 Research
 Open Access
Distributed coalitional game for friendly jammer selection in ultradense networks
 Ying Wang^{1}Email author,
 Zhongyu Miao^{1},
 Ruijin Sun^{1} and
 Lei Jiao^{2}
https://doi.org/10.1186/s1363801607025
© The Author(s) 2016
 Received: 11 March 2016
 Accepted: 19 August 2016
 Published: 6 September 2016
Abstract
Consider an ultradense heterogeneous network with one malicious eavesdropper intercepting macrolayer information. A portion of smallcell base stations (SBSs) acts as the friendly jammer to help improving macrousers’secrecy rate by transmitting interference signal on the wiretap channel. In return, the client macrouser pays to its jammers for the jamming power that they provide. Instead of transmitting noise as traditional jammers do, this paper proposes a modified spectrum leasing method, which allows SBSs to replace the thermal noise with their own traffic. This approach also permits the jamming SBSs to access extra spectrum in order to enhance the performance. In the considered scenario, the macrouser tries to find the SBSs that can mostly protect its confidential message, while each SBS decides whether to serve as a jammer or not. Once an SBS decides to be a jammer, it needs to choose the optimal client macrouser depending on the channel condition. This twoway selection problem between SBSs and macrousers is modeled as the coalition formation game with nontransferable utility, and a distributed scheme is proposed for this game, in which the players (macrousers and SBSs) individually make a decision and converge to a Nashstable partition in a selforganized manner. The simulation results show that the majority of macrouser equipments enjoy a fivefold increment in average secrecy rate and that the friendly jammer scheme effectively protects the macrousers from the eavesdropper. At the same time, the average capacity of smallcell layer also achieves a 16.92 % improvement.
Keywords
 Ultradense networks
 Friendly jammer
 Information security
 Coalition formation game
1 Introduction
With the proliferation of the smart and realtime devices, the demands for mobile data rise dramatically, which has promoted a large amount of hotspots in indoor areas. It is estimated that global wireless traffic will continue growing and reach a level that is 1000 times larger than nowadays in a decade [1]. The communication system is facing an unprecedented challenge. As a result, the technology of ultradense networks (UDNs) is introduced as a promising approach to satisfy the skyrocketed user demands and to improve indoor coverage and spectrum efficiency. A UDN is composed of a macrocell overlayed by a number of densely deployed lowpower, lowcost base stations which could provide high throughput for indoor and hotspot areas. The twotier architecture has the advantage of ensuring the overall coverage as well as satisfying the local traffic demand. UDN is viewed as one of the key technologies in 5G, and fruitful achievements have been made in fields of interference management [2–4], power control [5–7], energy efficiency [8–10], offloading [11, 12], network selection [13, 14], etc.
Meanwhile, information security is an important aspect in communications. However, there are few articles related to security and privacy in UDNs. Traditionally, the way to improve confidentiality of the information mainly relies on the encryption system at higher layers. However, the computational cost for either encryption or the decryption is usually so high that it may be a great burden for both the SBSs and macrocell user equipments (MUEs), especially for the smallcell infrastructures [15]. As the wireless and mobile network structure becomes more and more complicated, the key management is far more difficult as the number of nodes increases [16]. What is more, the broadcast nature of the wireless channel makes it unsafe for key distribution which is fatal to most of the encryption algorithms since the opponent can easily decipher the transferred message once the key is overheard. Hence, it is discovered in recent years that it may not be so efficient or suitable to rely on the upper layer encryption in wireless networks. A wiretap channel is introduced in [17], and it is proven that if the wiretap channel was worse than the main channel, the users could have a nonzero secrecy capacity. This pioneer work have enlightened the research on physical layer security, which has recently been discovered as an efficient way to fight against the malicious eavesdroppers. One of the basic methods of improving the legitimate users’ secrecy capacity in physical layer security is interfering the eavesdropper through the artificial noise, which is called friendly jamming or cooperative jamming. More specifically, in the regime of friendly jamming, there are numbers of friendly jammers in the network responsible for transmiting noise or the codewords on the same channel of the client user’s so as to confuse the eavesdropper and, thus, enhance the safety performance of the legitimate user.
