecy rates are improved with the help of trustworthysecondary users. Besides, typical examples of the Stackelberggame exist in analyzing the cooperation between the primaryand secondary transmitters in cognitive radio channels [24]and formulating the sources and the friendly jammers in atwo-way relay system as a power control scheme to achievethe optimized secrecy rate of the sources [25]. Moreover, anontransferable coalitional game [17], as one of cooperativegames, is applied to achieve secrecy rate gains from coop-eration in the presence of a cost for information exchange.The work in [26] formulates the relay selection in a two-stagedecode-and-forward cooperative network as a coalitionalgame with transferable utility, which decreases the compu-tation complexity in solving the distributed relay selectionproblem. In addition, the cooperative Kalai-Smorodinskybargaining game [16] is adopted, for transmitters with theMIMO channel, to find an operating point that balancesnetwork performance and fairness. This point allows trans-mitters to adjust their precoders appropriately and thusimproves the overall secrecy capacity of the network.

To the best of our knowledge, this paper is the first work tofocus on exploring secrecy rate adaptation in WSNs with theidea of EGT.Wefirst extend the classical secrecy rate equationto fit for the WSNs. We are thus able to understand whatparameters will influence the secrecy rate inWSNs.Then, wesolve the decision-making problem among sensor nodes formaximizing their utilities through a noncooperative secrecyrate game.The replicator dynamics is applied to illustrate theevolution process how sensor nodes select adaptively theirpower level strategies and achieve their adaptation of secrecyrate.

Compared with the related works mentioned herein,there are some distinctions in this paper. We concentrateclearly on the secrecy rate between a sensor node and itsresponsible cluster head in the clustered WSNs, while manyother channels are considered in [4–12, 46–48]. We employEGT to disclose the dynamics of secrecy rate adaptation forsensor nodes, while EGT is applied to other fields [34, 36–44, 49]. Our eventual objective is to maximize the fitnessof sensor nodes while several current works [16–21] makeuse of various games to improve PHY security for differentsurroundings. Therefore, our contributions lie mainly in thefollowing aspects:

(1) considering the interference environment, we con-struct an approach to compute the secrecy ratebetween a sensor node and its responsible clusterhead by developing the classical wire-tap channel [4,5], which is suitable for the clustered WSNs;

(2) we formulate a noncooperative secrecy rate gametowards WSNs, which is able to reflect interactionsamong sensor nodes and the cost of energy consumed

International Journal of Distributed Sensor Networks 3

for transmitting data. Solving the game will helpsensor nodes, by maximizing their utilities, to selecttheir power level strategies correctly;

(3) we attain, from the view of EGT, a selection dynamicsthat promotes sensor nodes to seek a power levelstrategy with higher fitness, as well as ESSs for thegame to describe which power level strategies willbe adopted by mutants in the end. The secrecy rateadaptation for sensor nodes is thus achieved.

The rest of this paper is organized as follows. In Section 2,we discuss the network surroundings to be studied andcompute the secrecy rate between a sensor node and itsresponsible cluster head in the clustered WSNs. In Section 3,a noncooperative secrecy rate game towards WSNs is setup. Moreover, secrecy rate dynamics and ESSs for the gameare explored. In addition, an algorithm is introduced todescribe how to adjust adaptively secrecy rate of sensornodes. In Section 4, by performing numerical experimentsin an example of the secrecy rate game, we illustrate theinfluence of a cost parameter, as well as the correspondingESSs. Moreover, the process of secrecy rate adaptation forsensor nodes is disclosed. Finally, a conclusion is provided inSection 5.

2. System Model

2.1. Interference among Sensor Nodes. Multiple-access chan-nels based WSNs are under consideration in this paper. Itis well known that multiaccess interference is a significantfactor to reduce the performance of such networks. Lack ofcoordination among sensor nodes will lead to a great numberof interfering power from their neighbors, even if each sensornode transmits signals at the lowest required power to itsdestination.

To attain the number of interferers for a particular sensornode, we use the assumptions as follows. We assume theWSNs are composed of static and identical sensor nodes thatare uniformly scattered with node density 𝜌 on a 2D grid.Each of sensor nodes is equipped with an omnidirectionaltransmitting and receiving antenna of the same gain andoperates in a half-duplex way; that is, receiving and trans-mitting simultaneously are not allowed. The receiving andinterfering ranges rely on the transmission powers of a sourceand the other sensor nodes in the same zone. Let 𝑟

𝑅be the

receiving distance that means the maximum distance fromwhich a receiving sensor node is able to recover correctlya signal transmitted, and let 𝑟

𝐼be the interfering distance

denoting the maximum distance from which a receivingsensor node can sense carriers. Typically, 𝑟

𝐼≈ 2𝑟𝑅[50].

Moreover, in order to sendmessages successfully, each sensornode is capable of selecting an appropriate power level.

For a given sensor node, we now describe the numberof its interferers. We employ Poisson distribution, due toits mathematical tractability, to estimate approximately thedistribution of sensor nodes. In general, the maximumnumber of interferers should be computed in terms of themean value and several times of standard deviation √𝜎. Forinstance, if we were to account for 99.6 percent of all sensor

Cluster headBase station

Member sensor nodeEavesdropper

· · · · · ·

· · ·

· · ·

H1Hn

Sm

Sm

1

ℰm

Sm

I𝑚

Figure 1: Network model.

nodes, then we have to consider 3 × √𝜎 in addition to themean [51]. For any sensor node, its interference zone is 𝑧

𝐼=

𝜋𝑟2

𝐼, and its mean number of sensor nodes in zone 𝑧

𝐼is 𝜌𝑧𝐼.

In addition, the variance 𝜎 for Poisson distribution is equalto 𝜌𝑧𝐼. Therefore, the maximum number of interferers to any

sensor node can be denoted by

𝐼 = 𝜌𝑧𝐼+ 3√𝜌𝑧

𝐼. (1)

2.2. Secrecy Rate in Clustered WSNs. The topology to bestudied, as depicted in Figure 1, is based on the clusteredstructure due to its popularity. By using clustering, all sensornodes are grouped into different clusters. Each cluster hasa coordinator, referred to as cluster head, and a number ofother sensor nodes as members, calledmember sensor nodes.This clustering leads to a two-tier hierarchy, where clusterheads construct the higher tier while member sensor nodesform the lower tier. In this hierarchy, a time-slotted datatransmission scheme is adopted. In the first time slot, datacaptured by member sensor nodes in the same cluster aretransmitted to the responsible cluster head, then in the secondtime slot the cluster head aggregates the data received andsends them to the base station through other cluster heads.Note that, for simplicity, our work is focused on the secrecyrate between a member sensor node and its responsiblecluster head in the first time slot.

