 Research
 Open Access
Nanoscale molecular communication networks: a gametheoretic perspective
 Chunxiao Jiang^{1}Email author,
 Yan Chen^{2} and
 K J Ray Liu^{2}
https://doi.org/10.1186/s1363401401884
© Jiang et al.; licensee Springer. 2015
 Received: 25 June 2014
 Accepted: 22 December 2014
 Published: 25 January 2015
Abstract
Currently, communication between nanomachines is an important topic for the development of novel devices. To implement a nanocommunication system, diffusionbased molecular communication is considered as a promising bioinspired approach. Various technical issues about molecular communications, including channel capacity, noise and interference, and modulation and coding, have been studied in the literature, while the resource allocation problem among multiple nanomachines has not been well investigated, which is a very important issue since all the nanomachines share the same propagation medium. Considering the limited computation capability of nanomachines and the expensive information exchange cost among them, in this paper, we propose a gametheoretic framework for distributed resource allocation in nanoscale molecular communication systems. We first analyze the intersymbol and interuser interference, as well as bit error rate performance, in the molecular communication system. Based on the interference analysis, we formulate the resource allocation problem as a noncooperative molecule emission control game, where the Nash equilibrium is found and proved to be unique. In order to improve the system efficiency while guaranteeing fairness, we further model the resource allocation problem using a cooperative game based on the Nash bargaining solution, which is proved to be proportionally fair. Simulation results show that the Nash bargaining solution can effectively ensure fairness among multiple nanomachines while achieving comparable social welfare performance with the centralized scheme.
Keywords
 Nanocommunication
 Molecular communication
 Game theory
 Noncooperative game
 Cooperative game
 Nash bargaining
1 Introduction
Nanotechnology, a manipulation of matter on an atomic and molecular scale, makes the design and fabrication of nanoscale components become a reality. Such nanoscale components can be used to assemble basic structural and functional devices, called nanomachines, which are able to perform basic and simple tasks at the nanolevel, such as computing, data storing, sensing, and actuation [1]. Recently, this promising technology catalyzes a new communication paradigm  ‘nanocommunications,’ arousing the researchers’ great interests in both industrial and academic fields. Nanocommunications refer to the information exchange between nanomachines, which can be realized through nanomechanical, acoustic, electromagnetic, and chemical or molecular communication means. It is expected that nanocommunication networks can be applied in many different scenarios including human health monitoring, food and water quality control, air pollution control, as well as aggressive chemical agent detection [2].
Since molecule migration naturally occurs within both living organisms and abiotic components, molecular communication is considered as the most promising approach for nanocommunication networks [3,4], where the transmission and reception of information are realized through molecules. In the molecular communication model, the nanotransmitter releases molecules, which are modulated and coded to carry information, into the fluid medium. The molecules propagate to the receiver through the medium, which are demodulated and decoded to restore the information that the transmitter intends to convey. There are mainly three molecule prorogation models: walkwaybased model, flowbased model, and diffusionbased model [5]. In the walkwaybased model, the molecules propagate through physical pathways connecting the transmitter to the receiver, such as molecular motors [6]. In the flowbased model, the propagation of the released molecules is controlled by the predefined flow and turbulence in the medium, such as pheromonal communication [2]. In the diffusionbased model, the molecules propagate through their spontaneous diffusion in the fluid medium, such as calcium signaling among cells [7]. In this paper, we focus on the diffusionbased model since it represents the most general and widespread molecular communication architecture found in nature.
In the literature, various topics about the diffusionbased molecular communications have been studied, including channel model [813], modulation and coding [1418], and receiver design [19,20]. The earlier works regarding diffusionbased molecular communications were mainly focused on channel capacity analysis [813], where a commonly accepted channel model is based on Brownian motion [21]. In [8], Eckford analyzed the achievable bound on information rate for the diffusionbased channel with onedimensional Brownian motion. As an extension, Brownian motion of molecules in a fluid medium with drift velocity instead of static environment was analyzed in [13]. In addition to the channel analysis, a comprehensive physical endtoend model, including molecule emission, diffusion, and reception, was proposed in [5]. Meanwhile, simulationbased approaches for exploring the diffusive molecules were conducted in [22] and [23]. As for channel noise and interference analysis, Pierobon and Akyildiz studied molecule sampling and counting noise in [16], as well as the intersymbol and cochannel interference in [17]; Kadloor et al. presented an additive inverse Gaussian noise channel model in [14]. To enhance system performance, various coding schemes were also introduced, including a forward error correction coding scheme in [15] and a ratedelay tradeoff network coding scheme in [18]. In terms of receiver design, an optimal receiver design based on weighted sum detectors was proposed in [19], and a ligandbinding reception model was studied in [20]. Moreover, the consensus problem and relaying role under diffusionbased molecular communication were recently studied in [24] and [25], respectively.
However, the resource allocation issue in molecular communication networks has not yet been investigated. In traditional electromagnetic wave wireless communications, when multiple users share the same resource, e.g., power and spectrum, how to allocate the resource among different users is very important to guarantee the high system efficiency while maintaining fairness [26,27]. Similarly, in the diffusionbased molecular communications, there can be multiple transmitters sharing the same fluid medium. In such a case, interference will occur since the molecules from different transmitters are identical and indistinguishable [13]. Therefore, how to perform resource allocation among multiple transmitters, molecule emission control, is an essential problem in practical molecular communication systems. Generally, the computational capability of nanomachines is low, and the information exchange cost among them is expensive. Thus, distributed resource allocation algorithms are strongly favorable in molecular communications. Moreover, different nanomachines may be fabricated by different manufacturers and may have different objectives, e.g., in human body monitoring nanosensor networks [3], different biomedical sensors may have different functionalities and may be deployed by different doctors. Considering these problems, in this paper, we propose a gametheoretic framework for distributed resource allocation in diffusionbased molecular communications. The gametheoretical model provides distributed molecule emission control solutions to achieve high system efficiency, while guaranteing fairness among different nanomachines. Game theory has been corroborated as an effective tool for modeling different problems in traditional distributed wireless communication networks [28]. To the best of our knowledge, this is the first work that uses game theory to study the interactions among nanomachines in molecular communication networks, which is expected to exemplify the application of game theory in the nanocommunication and networking field.
 1.
We analyze the intersymbol and interuser interference in the molecular communication system with amplitude modulation. Based on the interference analysis, we find the optimal decision threshold using the maximumaposterior (MAP) detection method and derive the corresponding bit error rate (BER) performances of different transmitters.
 2.
We propose a gametheoretic framework to model the resource allocation problem in nanoscale molecular communication systems. Specifically, we focus on the molecule emission control issue by using a twotransmitter and onereceiver system as an example. In the proposed gametheoretic framework, the players are the transmitters whose objectives are to maximize their own utilities, the strategy of each player is the number of emitted molecules, and the utility function is related with the BER performance.
 3.
Based on the gametheoretic framework, we formulate the molecule emission control problem as a noncooperative emission control game, where the Nash equilibrium (NE) is derived and proved theoretically as the unique NE. In order to improve the system efficiency while guaranteeing fairness, we further model the problem using a cooperative game based on the Nash bargaining solution, where we prove that the Nash bargaining solution of the proposed emission control game is a proportionally fair solution.
The rest of this paper is organized as follows. We first introduce the system model of molecular communication in Section 2. Then, we analyze the intersymbol and interuser interference and BER performance in Section 2. The noncooperative and cooperative molecule emission control games with simulation results are discussed in Section 2. Finally, conclusions are drawn in Section 2.
2 System model
2.1 Network entity