Most of the current studies on friendly jammer are carried out within relay [18–21] and cognitive scenarios [22–24]. Article [19] studies a twoway relay system with an untrusted relay node. The transmission pair improves its secrecy rate by buying proper jamming power from the friendly jammers. In [20], a coalition formation game is formulated to investigate the cooperation between relays and friendly jammers in order to assist data transmission. The drawback of those schemes is the requirement for installation and maintenance of the dedicated jammers, resulting in a significant cost to the operator. A friendly jamming paradigm using spectrum leasing is developed in [22], where the jammers are unsubscribed nodes that also have data to transmit. The subscribed user attracts the jammers’ cooperation by allowing them to use a fraction of its frame for their own data transmissions. In [23], a new cooperative scheme is introduced in a cognitive network with several relay nodes. The secondary users (SUs) are allowed to transmit simultaneously with the primary user (PU). In the first hop, the SU transmitter sends its information to the relay while the SU receiver acts as a friendly jammer. In the second hop, the relay passes the information to the SU receiver and the transmitter, in turn, takes the role of disturbing the eavesdropper. Nonetheless, most existing work on friendly jammer is based on a relatively simple network topology, which considers only one transmission pair (i.e., a source and a destination). Studies in [25–27] investigate the power allocation of the friendly jammer in a network with multiple sourcedestination links, but there is only one jammer in the system. All those abovementioned approaches and scenarios are not suitable for UDNs. In UDNs, several macro base station (MBS)MUE links need to keep their messages secret. Furthermore, the densely distributed SBSs can be employed as friendly jammers to help MUEs with secure communication, which avoids deploying dedicated jammers. Hence, friendly jammer via SBSs is an appropriate method to enhance the security of UDNs.
As mentioned previously, existing approaches do not apply directly to UDNs due to the fact that both the users and jammers have more than one candidate servers or clients. When there is only one sourcedestination link, the jammers do not need to consider which pair to choose, nor do the transmission pair in the situation where exists only one jammer. Nevertheless, in UDNs, the MUEs compete for the most effective jammers from multiple SBSs for themselves, while the SBSs carefully estimate the revenue from different MUEs and choose the one that brings the maximum benefits as a client user. This generates a twoway selection problem, making it more difficult to form the cooperative structure between the users and jammers. Therefore, extensions and modifications are needed to model the twoway selection among the multiple MUEs and jammers. Furthermore, jammers in previous studies have no resource to transmit data. They obtain the transmission opportunity as the resource reward for providing jamming service to the transmission pair, as in [22, 23]. More specifically, the jammers in the previous studies do not have any chance to serve their own traffic unless if they provide jamming service. Hence, the jammers in existing articles have strong motivation to perform cooperative jamming, which is not the case in UDNs. The SBSs have their own users and limited resource. Providing friendly jamming means a loss of performance for SBSs since they allocate part of their power to jamming and concentrate less on their own users. Consequently, the SBSs need to weigh the income against the performance loss and may not be so willing to help. A more powerful mechanism of reimbursement is required to encourage the SBSs to cooperate as well as to enhance their performance when they provide the jamming service. In addition, interference has always been a problem in UDNs. Though friendly jammer takes advantage of interference to protect privacy, a careful balance among all kinds of interferences is also required to guarantee the overall performance of the network. Therefore, the introduction of friendly jammer to UDNs is a more intricate problem. To our best knowledge, it is the first work to apply friendly jammer to UDNs.
The enormous number of nodes in UDNs makes it complicated for a centralized algorithm to handle such a large amount of data and computations. Game theory is a powerful tool for analyzing the interaction between various players. Each player in the game can, based on network condition, make a decision without the instruction of a centralized control node. Modeling interactions among users as a game and designing distributed algorithms accordingly have been widely applied in communication systems [5, 8, 11–13]. Coalition formation game is one of the most important classes in game theory, which can be used to form winwin or cooperative coalitions among the players to optimize the network performance and to improve their own benefits at the same time [20, 28, 29].
This paper tries to provide an insight about the future practical use of friendly jamming techniques and investigates the secure communication in UDNs with an eavesdropper, making use of the SBSs in the network to prevent the eavesdroppers from overhearing the information between MBS and MUEs. The MUEs compete for the jammers which can provide the maximum increase in secrecy capacity. On the other hand, the SBSs also have the freedom to decide whether to be a jammer or not and to choose its client user in order to optimize its own utilization. The interaction between MUEs and SBSs has been modeled as a coalition formation game. According to the role that SBSs and MUEs play in this friendly jammer system, we use SBS and jammer interchangeably in the following pages, as well as MUE and user (or client user).