Let S, H, and E be sets of 𝑀 member sensor nodes,𝑁 cluster nodes, and 𝐾 eavesdroppers, respectively. Thus,S = {𝑆

1, 𝑆2, . . . , 𝑆

𝑀}, H = {𝐻

1, 𝐻2, . . . , 𝐻

𝑁}, and E =

{𝐸1, 𝐸2, . . . , 𝐸

𝐾}. For any member sensor node 𝑆

𝑚, 𝑆𝑚

∈ S,there exists a uniquely responsible cluster head 𝐻

𝑛, 𝐻𝑛∈ H,

where 𝑛 is in fact determined by 𝑚; there also exist severaleavesdroppers, each of which is denoted by 𝐸

𝑘, 𝐸𝑘

∈ E𝑚,

E𝑚

⊆ E, where E𝑚

is the set of eavesdroppers capableof listening in on data sensed by 𝑆

𝑚. Channel gains from

𝑆𝑚to 𝐻𝑛and 𝐸

𝑘are denoted by 𝐺

𝑆𝑚

𝐻𝑛

and 𝐺𝑆𝑚

𝐸𝑘

, respectively.For simplicity, all thermal noise powers at cluster heads andeavesdroppers are denoted by 𝜂

2. In addition, each channelbandwidth is 𝑊.


Due to the broadcast nature of channels, a cluster headendures a collision of signals from some member sensornodes that are transmitting their data simultaneously. Wetreat these signals from member sensor nodes other thanthe target source as additive interference and compute thenumber of these interferers according to (1). For any membersensor node 𝑆

𝑚, 𝑆𝑚

∈ S, let S𝑚be the set of its interferers

whose element is denoted by 𝑆𝑚

𝑖, 𝑖 ∈ {1, 2, . . . , 𝐼

𝑚}, where

𝐼𝑚

denoting the number of interferers of 𝑆𝑚

comes from(1). Thus, S

𝑚= {𝑆𝑚

1, 𝑆𝑚

2, . . . , 𝑆

𝑚

𝐼𝑚

}. Following the idea of theclassical wire-tap channel [4, 5], the channel capacity frommember sensor node 𝑆

𝑚to its corresponding cluster head𝐻

𝑛,

denoted by 𝐶𝑆𝑚

𝐻𝑛

, can be given by

𝐶𝑆𝑚

𝐻𝑛

= 𝑊 log2(1 +

𝑃𝑚𝐺𝑆𝑚

𝐻𝑛

∑𝐼𝑚

𝑖=1��𝑚

𝑖𝐺𝑆𝑚

𝑖

𝐻𝑛

+ 𝜂2) , (2)

where 𝑃𝑚is the transmission power adopted by 𝑆

𝑚, 𝑆𝑚

∈

S, ��𝑚𝑖

is the interference power adopted by 𝑆𝑚

𝑖, 𝑆𝑚𝑖

∈ S𝑚,

and 𝐺𝑆𝑚

𝑖

𝐻𝑛

is the channel gain between interferer 𝑆𝑚

𝑖and the

interfered cluster head 𝐻𝑛.

Similarly, the channel capacity frommember sensor node𝑆𝑚to eavesdropper 𝐸

𝑘, 𝐸𝑘

∈ E𝑚, denoted by 𝐶

𝑆𝑚

𝐸𝑘

, can begiven by

𝐶𝑆𝑚

𝐸𝑘

= 𝑊 log2(1 +


𝐸𝑘

∑𝐼𝑚

𝑖=1��𝑚

𝑖𝐺𝑆𝑚

𝑖

𝐸𝑘

+ 𝜂2) , (3)

where 𝐺𝑆𝑚

𝑖

𝐸𝑘

is the channel gain between interferer 𝑆𝑚

𝑖and

eavesdropper𝐸𝑘,𝐸𝑘∈ E𝑚.The secrecy rate betweenmember

sensor node 𝑆𝑚and its responsible cluster head𝐻

𝑛, thus, can

be defined as (in fact, this scenario can be regarded as 𝐼𝑚-

user interference channel with |E𝑚| external eavesdroppers,

where |E𝑚| is the number of eavesdroppers who are able to

listen in data sensed by 𝑆𝑚)

𝐶 (𝑃𝑚) = (𝐶

𝑆𝑚

𝐻𝑛

− max𝐸𝑘∈E𝑚

𝐶𝑆𝑚

𝐸𝑘

)

+

, (4)

where (𝑥)+= max{𝑥, 0}.

3. Joint Secrecy Rate and EGT

3.1. ACompendiumof EGT. Briefly, EGT ismainly concernedabout a dynamic environment where players are continuallyinteracting with others and adapting their strategies basedon their expected payoffs they attain. In EGT, the concept ofpopulation consisting of a group of individuals (i.e., players)is introduced, and expected payoffs are replaced by fitnessof individuals. An individual in an evolutionary game isable to observe the actions of other individuals, learn fromthese observations, and slowly adjust its strategy to attainthe solution in the end. In this manner, we are able tounderstand the dynamics of interactions among individualsin a population.

In fact, the evolutionary process is determined bytwo noticeable elements: a mutation mechanism to providemutants and a selection mechanism to promote somemutantswith higher fitness over others [52]. While the mutationmechanism is described by evolutionary stable strategy (ESS),the selection mechanism is highlighted by replicator dynam-ics. The concept of ESS is to require that the final equilibriummust be capable of repelling all invaders; that is, if a strategyis evolutionarily stable, then it must keep such a quality thatalmost all individuals in the same population follow thisstrategy and mutants hardly invade successfully. This ESSis in fact a refinement of Nash equilibrium. On the otherhand, the replicator dynamics is able to explain and predictthe changeable trend towards population shares associatedwith different pure or mixed strategies. It selects individualsin a fitness proportional fashion. Thus, subpopulations withbetter fitness than the average will grow, while those withworse fitness than the average will be reduced in theirnumbers.

3.2. Secrecy Rate Game towards WSNs

Definition 1. The secrecy rate game towards WSNs that issymmetric consists of a 3-tuple G = (S,P,U), where

(i) S = {𝑆1, 𝑆2, . . . , 𝑆

𝑀} is a set of member sensor nodes

(individuals) in the same WSNs;(ii) P = ∏

𝑀

𝑚=1P𝑚

is a set of strategy profiles for allmember sensor nodes, where P

𝑚= {𝑃𝑚

| 𝑃𝑚

1, 𝑃𝑚

2,

. . . , 𝑃𝑚

𝐿}, 𝑚 ∈ {1, 2, . . . ,𝑀}. Here P

𝑚and 𝐿, respec-

tively, are the set and number of pure power levelstrategies available to member sensor node 𝑆

𝑚, 𝑆𝑚

∈

S;(iii) U = {𝜇(𝑃

𝑚, 𝑃��) | 𝑃𝑚

∈ P𝑚, 𝑃��

∈ P��, 𝑚, �� ∈

{1, 2, . . . ,𝑀}} is a set of utility attained by membersensor node 𝑆

𝑚adopting power level strategy 𝑃

𝑚

when its opponent adopting 𝑃��.