This paper extends the application of friendly jammer to a more realistic network scenario with multiple users (i.e., MUEs) and multiple jammers (i.e., SBSs). Furthermore, a noveldistributed scheme is proposed to solve the twoway selection problem among users and jammers, by exploiting the nontransferable utility (NTU) coalition formation game.

Since it is reasonable for SBSs to attach more importance to performance than to the money paid by MUEs, in addition to compensating the SBSs for the jamming power, a modified stage combined spectrum leasing (SCSL) is proposed to effectively motivate the SBSs to cooperate. Besides, SCSL allows jammers to replace noise with useful information as the interference signal to eavesdropper, which makes best use of the resource and greatly improves the efficiency.
The rest of this paper is organized as follows. In Section 2, the system model is presented. Section 3 formulates the problem in a coalitional game approach and solves it in a distributed manner. The property of stabilization is also proven in this section. Numerical results are displayed and discussed in Section 4 before we conclude the paper in Section 5.
2 System model
In this section, we first describe the network model of the proposed scheme. Then, the mechanism of the friendly jammer is introduced and a brief analysis of the mechanism is presented. At last, we explain the SCSL and discuss how it can improve the performance compare with the spectrum leasing [22, 30].
2.1 Network model
Consider an ultradense network with an eavesdropper, an MBS, densely deployed small cells, and several MUEs. We assume that the index of the MBS is 0 sand that of the eavesdropper is e. Let \(\mathbb {M}=\{1,2,\ldots,M\}\) and \(\mathbb {K}=\{1,2,\ldots,K\}\) be the set of M MUEs and K SBSs in the network, respectively, where \(M=\mathbb {M}\) and \(K=\mathbb {K}\). Each SBS k serves only one smallcell user equipment (SUE), denoted by ν _{ k }. The spectrum access strategies between the two layers can be divided into three classes: (1) shared, i.e., the small cells are allowed to reuse the entire bandwidth of MBS, which has a high level of resource utilization but relatively severe cross and colayer interference; (2) dedicated, i.e., a dedicated spectrum that are orthogonal to that of macrocell’s is allocated to the small cells, which eliminates the crosslayer interference at the cost of a lower level of resource utilization; (3) hybrid, i.e., a portion of the macrocell’s spectrum is allocated to a small cell, which is a compromise of interference and resource reuse. To eliminate the crosslayer interference, we assume a dedicated mode as the basic spectrum access strategy between the two layers. Assuming that there are N orthogonal subchannels of bandwidth W and let \(\mathbb {N}=\{1,2,\ldots,N\}\) represent all the frequency resource available in the system. In a dedicated spectrum allocation, the \(\mathbb {N}\) is divided into two disjoint sets N _{ m } and N _{ k }, where N _{ m }∩N _{ k }= ∅ and N _{ m }∪N _{ k }=\(\mathbb {N}\). The MBS chooses one subchannel n _{ m } from N _{ m } to serve MUE m, \(m \in \mathbb {M}\) and SBS k, \(k \in \mathbb {K}\) transmits on n _{ k }∈N _{ k } to serve its user. Since there are far more SBSs than the subchannels that dedicated to SBSs, it is indispensable to reuse the subchannels in N _{ k } and thus the colayer interference among cochannel SBSs is generated. Denote by I(n) the set of SBSs that use the same channel n∈N _{ k }, \(I(n)=\{n_{k}=n, k\in \mathbb {K}\}\). The channel model includes the path loss and Rayleigh fading [20]. Let \(h_{i,j}^{n}\) be the channel gain between the transmitter i and the receiver j on subchannel n. We assume that the channel gains in the system can be measured (including those of channels to the eavesdropper) [22]. The values of maximum power of MBS and SBS are P _{0} and P _{ sbs }, respectively. The transmission power of SBS k on subchannel n is denoted by \({P_{k}^{n}}\). The power of thermal noise is σ ^{2}.
where [x]^{+} represents max(0,x). When the data rate received by the eavesdropper is higher than that of the MUE m, all the information will be wiretapped, resulting in a zero secrecy capacity. {m} is the coalition that MUE m belongs to and in the initial noncooperative condition, MUE m lies in a singleton coalition which is formed by itself.
where i∈I(n _{ k }),i≠k, is the set of SBSs that interfere with k.
2.2 Friendly jammer
Note that the interference direct to the eavesdropper will also affect the communication between MUE and MBS. Friendly jammer makes use of the disparate value of interference that a jammer brings to the eavesdropper and the MUE to increase the secrecy capacity. Only if a jammer causes more interference to the eavesdropper than to MUE m can it possibly play a positive role and, in other words, have the qualification of being MUE m’s jammer.
Proposition 1
The necessary condition that SBS k is able to improve MUE m’s secrecy capacity is that the channel gain between SBS k and MUE m is less than that from SBS k to the eavesdropper, i.e. \(h_{k,e}^{n_{m}}>h_{k,m}^{n_{m}}\).
A similar conclusion is drawn in [20], but we will proof it in a more mathematical way.
Proof
The problem can be divided into two cases according to the initial secrecy capacity of the MUE.
I. MUE m originally has a nonzero secrecy capacity.
It can be easily obtained that \(\left (\sigma ^{2}+\alpha P_{k}h_{k,e}^{n_{m}}\right)/\left (\sigma ^{2}+\alpha P_{k}h_{k,m}^{n_{m}}\right)>1\), i.e., \( h_{k,e}^{{n_{m}}} > h_{k,m}^{{n_{m}}}\).
II. MUE m has a zero secrecy capacity in the beginning.
the condition \(h_{k,e}^{{n_{m}}} > h_{k,m}^{{n_{m}}}\) holds. □
Based on Proposition 1, one understands that only a fraction of SBSs are qualified to enhance the secrecy capacity of a certain MUE. We assume that every MUE can select one or more jammers on its eligible jammer list, while the SBS can only choose to serve one MUE. The rationale behind this is that the SBS concerns most about its own traffic and has to guarantee the performance of its own SUE ν _{ k }. It is injudicious to utilize much of its energy to protect the security of macrolayer, putting its own user at a reduced performance.
2.3 Stage combined spectrum leasing
2.4 Twoway selection as an optimization problem
where R _{ m } denotes the secrecy capacity of MUE m and accrodingly, R _{ k } is the capacity of SBS k, which are defined in (7) and (16), respectively.
Nevertheless, the huge number of communication nodes in UNDs makes it almost impossible to solve the above problem in a centralized method due to the enormous network information to collect as well as the astounding computational complexity. The likely number of \(\prod \) is given by the Bell Number which has reached to 115975 when there’s only 10 nodes in the network. Hence, we turn to the distributed and practical methods and model the problem as the a coalition formation game to be detailed in the next section.
3 Coalitional game
In this section, we model the twoway selection problem between MUEs and SBSs as the coalition formation game in partition form with nontransferable utility (NTU). Through the game, the players choose the best allies to maximize their utilities in a distributed manner, the result of which satisfies the Nash stability and individual stability.
3.1 Game formulation
The coalitional game theory provides a convenient analytical tool for studying the interaction among the players to form the cooperative groups, and it has been applied to wireless communication system in many articles [28–31]. To further explain the concept of coalition formation game in partition form, we first introduce the definition of the coalition partition [32].
Definition 1
A coalition partition or coalitional structure is defined as the set \(\prod _{\Omega }=\{S_{1},~S_{2},\ldots,~S_{L}\}\) which partitions the player set Ω (in our game Ω is the collection of all MUEs and SBSs, in other words, \(\Omega =\mathbb {M}\cup \mathbb {K}\)), i.e., ∀k,S _{ k }⊆Ω are disjoint coalitions such that \(\bigcup \nolimits _{l = 1}^{L} {{S_{l}} = \Omega }\).
The coalition formation game in partition form is introduced in [33] as a class of game in which the profit of a coalition S, and its members have a strong dependence on the coalition partition and the way that the player in Ω∖S is organized. The coalitional game in partition form with NTU can be defined as follows [34]:
Definition 2
A coalitional game in partition form with NTU is defined by a pair (Ω,U), where Ω is the set of players while U is a partition function that maps any coalition S _{ l }⊂Ω,S _{ l }∈Π _{ Ω } to a closed convex subset of \(\mathbb {R}^{S_{l}}\). U(Π _{ Ω },S _{ l }) is the payoff vector for every player in S _{ l }.
In the proposed game, the performance of the members in any coalition S _{ l } is affected by the coalitional structure of the players outside the S _{ l }. Hence, it has the following property:
Property 1
The game between the MUEs and SBSs is indeed in partition form.