In the secrecy rate game towards WSNs, we consider allmember sensor nodes form a population and regard themas individuals. These individuals are intelligent agents who,during interactions with others, are able to self-adjust theirstrategies according to their current fitness. Their objectivesare to adapt their secrecy rates by maximizing individualfitness.

Due to the interference among sensor nodes, an optimalpower must be allocated to all member sensor nodes. From(4), the secrecy rate of member sensor node 𝑆

𝑚may be

maximized if 𝑆𝑚

transmits its data at its full power. Themember sensor node 𝑆

𝑚, if doing so, will however result

in more interference to its neighboring sensor nodes andwill also consume its own battery largely. The performanceof the WSNs will thus be reduced. Therefore, to meet theseinteractions, we introduce the utility function for membersensor node 𝑆

𝑚as

𝜇 (𝑃𝑚, 𝑃��) = 𝐶 (𝑃

𝑚) − 𝛼𝑃

𝑚, (5)

where 𝐶(𝑃𝑚) is from (4), and 𝛼 is a cost parameter that

reflects the degree of a member sensor node consuming


energy for data transmission. Note that the member sensornode adopting 𝑃

��has been included as one of interferers in

the first term of (5). Next, the goal of the secrecy rate gameis to help all individuals select their strategies adaptively sothat the maximum fitness of member sensor nodes can beattained with the minimum expense of their power. Thus,secrecy rate adaptation of a member sensor node, along withthis evolutionary process, is to be achieved.

3.3. Dynamics Analysis on Secrecy Rates of Member SensorNodes. Wenow explore the secrecy rate dynamics ofmembersensor nodes using the replicator dynamics. To begin with, allmember sensor nodes, for transmitting data, select randomlyone of available power levels. Each member sensor node,from the view of noncooperative game theory, expects tomaximize its own fitness. It therefore adjusts periodically itspower level (i.e., its secrecy rate is adjusted) and compares thecorresponding fitness with the average fitness of the wholeWSNs. If a higher value is attained, the member sensor nodechanges its current power level (i.e., its current secrecy rateis changed) to a new one; otherwise, it keeps its currentstrategy unchanged. Letting 𝜃

𝑗(𝑡) (note that here, ∀𝑚, �� ∈

{1, 2, . . . ,𝑀}, we assume P𝑚

= P��, which is rational due

to the same characteristic of all member sensor nodes. Thisassumption means all member sensor nodes have the sameset of power level strategies; i.e., ∀𝑚, �� ∈ {1, 2, . . . ,𝑀}, ∀𝑗 ∈

{1, 2, . . . , 𝐿}, 𝑃𝑚𝑗

= 𝑃��

𝑗is satisfied, where 𝑃

𝑚

𝑗and 𝑃

��

𝑗denote

member sensor nodes 𝑆𝑚and 𝑆

��adopting the same power

level strategy 𝑗) be the fraction ofmember sensor nodes usingpower level strategy 𝑗 at time 𝑡, we have

∑

𝑗∈P𝑚

𝜃𝑗 (𝑡) = 1. (6)

The state of the WSNs at time 𝑡, denoted by 𝜃(𝑡), is then𝜃(𝑡) = [𝜃

𝑃𝑚

1

(𝑡), 𝜃𝑃𝑚

2

(𝑡), . . . , 𝜃𝑃𝑚

𝐿

(𝑡)] that can be consideredas a mixed strategy of the WSNs. Let 𝑙 be the power levelstrategy adopted by the opponent 𝑆

��. Thus, referring to [53],

the fitness of member sensor node 𝑆𝑚adopting power level

strategy 𝑗 at time 𝑡 is

𝜇𝑗 (𝑡) = ∑

𝑙∈P��

𝜃𝑙 (𝑡) 𝜇 (𝑗, 𝑙) , (7)

where 𝜇(𝑗, 𝑙) is from (5). The average fitness of the wholeWSNs at time 𝑡 is

𝜇 (𝑡) = ∑

𝑗∈P𝑚

𝜃𝑗 (𝑡) 𝜇𝑗 (𝑡) . (8)

Correspondingly, we define the expected secrecy rate ofmember sensor node 𝑆

𝑚adopting power level strategy 𝑗 at

time 𝑡 as

𝜍𝑗 (𝑡) = ∑

𝑙∈P��

𝜃𝑙 (𝑡) 𝐶 (𝑗) , (9)

where 𝐶(𝑗) is from (4). The average secrecy rate of the wholeWSNs at time 𝑡 is defined as

𝜍 (𝑡) = ∑

𝑗∈P𝑚

𝜃𝑗 (𝑡) 𝜍𝑗 (𝑡) . (10)

To make use of the idea of the replicator dynamics,we require the rate at which member sensor nodes changetheir strategies. This rate is determined by the performanceof current strategies of member sensor nodes and by thepopulation state. Let 𝑟

𝑗(𝜃) be the average review rate of

member sensor nodes using power level strategy 𝑗, 𝑝𝑗𝑞(𝜃) (in

particular, 𝑝𝑗𝑗(𝜃) is the probability that the reviewingmember

sensor nodes adopting power level strategy 𝑗 do not changetheir strategy) be the probability that the reviewing membersensor nodes change from power level strategy 𝑗 to 𝑞. Thus,in the whole WSNs the fraction of the reviewing membersensor nodes changing from power level strategy 𝑗 to 𝑞 is𝜃𝑗(𝑡)𝑟𝑗(𝜃)𝑝𝑗

𝑞(𝜃). As a result, the outflow from the power level

strategy 𝑗 is

∑

𝑞,𝑞 =𝑗

𝜃𝑗 (𝑡) 𝑟𝑗 (𝜃) 𝑝

𝑗

𝑞(𝜃) = 𝜃

𝑗 (𝑡) 𝑟𝑗 (𝜃) ∑

𝑞,𝑞 =𝑗

𝑝𝑗

𝑞(𝜃)

= 𝜃𝑗 (𝑡) 𝑟𝑗 (𝜃) (1 − 𝑝

𝑗

𝑗(𝜃)) ,

(11)

while the inflow to the power level strategy 𝑗 is∑𝑞,𝑞 =𝑗

𝜃𝑞(𝑡)𝑟𝑞(𝜃)𝑝𝑞

𝑗(𝜃). Subtracting the outflow from the

inflow, we can then attain the differential equation as

𝜃𝑗 (𝑡) = ∑

𝑞,𝑞 =𝑗

𝜃𝑞 (𝑡) 𝑟𝑞 (𝜃) 𝑝

𝑞

𝑗(𝜃) − 𝜃

𝑗 (𝑡) 𝑟𝑗 (𝜃) (1 − 𝑝𝑗

𝑗(𝜃))

= ∑

𝑞

𝜃𝑞 (𝑡) 𝑟𝑞 (𝜃) 𝑝

𝑞

𝑗(𝜃) − 𝜃

𝑗 (𝑡) 𝑟𝑗 (𝜃) ,

(12)

which constructs the replicator dynamics determining whichpower level strategies will be adaptively selected by amembersensor node.