For example, if an SBS k decides to act as a jammer after evaluating the tradeoffs and merges into a coalition S which contains MUE m, a fraction of its power will be transferred to the subchannel n _{ m }, attenuating the colayer interference on subchannel n _{ k } suffered by the SBSs outside S. As a result, the performance of a player is closely related to the organization of players from other distinct coalitions and has a dependence on the network structure.
where h(m) is the record of the historical coalitions that m has been to. The players are not permitted to revisit the coalition that they had left previously and will acquire a negative infinite utility if they do so. This rule is adopted by many studies [28, 29, 31] as an effective measure to guarantee a faster convergence of the algorithm. Note that the singleton coalition {m} will never be recorded since a player is allowed to quit a coalition whenever it discovers that it is better to be single. In (20), Π is the current partition while S is the coalition that MUE m belongs to. R _{ m }(S) is given by (7) and represents the secrecy capacity that MUE m obtains with the help of the jammers in S. C _{ m }(S) is the total cost for the jamming power which is defined in (15).
The main rationale behind the utility function u _{ m } is that in the proposed scheme, the purpose of an MUE to join the game is to find the jammers who are keen on supporting its secret communication, making up a coalition in which the jammers have the right to send messages on the MUE’s channel in a relatively low power. Henceforth, we summarize the two features of a legal coalition which contains the MUE m. First, a nonsingleton coalition is formed by the MUE m and its jammers only. Second, all the members in the coalition transmit on the subchannel n _{ m } with full (for MUE m) or a portion (for jammers, if there’s any) of their power. In the condition that \(S\cap \mathbb {M}\!>\) 1, more than one MUE exists in the coalition S, violating the two features above. The reasons are as follows. Firstly, due to the fact that merely the SBSs have the function of jamming, such a coalition can only be formed by one MUE and several SBSs. The first rule is broken by adding additional MUEs into the coalition since the MUE cannot be a friendly jammer of another MUE. Secondly, each MUE in the system occupies a different subchannel so that no interference exists within the macrolayer. If there are some MUEs other than MUE m that stay in the coalition, the extra MUEs are working on subchannels distinct from that of MUE m’s and are banned to transmit on subchannel n _{ m }, which conflicts the second feature. From what has been discussed, we set a negative benefit of forming a coalition with another MUE to prevent such event. For the case that a legal coalition is formed, the goal of MUE m is to pay less for better secret transmission via carefully selecting its allies among the potential jammers. When there is only one MUE m in S, i.e., S=1, with no assist of any SBS, it has no need to pay for the jamming power. The utility of m is simply its secrecy capacity in this case. Under the circumstances that at least one jammer comes to help, the profit of m is the secrecy performance minus the expense.
where ρ _{ k }, given in (14), is the monetary emolument for unit jamming power payed by client MUE. Similar to h(m), h(k) is the historical record of the coalitions that SBS k has once joined in. R _{ k }({k}) and R _{ k }(S) are given by (4) and (16), respectively, as the capacity of SBS k in different situations.
Similar to MUE players, the utility function of SBSs includes their capacity and the money they receive (if any), as presented in the last two lines of (21), when they form a legal coalition. There are also some coalitions considered as illegal. When \(S \cap \mathbb {M}=0\) and S>1 holds, all members in S are SBSs, in other words, jammers, which makes no sense due to the absence of an MUE. The impetus for an SBS to assist the secret communication of the macrolayer is money and resource incentive that lead to the boost of both performance and income. As is analyzed, a nonsingleton coalition is supposed to have one MUE that needs friendly jamming and takes the role of offering the bonus to SBSs. Consequently, there is no inspiration for an SBS to participate in a group without MUE. The utility of merging into a coalition with pure SBSs is thus set to negative infinite, showing that it is extremely unwise to do so. Other than the unit price of jamming power, ρ _{ k } also presents the marginal gain of the secrecy capacity that SBS k brings to its client MUE. The sign of ρ _{ k } represents the impact of SBS k on its client MUE in the presence of the other jammers in the coalition. A positive ρ _{ k } implies that the existence of SBS k is meaningful since k indeed elevates the MUE’s secrecy performance. On the contrary, a negative ρ _{ k } means that the client MUE would have a higher secrecy capacity without SBS k. As the paramount goal of this game is to protect the macrolayer’s information from the malicious eavesdropper, any collaborator that may impair the MUE’s secrecy capacity will be declined and eliminated since it no longer performs as an effective jammer. The profit is set to negative in virtue of the MUE’s rejection of cooperation, compelling SBS k to split from the coalition.