For a member sensor node, whether or not to switch itspower level is based on the corresponding fitness. Amongthe notations above, we suppose that 𝑗 and 𝑞 denote thepower level strategies adopted by the original individuals anda small group of mutants, respectively. Then, the reviewingmember sensor nodes change to the mutant strategy if andonly if the fitness difference observed is positive; that is,𝜇𝑞(𝑡) > 𝜇

𝑗(𝑡). This difference between random variables

𝜇𝑗(𝑡) and 𝜇

𝑞(𝑡) has a continuously differentiable probability

distribution function 𝜙 : R → [0, 1]. The conditionalprobability that member sensor nodes will change to powerlevel strategy 𝑞, given that its original strategy is 𝑗, is thus𝜙(𝜇𝑞(𝑡) − 𝜇

𝑗(𝑡)). Furthermore, since the probability that

member sensor nodes will select power level strategy 𝑞 attime 𝑡 is 𝜃

𝑞(𝑡), we can attain the resulting conditional choice

probability as

𝑝𝑗

𝑞(𝜃) =

{{

{{

{

𝜃𝑞 (𝑡) 𝜙 (𝜇

𝑞 (𝑡) − 𝜇𝑗 (𝑡)) , if 𝑞 = 𝑗,

1 − ∑

𝑞 =𝑗

𝜃𝑞 (𝑡) 𝜙 (𝜇

𝑞 (𝑡) − 𝜇𝑗 (𝑡)) , if 𝑞 = 𝑗.

(13)

In fact, the choice probabilities in (13) result from indi-vidual differences in preferences acrossmember sensor nodesand not from observation errors made by member sensor


nodeswith identical preferences. To isolate the effect of choiceprobabilities, for simplicity, we assume that all review rates areconstantly equal to one; that is,

∀𝑗 ∈ P𝑚, 𝑟𝑗 (𝜃) ≡ 1. (14)

Substituting (13) and (14) into (12), we can, formember sensornodes, get the selection dynamics as

𝜃𝑗 (𝑡) = 𝜃

𝑗 (𝑡) ∑

𝑞,𝑞 =𝑗

𝜃𝑞 (𝑡)

⋅ (𝜙 (𝜇𝑗 (𝑡) − 𝜇

𝑞 (𝑡)) − 𝜙 (𝜇𝑞 (𝑡) − 𝜇

𝑗 (𝑡))) .

(15)

This selection dynamics, determining the selection prob-ability of a power level strategy, will be used to adapt thedecision of a member sensor node and is at the same timeaffected by decisions of the other member sensor nodes.

3.4. Convergence and Stability Analysis on Secrecy Rate Gametowards WSNs

Lemma 2. If a power level strategy 𝑗 is strictly dominant, thenlim𝑡→∞

𝜃𝑗(𝑡) = 1.

Proof. Thecase that a power level strategy is strictly dominantmeans member sensor nodes with this strategy, irrespectiveof which strategy any of other member sensor nodes chooses,can gain a strictly higher fitness than with any of otherstrategies. This will result in increasing the correspondingfraction in the population. Member sensor nodes employinga strictly dominant strategy will then occupy the wholepopulation over time. Lemma 2 therefore follows.

From Lemma 2, we can attain that member sensor nodeswhose strategies are strictly dominated will vanish eventuallyfrom the population; that is, if a power level strategy 𝑗 isstrictly dominated, then lim

𝑡→∞𝜃𝑗(𝑡) = 0.

Theorem3. TheWSNs population state 𝜃(𝑡) converges to equi-librium.

Proof. It is obvious that member sensor nodes employing dif-ferent power level strategies will attain different fitness underthe same WSNs surroundings. This fact means only onepower level strategy, correspondingly, is endowed with thehighest fitness among all strategies. Ranking all fitness valuesin descending order, we express the relationship of thesefitness as 𝜇

1(𝑡) > 𝜇

2(𝑡) > ⋅ ⋅ ⋅ > 𝜇

𝐿(𝑡), and let lim

𝑡→∞𝜃(𝑡) =

[𝜃1(𝑡), 𝜃2(𝑡), . . . , 𝜃

𝐿(𝑡)] be the corresponding population state.

According to Lemma 2 theWSNs population state, subject tothe constraints 𝜃

𝑗(𝑡) ≥ 0 and ∑

𝑗∈P𝑚

𝜃𝑗(𝑡) = 1, will ultimately

converge to

lim𝑡→∞

𝜃 (𝑡) = [𝜃1 (𝑡) , 𝜃2 (𝑡) , . . . , 𝜃𝐿 (𝑡)] = [1, 0, . . . , 0] (16)

as equilibrium.Theorem 3 therefore follows.

Theorem 4. The equilibrium of the secrecy rate game towardsWSNs is evolutionarily stable.

Proof. Following Theorem 3, we can, for the secrecy rategame towards WSNs, attain equilibrium as {1, 0, . . . , 0} of 𝐿-dimension. At this equilibrium point the set of differentialequations denoted in (15), by substituting

𝜃1 (𝑡) = 1 − 𝜃

2 (𝑡) − ⋅ ⋅ ⋅ − 𝜃𝐿 (𝑡) , (17)

can be changed into

𝜗𝑗 (𝑡) = 𝜗

𝑗 (𝑡)(𝜑𝑗𝑞

(1 − 𝜗𝑗 (𝑡)) +

𝑃𝑚

𝐿

∑

𝑖=𝑃𝑚

2,𝑖 =𝑗

𝜗𝑖 (𝑡) 𝜑𝑗𝑖) ,

𝑗 = 𝑃𝑚

2, . . . , 𝑃

𝑚

𝐿,

(18)

where 𝜑𝑗𝑞is equal to 𝜙(𝜇

𝑗(𝑡) − 𝜇

𝑞(𝑡)) − 𝜙(𝜇

𝑞(𝑡) − 𝜇

𝑗(𝑡)), and

𝜗 (𝑡) = [𝜗2 (𝑡) , 𝜗3 (𝑡) , . . . , 𝜗𝐿 (𝑡)] (19)

denotes the corresponding downsized population state,which has equilibrium as 𝜗

∗(𝑡) = [0, 0, . . . , 0] of (𝐿 − 1)-

dimension.We now illustrate, in order to justify the equilibrium is

evolutionarily stable, that eigenvalues of the Jacobian matrixof the downsized population state all have negative real part.Here an element of the Jacobianmatrix that is a (𝐿−1)×(𝐿−1)

matrix, denoted by 𝐽𝑗𝑞, is a partial derivative as

𝐽𝑗𝑞

= [𝜕 𝜗𝑗(𝑡)

𝜕𝜗𝑞(𝑡)

]

𝜗(𝑡)=𝜗∗(𝑡)

, 𝑗, 𝑞 = 𝑃𝑚

2, . . . , 𝑃

𝑚

𝐿. (20)

The Jacobian matrix denoted by J is therefore attained by

J =

[[[[[[[

[

𝜑𝑃𝑚

2𝑃𝑚

1

0 ⋅ ⋅ ⋅ 0

0 𝜑𝑃𝑚

3𝑃𝑚

1

⋅ ⋅ ⋅ 0

.