From the description of utility function we can clearly see that it is an individual performance measurement that cannot be transferred among the MUEs and SBSs, showing the NTU property of the game.
Note that θ(S) also means the entire capacity (for SBSs) and secrecy capacity (for MUEs) coalition S achieves.
Property 2
The proposed coalition formation game (Ω,U) is nonsuperadditive.
Therefore, this game does not meet the nature of superadditive.
Property 3
The grand coalition which consists of all the players will never form in the proposed game.
This property is obvious due to the nature of nonsuperadditive.
3.2 The algorithm and distributed implementation
Up to now, we have modeled the twoway selection problem between MUEs and SBSs as coalition formation game in partition form. This type of game is comprehensive to solve but can capture the intercoalition effects in many realistic situations [30, 35]. In this section, we develop a decentralized algorithm to endow the players with the capability of automatically finding their best collaborators in terms of the network environment, applying the mergeandsplit method [36].
where S _{1}, S _{2} are two different coalitions potentially joined by player i. Π _{1} and Π _{2} are the initial partition and the new partition after player i switching to S _{2}, respectively. Based on this preference order, we define the following switching rule for coalition formation:
Definition 3
Given a partition Π _{1}={S _{1},S _{2},…,S _{ L }} of the player set Ω, player i∈Ω decides to split from its current coalition S _{ l }, l∈1,2,…,L to join another coalition S _{ m }=Π _{1}∪{∅}, S _{ m }≠S _{ l } if and only if S _{ m }∪{i}≻_{ i } S _{ l }. After switching, Π _{1} is modified into a new partition Π _{2}=(Π _{1}∖{S _{ m },S _{ l }})∪(S _{ l }∖{i},S _{ m }∪{i}).
The switching rule establishes the principle for players in the coalition formation process. At any moment a player discovers a coalition that can strictly improve its income, it will leave the current coalition and participate in the new one. By repeating the switching operation, the players are able to ameliorate their performance step by step until the network becomes stable. From what has been presented above, we can observe that the players leave the lowpaying coalition and switch to a highpaying one, regardless of the effect on other members except the MUE. Hence, in the proposed game, the players adopt a relatively selfish strategy to maximize their own benefits rather than an altruistic one. Moreover, whenever a player i makes a move, it needs to update the historical record h(i), adding the coalition that it newly leaves to the history set.
In each loop, in order to find the top preferred coalition, every player has to examine all the remaining coalitions, which is quite heavy workload and timeconsuming for the player. From what can be seen in (20) and (21), an SBS will achieve no benefit joining a coalition composed of SBSonly and no two MUEs can coexist in the same coalition. As a result, there are only three possible types of coalitions in the final network structure: singleton coalitions with one MUE which cannot find a proper jammer; coalitions with one MUE and its jammers in which the MUE is chosen as the coalitionhead, taking the role of negotiating with SBSs in and out of its coalition; and singleton coalitions with one SBS that is unwilling to provide jamming service or being rejected by its possible client MUEs. In this context, an SBS in the network only has to make a decision either trying to be some MUE’s jammer or working alone, with no need to negotiate with other SBSs. As for MUEs, they can just negotiate with SBSs to see whether the collaboration is profitable for both sides. To further confine the possible negotiators, we use Proposition 1 in Section 2 to exclude the unqualified allies for both MUEs and SBSs. If SBS k is not able to improve MUE m’s secrecy capacity, there is no need for them to negotiate. In this case, SBS k has no necessity to consider the coalition that contains MUE m and vice versa. Thus, from an SBS’s perspective for determining the switching operation, it simply negotiates with the MUEs sequentially on its partner list to see the possibility of switching. Similar to SBSs, an MUE negotiates with the SBSs on its list. The difference is that there is still space to further refine its potential partners. From an MUE’s perspective, it only needs to consider about those SBSs whose coalition lies in the third class, i.e., singleton coalition with only an SBS. The reason for this is that an MUE has no benefit switching to a coalition that contains the MUE, ruling out the first two kinds of coalitions.
Simplified coalition formation algorithm for twoway friendly jammer selection problem
Initialization 
The coalitional structure is initialized as \(\Pi _{0}=\Omega =\mathbb {M}\cup \mathbb {K}\) and each player i’s history set h(i) is set empty. 