.

.... d

.

.

.

0 0 ⋅ ⋅ ⋅ 𝜑𝑃𝑚

𝐿𝑃𝑚

1

]]]]]]]

]

. (21)

In (21) 𝜑𝑃𝑚

2𝑃𝑚

1

, 𝜑𝑃𝑚

3𝑃𝑚

1

, . . . , 𝜑𝑃𝑚

𝐿𝑃𝑚

1

are in fact the eigenvaluesof Jacobian matrix J. From Lemma 2, for any 𝑗 = 2, . . . , 𝐿,𝜙(𝜇𝑗(𝑡) − 𝜇

𝑞(𝑡)) is equal to zero. We can thus attain

𝜑𝑗1

= 𝜙 (𝜇𝑗 (𝑡) − 𝜇

𝑞 (𝑡)) − 𝜙 (𝜇𝑞 (𝑡) − 𝜇

𝑗 (𝑡))

= −𝜙 (𝜇𝑞 (𝑡) − 𝜇

𝑗 (𝑡)) < 0.

(22)

FromTheorem 2.7.3 in [54], the equilibrium point 𝜗∗(𝑡)is evolutionarily stable. Theorem 4 therefore follows.

3.5. Secrecy Rate Adaptation Algorithm. The algorithm ofsecrecy rate adaptation for WSNs is in fact to describethe process how member sensor nodes select adaptivelytheir power level strategies. During this iterative process,periodically, a member sensor node observes its fitness inan entirely distributed mode and evaluates its current fitnesswith the average fitness of the wholeWSNs. If the difference islarger than a predefined maximum bound, then the member


sensor node selects a new power level strategy according tothe selection dynamics from (15). This interactive processgoes on until the ESS of the secrecy rate game towards WSNsis achieved. All member sensor nodes thus find a satisfac-tory solution to secrecy rate adaptation, and no membersensor nodes can benefit by switching their current powerlevel strategies while the other ones keep their strategiesunchanged. From now on, we can show the algorithm asfollows.

Algorithm 1.

(1) Initialize all coefficients including 𝑊, 𝐼, 𝜂2 and all

channel gains;

(2) 𝑡 ← 0;

(3) select a power level strategy 𝑗with the selection prob-ability 𝜃(𝑡) = [1/𝐿, 1/𝐿, . . . , 1/𝐿]. This probabilityassures eachmember sensor node has the same fitnessat the beginning of the game;

(4) compute the fitness 𝜇𝑗(𝑡) and the average fitness 𝜇(𝑡)

according to (7) and (8), respectively;

(5) compute the expected secrecy rate 𝜍𝑗(𝑡) and the

average secrecy rate 𝜍(𝑡) according to (9) and (10),respectively;

(6) DOWHILE .T.

//𝜏 means a predefined minimum bound;

(7) IF |𝜇(𝑡) − 𝜇𝑗(𝑡)| < 𝜏;

(8) EXIT;

(9) ENDIF

//𝜏 means a predefined maximum bound;

(10) IF |𝜇(𝑡) − 𝜇𝑗(𝑡)| > 𝜏;

(11) get 𝜃(𝑡 + 1) by solving (15) with thestandard numerical solver;

(12) select a newpower level strategy 𝑗withthe new selection probability 𝜃(𝑡 + 1);

(13) compute the fitness 𝜇𝑗(𝑡 + 1) and the

average fitness 𝜇(𝑡 + 1) according to (7) and (8),respectively;

(14) compute the expected secrecy rate𝜍𝑗(𝑡+1) and the average secrecy rate 𝜍(𝑡+1) according

to (9) and (10), respectively;

(15) ENDIF;

(16) 𝑡 ← 𝑡 + 1;

(17) ENDDO;

(18) RETURN the array of 𝜍𝑗and 𝜍.

4. Experiments

Since member sensor nodes are generally constrained inlimited computation, memory, and battery life, we simplifythe number of power level strategies to introduce an exampleof the secrecy rate game towards WSNs. We consider thata member sensor node 𝑆

𝑚, when transmitting data, has to

select a strategy in P𝑚

= {𝑃𝐻, 𝑃𝐿}, where 𝑃

𝐻and 𝑃

𝐿denote

the high and low power levels, respectively. That is, for any𝑚 ∈ {1, 2, . . . ,𝑀}, P

𝑚= {𝑃𝐻, 𝑃𝐿}. The utility of a member

sensor node, from (5), can thus be rewritten as

𝜇 (𝑃𝑚, 𝑃��)

= 𝑊(log2(1 +


𝐻𝑛

𝑃��𝐺𝑆��

𝐻𝑛

+ ∑𝐼𝑚

𝑖=1��𝑚

𝑖𝐺𝑆𝑚

𝑖

𝐻𝑛

+ 𝜂2)

− max𝐸𝑘∈E𝑚

log2(1 +


𝐸𝑘

𝑃��𝐺𝑆��

𝐸𝑘

+ ∑𝐼𝑚

𝑖=1��𝑚

𝑖𝐺𝑆𝑚

𝑖

𝐸𝑘

+ 𝜂2))

+

− 𝛼𝑃𝑚,

(23)

where 𝑃𝑚, 𝑃��

∈ P𝑚, 𝑆𝑚, 𝑆��

∈ S. For simple description, let𝜇𝐻𝐻

, 𝜇𝐻𝐿

, 𝜇𝐿𝐻

, and 𝜇𝐿𝐿

be 𝜇(𝑃𝐻, 𝑃𝐻), 𝜇(𝑃

𝐻, 𝑃𝐿), 𝜇(𝑃

𝐿, 𝑃𝐻),

and 𝜇(𝑃𝐿, 𝑃𝐿), respectively.

Obviously, for the example of the secrecy rate gametowards WSNs, we can get (a) 𝑃

𝐻is an ESS if and only if

𝜇𝐻𝐻

> 𝜇𝐿𝐻

and 𝜇𝐻𝐿

> 𝜇𝐿𝐿, (b) 𝑃

𝐿is an ESS if and only if

𝜇𝐻𝐻

< 𝜇𝐿𝐻

and 𝜇𝐻𝐿

< 𝜇𝐿𝐿, and (c) ((𝜇

𝐿𝐿−𝜇𝐻𝐿

)/(𝜇𝐻𝐻

−𝜇𝐻𝐿

−

𝜇𝐿𝐻

+𝜇𝐿𝐿

), (𝜇𝐻𝐻

−𝜇𝐿𝐻

)/(𝜇𝐻𝐻

−𝜇𝐻𝐿

−𝜇𝐿𝐻

+𝜇𝐿𝐿

)) is a mixedESS if and only if 𝜇

𝐻𝐻< 𝜇𝐿𝐻

and 𝜇𝐻𝐿

> 𝜇𝐿𝐿.

Next, we initialize all coefficients required in the exampleof the secrecy rate game towards WSNs for performingnumerical experiments. According to IEEE 802.15.4 physicallayer specifications, we let 𝜌 = 0.01, 𝑊 = 2MHz, 𝑃

𝐻=

30mW, 𝑃𝐿

= 10mW, and 𝜎2

= −112 dBm, respectively. Wealso let the channel gain between a member sensor node andits responsible cluster head be 1, and we let the channel gainbetween a member sensor node and an eavesdropper be 0.6.Since different power levels will lead to different interferencezones, we let 𝑟

𝐻= 50m and 𝑟

𝐿= 10m, where 𝑟

𝐻and 𝑟𝐿

denote the interference distance of a member sensor nodeadopting 𝑃

𝐻and 𝑃

𝐿, respectively. In addition, we assume

the working probability of interferers is 0.01 based on theempirical value. The number of interferers adopting the high(resp., low) power level strategy 𝑃

𝐻(resp. 𝑃

𝐿), therefore, is

0.01 × (𝜌𝜋𝑟2

𝐻+ 3√𝜌𝜋𝑟2

𝐻) (resp., 0.01 × (𝜌𝜋𝑟

2

𝐿+ 3√𝜌𝜋𝑟2

𝐿)).

4.1. Effect of 𝛼. The ESSs of the secrecy rate game towardsWSNs are dependent on the cost parameter 𝛼. As depicted inFigure 2, utilities of all strategy profiles are decreasing linearlywith different speeds while 𝛼 is increasing gradually. Thisfact means the influence of a power level strategy becomeslarger than that of secrecy rate on the utility of a membersensor node. We can see lines of 𝜇

𝐻𝐻and 𝜇𝐿𝐻

intersect when𝛼 ≈ 4.2267, which implies 𝜇

𝐻𝐻> 𝜇𝐿𝐻

if 𝛼 < 4.2267,


1 2 3 4 5 6 7 8 9 10

0

50

100

150

Util

ity

𝛼

−50

−100

−150

−200

𝜇HH 𝜇LH𝜇HL 𝜇LL

Figure 2: Effect of 𝛼 on ESSs of the example of the secrecy gametowards WSNs.

otherwise 𝜇𝐻𝐻

< 𝜇𝐿𝐻

. Similarly, if 𝛼 < 5.0919 then 𝜇𝐻𝐿

>

𝜇𝐿𝐿; otherwise 𝜇

𝐻𝐿< 𝜇𝐿𝐿. Therefore, 𝑃

𝐻is an ESS if and

only if 𝛼 < 4.2267; 𝑃𝐿is an ESS if and only if 𝛼 > 5.0919;

((𝜇𝐿𝐿

−𝜇𝐻𝐿

)/(𝜇𝐻𝐻

−𝜇𝐻𝐿

−𝜇𝐿𝐻

+𝜇𝐿𝐿

), (𝜇𝐻𝐻

−𝜇𝐿𝐻

)/(𝜇𝐻𝐻

−𝜇𝐻𝐿

−

𝜇𝐿𝐻

+𝜇𝐿𝐿

)) is a mixed ESS if and only if 4.2267 < 𝛼 < 5.0919.

4.2. ESSs for the Example of the Secrecy Rate Game towardsWSNs. To illustrate the evolution of member sensor nodesselecting different power level strategies, the probability dis-tribution function 𝜙 in (15) should be first defined. Accordingto the idea [52] through linearizing the function 𝜙, wesuppose

𝜙 (𝑥) = 𝛽 + 𝛾𝑥, (24)

where 𝛽, 𝛾 ∈ R, such that 0 ≤ 𝛽 + 𝛾𝑥 ≤ 1. Substituting(24) into (15), we can attain the selection dynamics ofmembersensor nodes as

𝜃𝑗 (𝑡) = 2𝛾𝜃

𝑗 (𝑡) ∑

𝑞,𝑞 =𝑗

𝜃𝑞 (𝑡) (𝜇𝑗 (𝑡) − 𝜇

𝑞 (𝑡)) , (25)

where 𝛾 is in fact a parameter that will influence theconvergence speed of ESS. Let 𝛾 be 0.5 (in practice, 𝛾 maybe an arbitrary real number); we attain different ESSs ofthe example of the secrecy rate game under different costparameter values 3, 4.5, and 6, which are illustrated in Figures3, 4, and 5, respectively. Note that these cost parameter valuesdetermine three different ESSs. As illustrated in Section 4.1,𝛼 = 3 specifies that 𝑃

𝐻is an ESS, 𝛼 = 4.5 specifies that 𝑃

𝐿is

an ESS, and 𝛼 = 6 specifies that there is a mixed ESS.In Figure 3, when the initial value of (25) is 0.005, that is,

if only 0.5%member sensor nodes select the high power levelstrategy 𝑃

𝐻in the beginning, the fraction of member sensor

nodes selecting 𝑃𝐻, after about 51 times of playing the game,

will be stable at 1; that is, all member sensor nodes select𝑃𝐻as

their optimal strategy. We can also see that the higher initialfraction of member sensor nodes selecting 𝑃

𝐻is, the faster

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Times

𝜃H(0) = 0.005

𝜃H(0) = 0.255𝜃H(0) = 0.555

𝜃H(t)

Figure 3: 𝑃𝐻is an ESS when 𝛼 = 3.

0 10 20 30 40 50Times

𝜃H(0) = 0.995

𝜃H(0) = 0.755𝜃H(0) = 0.455

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1𝜃H(t)

Figure 4: 𝑃𝐿is an ESS when 𝛼 = 6.

convergence speed of achieving ESS𝑃𝐻can be attained.These

results reflect the fact that 𝑃𝐻is an ESS when 𝛼 = 3.

In Figure 4, even if 99.5% member sensor nodes select𝑃𝐻in the beginning, we can find that, after about 37 times

of playing the game, the fraction of member sensor nodesselecting 𝑃

𝐻will be stable at 0; that is, all member sensor

nodes select the low power level strategy 𝑃𝐿in the end. In

addition, the convergence speed of achieving ESS𝑃𝐿becomes

faster and faster as decreasing the initial fraction of membersensor nodes selecting 𝑃

𝐻. The fact that 𝑃

𝐿is an ESS when

𝛼 = 6 is thus reflected.In Figure 5, all curves of member sensor nodes selecting

𝑃𝐻, irrespective of different initial value of (25), converge

to a limit 0.6841 or so. At this equilibrium point, about


0 10 20 30 40 50Times

𝜃H(0) = 0.155

𝜃H(0) = 0.500𝜃H(0) = 0.855

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

𝜃H(t)

Figure 5: A mixed ESS (0.6841, 0.3159) exists when 𝛼 = 4.5.