Phase 1 : Qualification Confirmation 
a Each SBS k measures the channel gain to the eavesdropper and MUEs, i.e., \(h_{k,e}^{n_{m}}\), \(h_{k,m}^{n_{m}}\), \(m\in \mathbb {M}\). 
b MBS broadcasts the collected channel information to MUEs. 
c Each player i calculates its potential partner list. For SBS k, \({\Psi _{k}} = \{ mh_{k,e}^{{n_{m}}} / h_{k,m}^{{n_{m}}} > 1,m \in \mathbb {M}\}\), For MUE m, \({\Psi _{m}} = \{ kh_{k,e}^{{n_{m}}} / h_{k,m}^{{n_{m}}} > 1,k \in \mathbb {K}\}\). 
Phase 2 : Coalition Formation 
Loop: 
Given the current partition Π _{ current } (in the beginning, Π _{ current }=Π _{0}), for every player i in coalition S∈Π _{ current } 
a Update the utility in S, U _{ current }=u _{ k }(Π _{ current },S) 
b For SBS, find the coalitions that contain the MUEs in Ψ _{ i }, S _{ Ψ }={S ^{′}S ^{′}∈Π _{ current }∖S,m∈Ψ _{ i },m∈S ^{′}} For MUE, find the SBSs who are still alone in Ψ _{ i }, S _{ Ψ }={S ^{′}k∈Ψ _{ i },k∈S ^{′},S ^{′}=1} 
c Investigate the possible switching operation among S ^{′}∪{∅}, according to the switching rule. Record the maximal utility U _{ max } and the corresponding coalition 
d If U _{ max }>U _{ current } 
Player i stores the current coalition S into history set h(i), and joins the new coalitionelse Player i stays in the current coalition and the coalition partition does not changeend 
Until: No player deviate from its coalition. 
Phase 3 : Friendly Jamming 
The SBS in nonsingleton coalition allocates α of the total power on its client MUE m’s subchannel n _{ m } to transmit the information as the jamming noise. MUE pays to its jammers according to (14) and (15). 
Note that although we assume a full buffer traffic, the proposed game still works under the burst traffic scenario for that the coalition formation process needs not to be changed. The only difference is that an MUE needs to send a control message at the beginning as well as the end of the communication to its jammers so that the jammers could know when to start and finish their jamming service.
3.3 Convergence and stability
Convergence is of great importance in the research of coalition formation algorithm. The convergence of the proposed algorithm can be guaranteed, as follows.
Proposition 2
From the initial state of the coalition partition, the convergence of the proposed coalition formation game is guaranteed.
Proof
With a certain number of players in the network, the number of partitions that can be formed is finite, given by the Bell number B _{Ω}. As we regulate a negative payoff for players going back to the history coalitions in h(i), each deviation of the players leads to a new partition of the network. Since the number of the possible partitions is bounded, our algorithm surely converges in a finite number of iterations. □
To study the stability of the algorithm, we use the concepts of individually stable and Nashstable from [32]:
Definition 4
A partition Π={S _{1},S _{2},…,S _{ L }} is individually stable if there do not exist a player i∈Ω,i∈S _{ l } and a coalition S _{ k }∈Π∪{∅} such that S _{ k }∪{i}≻_{ i } S _{ l } and S _{ k }∪{i}≽_{ j } S _{ k },∀j∈S _{ k } holds.
Definition 5
A partition Π={S _{1},S _{2},…,S _{ L }} is Nashstable if \(\forall i \in \Omega,i \in S_{l},{S_{l}}{\underline \succ _{i}}{S_{k}}\cup \{ i\} \) for all S _{ k }∈Π∪ {∅}.
The relationship between the two stability concept is that a Nashstable is individually stable [32]. From what can be seen, the Nashstable is stronger than the individually stable. The essence of the Nashstable is that in the final partition Π _{ f }, no player can find a coalition that can further improve its payoff; thus, the player has no incentive to deviate from its current coalition.
Proposition 3
The final partition Π _{ f } resulted from the proposed coalition formation game is Nashstable as well as individually stable.
Proposition 3 is obvious and a similar proof can be found in [29] and [31].
The communication and computational complexity for a player to detect the switching operation in each loop is O(K) (if it is an MUE) or O(M) (if it is an SBS) in the worst case where all players behave in a noncooperated way and any one of the SBSs has the qualification of staying on any MUE’s potential partner list. In fact, the actual complexity is much less in realistic situation.