68.41% member sensor nodes select 𝑃𝐻while about 31.59%

member sensor nodes selecting𝑃𝐿, as their respective optimal

strategies. The other interpretation is that member sensornodes switch their strategies between strategies 𝑃

𝐻and 𝑃

𝐿,

that is, playing 𝑃𝐻

for 68.41% of the time and 𝑃𝐿for the

remainder of the time. In fact, under this situation the casethat all member sensor nodes select 𝑃

𝐻(resp. 𝑃

𝐿) cannot be

evolutionarily stable. This is because the utility of a membersensor node adopting 𝑃

𝐿(resp., 𝑃

𝐻), when encountering one

adopting 𝑃𝐻(resp. 𝑃

𝐿), is better than that of a member sensor

node adopting𝑃𝐻(resp.𝑃

𝐿).These results lead to the fact that

there is a mixed ESS (0.6841, 0.3159) when 𝛼 = 4.5.

4.3. Secrecy Rate Adaptation. Our mechanism of secrecy rateadaptation in WSNs is realized when member sensor nodesselect adaptively their power level strategies for maximizingtheir fitness. For convenience, let 𝜃

𝐻(𝑡) be the fraction of

member sensor nodes adopting𝑃𝐻at time 𝑡; then the fraction

of nodes adopting 𝑃𝐿at time 𝑡 is 1 − 𝜃

𝐻(𝑡). We denote the

fitness of member sensor nodes adopting 𝑃𝐻and 𝑃𝐿at time 𝑡

by 𝜇𝐻(𝑡) and 𝜇

𝐿(𝑡), respectively, where 𝜇

𝐻(𝑡) and 𝜇

𝐿(𝑡), from

(7), are given by

𝜇𝐻 (𝑡) = 𝜃

𝐻 (𝑡) 𝜇𝐻𝐻 + (1 − 𝜃𝐻 (𝑡)) 𝜇𝐻𝐿, (26)

𝜇𝐿 (𝑡) = 𝜃

𝐻 (𝑡) 𝜇𝐿𝐻 + (1 − 𝜃𝐻 (𝑡)) 𝜇𝐿𝐿. (27)

From (8), we can attain that the average fitness of the wholeWSNs is

𝜇 (𝑡) = 𝜃𝐻 (𝑡) 𝜇𝐻 (𝑡) + (1 − 𝜃

𝐻 (𝑡)) 𝜇𝐿 (𝑡) . (28)

Similarly, we denote the expected secrecy rate of amember sensor node adopting 𝑃

𝐻and 𝑃

𝐿at time 𝑡 by 𝜍

𝐻(𝑡)

0 10 20 30 40 5010

20

30

40

50

60

70

80

Times

Fitn

ess

𝜇(t)𝜇H(t)

𝜇L(t)

Figure 6: Different fitness when 𝛼 = 3.

and 𝜍𝐿(𝑡), respectively, where 𝜍

𝐻(𝑡) and 𝜍

𝐿(𝑡), from (9), are

given by

𝜍𝐻 (𝑡) = 𝜃

𝐻 (𝑡) 𝜍𝐻𝐻 + (1 − 𝜃𝐻 (𝑡)) 𝜍𝐻𝐿, (29)

𝜍𝐿 (𝑡) = 𝜃

𝐻 (𝑡) 𝜍𝐿𝐻 + (1 − 𝜃𝐻 (𝑡)) 𝜍𝐿𝐿, (30)

and the average secrecy rate of the whole WSNs, 𝜍(𝑡), from(10), is given by

𝜍 (𝑡) = 𝜃𝐻 (𝑡) 𝜍𝐻 (𝑡) + (1 − 𝜃

𝐻 (𝑡)) 𝜍𝐿 (𝑡) . (31)

Here 𝜍𝑢V, 𝑢, V ∈ {𝐻, 𝐿}, denotes the secrecy rate of a

member sensor node adopting 𝑃𝑢when encountering one

adopting 𝑃V, which can be computed by the first term in (23).We are now able to, under different cost values 3, 4.5, and

6, show changeable trends of fitness depicted in Figures 6,7, and 8, as well as expected secrecy rates in Figures 9, 10,and 11, respectively. In Figures 6 and 9 where 𝑃

𝐻is an ESS,

the fitness (resp., expected secrecy rate) of a member sensornode adopting 𝑃

𝐻is always higher than that of one adopting

𝑃𝐿. As member sensor nodes play the game continuously,

both fitness values (resp., expected secrecy rates) decreasegradually while the average fitness (resp., secrecy rate) ofthe whole WSNs increases. After about 51 times of playingthe game, the curves of both 𝜇

𝐻(𝑡) (resp., 𝜍

𝐻(𝑡)) and 𝜇(𝑡)

(resp., 𝜍(𝑡)) converge to a limit 42.9739 (resp., 132.9091) orso. Arriving at this equilibrium point implies all membersensor nodes, through switching their power level strategiesadaptively, select 𝑃

𝐻in the end due to its higher fitness.

The secrecy rate attained by a member sensor node is alsoadjusted adaptively. In Figures 7 and 10 where𝑃

𝐿is an ESS, we

can see some converse phenomena. The fitness of a membersensor node adopting 𝑃

𝐿is higher than that of a member

sensor node adopting 𝑃𝐻, whereas the expected secrecy rate

of a member sensor node adopting 𝑃𝐿is lower. Moreover,

the average fitness of the whole WSNs is increasing whilethe average secrecy rate is decreasing. In the end, when

10 International Journal of Distributed Sensor NetworksFi

tnes

s

0 10 20 30 40 50

0

Times

−10

−15

−20

−25

−30

−35

−40

−45

−50

−5

𝜇(t)𝜇H(t)

𝜇L(t)

Figure 7: Different fitness when 𝛼 = 6.

Fitn

ess

0 10 20 30 40 506

8

10

12

14

16

18

20

22

Times

𝜇(t)𝜇H(t)

𝜇L(t)

Figure 8: Different fitness when 𝛼 = 4.5.

all member sensor nodes select 𝑃𝐿as their optimal power

level strategy, the curves of both 𝜇𝐿(𝑡) and 𝜇(𝑡), after about

37 times of playing the game, converge to a limit −1.0675or so. Correspondingly, the expected secrecy rate attainedby a member sensor node is adjusted adaptively to a limit59.9593 or so. In Figures 8 and 11 where a mixed ESS existsat (0.6841, 0.3159) or so, there is a similar trend among curvesof 𝜇𝐻(𝑡), 𝜇𝐿(𝑡), and 𝜇(𝑡). After about 36 times of playing the

game, all curves in Figure 8 converge to a limit 6.8323 or so.That is, after achieving the mixed equilibrium point, bothstrategies 𝑃

𝐻and 𝑃

𝐿coexist permanently and all member

sensor nodes have the samefitness regardless of their differentstrategies. On the other hand, in Figure 11 we can see expectedsecrecy rates of member sensor nodes adopting 𝑃

𝐻and 𝑃

𝐿

are decreasing slowly. The average secrecy rate of the whole

0 10 20 30 40 5040

60

80

100

120

140

160

180

Times

Expe

cted

secr

ecy

rate

𝜍(t)

𝜍H(t)

𝜍L(t)

Figure 9: Different expected secrecy rates when 𝛼 = 3.

0 10 20 30 40 5040

60

80

100

120

140

160

180

Times

Expe

cted

secr

ecy

rate

𝜍(t)

𝜍H(t)

𝜍L(t)

Figure 10: Different expected secrecy rates when 𝛼 = 6.

WSNs, however, is increasing little by little. This adaptationof average secrecy rate results in equilibrium point 113.3959or so in the end.

5. Conclusion

We have demonstrated the evolution process of secrecy rateadaptation for sensor nodes in the clustered WSNs. For thisaim, the classical wiretap channel has been extended to fitfor the clustered WSNs and the corresponding equation tocompute secrecy rate has been attained. The secrecy rategame we have constructed is able to reflect the interactionsamong member sensor nodes. With EGT, we have attained aselection dynamics to determine how member sensor nodes


0 10 20 30 40 5040

60

80

100

120

140

160

180

Times

Expe

cted

secr

ecy

rate

𝜍(t)

𝜍H(t)

𝜍L(t)

Figure 11: Different expected secrecy rates when 𝛼 = 4.5.

for maximizing their fitness will select their power levelstrategies.We have also proved the stability of the secrecy rategame to illustrate which power level strategy will be adoptedin the end.The results from experiments for an example of thesecrecy rate game towards WSNs have verified all ESSs andhave revealed the process of expected secrecy rate adaptationfor member sensor nodes. As a result, this disclosure ofsecrecy rate adaptation will be beneficial to employ PHYsecurity to guarantee the secrecy of transmission in WSNs.

Notations

𝜌: Node density𝑟𝑅: Receiving distance

𝑟𝐼: Interfering distance

√𝜎: Standard deviation𝑧𝐼: Interference zone

𝐼: Maximum number of interferersto a sensor node

S,H, and E: Sets of 𝑀 member sensor nodes,𝑁 cluster nodes, and 𝐾

eavesdroppers, respectivelyE𝑚: Set of eavesdroppers capable of

listening in data sensed bymember sensor node 𝑆

𝑚

𝐺𝑆𝑚

𝐻𝑛

: Channel gain between membersensor node 𝑆

𝑚and its responsible

cluster head 𝐻𝑛

𝐺𝑆𝑚

𝐸𝑘

: Channel gain between membersensor node 𝑆

𝑚and eavesdropper

𝐸𝑘, 𝐸𝑘∈ E𝑚

𝜂2: Thermal noise power at cluster

heads and eavesdroppers,respectively

𝑊: Channel bandwidth

𝑆𝑚

𝑖: An interferer to member sensor node 𝑆

𝑚,

𝑖 ∈ {1, 2, . . . , 𝐼𝑚}

S𝑚: Set of interferers to member sensor node

𝑆𝑚

𝐶𝑆𝑚

𝐻𝑛

: Channel capacity from member sensornode 𝑆

𝑚to its responsible cluster head 𝐻

𝑛

𝑃𝑚: Transmission power adopted by member

sensor node 𝑆𝑚

��𝑚

𝑖: Interference power adopted by interferer

𝑆𝑚

𝑖

𝐺𝑆𝑚

𝑖

𝐻𝑛

: Channel gain between interferer 𝑆𝑚𝑖and

the interfered cluster head 𝐻𝑛

𝐶𝑆𝑚

𝐸𝑘

: Channel capacity from member sensornode 𝑆

𝑚to eavesdropper 𝐸

𝑘, 𝐸𝑘∈ E𝑚

𝐺𝑆𝑚

𝑖

𝐸𝑘

: Channel gain between interferer 𝑆𝑚𝑖and

eavesdropper 𝐸𝑘, 𝐸𝑘∈ E𝑚

𝐶(𝑃𝑚): Secrecy rate between member sensor node

𝑆𝑚adopting a power level 𝑃

𝑚and its

responsibleG = (S,P,U): Our secrecy rate game towards WSNsP: Set of strategy profiles for all member

sensor nodesP𝑚: Set of pure power level strategies available

to member sensor node 𝑆𝑚

U: Set of utilities𝜇(𝑃𝑚, 𝑃��): Utility function for member sensor node

𝑆𝑚adopting power level 𝑃

𝑚when its

opponent𝛼: Cost parameter reflecting the degree of a

member sensor node consuming energyfor transmission

𝜃𝑗(𝑡): Fraction of member sensor nodes using

power level strategy 𝑗 at time 𝑡

𝜃(𝑡): State of the whole WSNs at time 𝑡

𝜇𝑗(𝑡): Fitness of member sensor nodes adopting

power level strategy 𝑗 at time 𝑡

𝜇(𝑡): Average fitness of the whole WSNs at time𝑡

𝜍𝑗(𝑡): Expected secrecy rate of member sensor

nodes adopting power level strategy 𝑗 attime 𝑡

𝜍(𝑡): Average secrecy rate of the whole WSNs attime 𝑡

𝑟𝑗(𝜃): Average review rate of member sensor

nodes using power level strategy 𝑗

𝑝𝑗

𝑞(𝜃): Probability of a reviewing member sensor

node changing from power level strategy 𝑗

to 𝑞

𝜙: Probability distribution function𝜗𝑗(𝑡): Fraction of member sensor nodes in the

downsized population using power levelstrategy 𝑗 at time 𝑡

𝜗(𝑡): State of the downsized population at time 𝑡

𝜑𝑗𝑞: Notation simple to denote equation

𝜙(𝜇𝑗(𝑡) − 𝜇

𝑞(𝑡)) − 𝜙(𝜇

𝑞(𝑡) − 𝜇

𝑗(𝑡))

𝐽𝑗𝑞: Element of Jacobian matrix J


𝑃𝐻and 𝑃

𝐿: High and low power levels in the

example of the secrecy rate game,respectively

𝜇𝐻𝐻

, 𝜇𝐻𝐿

, 𝜇𝐿𝐻

, and 𝜇𝐿𝐿: Notations simple to denote𝜇(𝑃𝐻, 𝑃𝐻), 𝜇(𝑃𝐻, 𝑃𝐿), 𝜇(𝑃𝐿, 𝑃𝐻),

and 𝜇(𝑃𝐿, 𝑃𝐿), respectively.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work was supported by National Natural Science Foun-dation of China under Grant no. 61272034, by ZhejiangProvincial Natural Science Foundation of China underGrants LY13F030012 and LY13F020035, and by Science Foun-dation of Shaoxing University under Grant no. 20145021.

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