3.4 Adapting to the dynamic feature of wireless network
One distinctive nature of the wireless network is the ever changing environment which is caused by the time varying channel states, the mobility of the users, the plugin feature of the SBSs, etc. Such changes observably modify the utilities of the players so that a reorganization of the network partition may be necessary in order to guarantee an optimal performance for all the SBSs and MUEs. Due to the abovementioned changes in the network, the proposed algorithm presented in Table 1 can be executed periodically as a response. According to our previous analyses, the algorithm is definite to converge within a finite number of steps because the possibility of switch operation is limited, no matter what the initial state is. Hence, the convergence and stability nature of the algorithm still hold, regardless of the variation of user’s location as well as the randomness of the wireless channel. Note that the period of executing the proposed algorithm can be chosen according to the frequency of the changes in the network, which is beyond the scope of this paper.
4 Numerical results and discussions
Primary simulation parameters
Radius of MBS, R _{ m }  400 m 
Number of MUEs, M  10 
Subchannels allocated to SBSs, N _{ m }  30 
Maximum power of MBS, P _{ m }  46 dBm 
Maximum power of SBS, P _{sbs}  23 dBm 
Power of white noise, σ ^{2}  −174 dBm/Hz 
Bandwidth of an subchannel, W  180 kHz 
Performance comparison between cooperative and noncooperative system (M = 10, K=80)
Parameters  Noncooperative  Cooperative  Gain (%) 

Mean number of MUEs that have positive SC ^{a}  3.88  9.76  151.55 
Mean SC of MUE(bps)  9.35 ×10^{4}  6.09 ×10^{5}  551.34 
Mean SC of MUEs in Ξ _{ m }(bps)  2.43 ×10^{5}  7.55 ×10^{5}  210.70 
Mean Capacity of SBS(bps)  2.01 ×10^{6}  2.35 ×10^{6}  16.92 
Figure 7 also shows the impact of the number of SBSs on the secrecy capacity of MUEs. Firstly, we can see that the MUEs have a poor secrecy performance in noncooperative condition, i.e., lower than 100 kbps. In addition, the original secrecy capacity is not influenced when the quantity of SBSs increases. The reason is that in noncooperative scenario, the two layers work on the orthogonal sets of frequencies that would not affect each other. Hence, the MUEs’ performance is independent of the small cells in the system and stays essentially constant as the number of the SBSs increases. However, the secrecy capacity in the proposed scheme is highly correlated to the number of SBSs in the system. We can observe an obvious enhancement of secrecy capacity along with the augment of small cells. The average secrecy capacity of MUEs improves by 35.17 % as the number of SBSs increases from 40 to 120. The reason is that the larger number of SBSs gives a wider choice of candidate jammers, making it easier for MUEs to select proper jammers. Additionally, according to Proposition 1, the SBSs locating around the eavesdropper have more impact on disturbing and can thus better protect the MUE’s privacy. Consequently, those SBSs become the competitive focus of the MUEs. When there are not many small cells, the quantity of such SBSs that can provide high quality of jamming service is relatively small. As the number of SBSs gradually rises, the density of the small cells grows, resulting in a larger amount of SBSs that lie in the vicinity of the eavesdropper. Besides the capable jammers, the ordinary jammers on the MUE’s potential partner list will also increase, providing a higher possibility to find the SBSs that are willing to cooperate.
5 Conclusions
In this paper, we consider the secrecy communication in ultradense networks and extend the study from a singleuser case to multiuser and multijammer scenario. In order to protect the MUEs from being overheard by the eavesdropper, we exploit the SBSs in the system to provide jamming for MUEs. The interaction between the MUEs and SBSs has been formulated as a coalition formation game with nontransferable utility in partition form. In addition, we propose an SCSL scheme to effectively encourage the SBSs to cooperate. We study the properties and convergence of the game and propose a novel algorithm to solve it in a distributed manner. The simulation result shows that more than half of the MUEs would be completely intercepted by the eavesdropper in a noncooperative condition. By adopting the proposed algorithm, almost all the MUEs can obtain a nonzero secrecy capacity and the secrecy capacity of the macrolayer is increased by four times. In addition, the capacity in smallcell layer also gains a 16.92 % improvement on average.
Declarations
Acknowledgement
This work is supported by National 863 Project (2014AA01A701) and National Nature Science Foundation of China (61372113,61421061).
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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