Transmitter: Each transmitter can independently generate and emit molecules into the propagation medium. It is responsible for the modulation of a series of input symbols s(t)∈{0,1} by controlling the speed, number, or time of the emitted molecules. We assume that the transmitters can perfectly control the release time and number of the molecules, while having no control on the motion of the molecules once the molecules escape from the transmitter [5,9].

Molecule: A molecule is an indivisible object that can be released by the transmitter and absorbed by the receiver by means of chemical reactions. The molecules carry information of the transmitters and randomly diffuse in the propagation medium. They are considered as identical and undistinguishable between each other. Moreover, the interactions among molecules are not taken into account in general [14]. Therefore, the trajectories of all molecules in the medium are independent of each other.

Propagation medium: The propagation medium is made of some kind of fluid, where the molecules can freely diffuse inside. Relative to a single molecule, the space of the medium is considered as infinite in any dimension. The Brownian motion is a generally accepted model for the propagation of molecules in the medium, which can be characterized by two parameters: drift velocity and diffusion constant determined by the physical properties of the fluid medium [29]. In this paper, we only consider the diffusion effect of molecules. Note that [13] considered both diffusion and drift effects, and all our analyses and results can be easily extended to that case.

Receiver: When the molecules arrive at the receiver, they are absorbed by the receiver and disappear from the medium. It is assumed that the receiver can perfectly measure the time when it absorbs a single molecule and use such information to determine the information sent by the transmitter. Moreover, the synchronization between the transmitter and receiver is also assumed to be perfect [3032]. Note that the focus of this paper is the interference and resource allocation in molecular communication networks, while the synchronization issue is out of the scope. In molecular communication networks, the receiver is usually more powerful than the transmitter, having a larger size, which is usually designed to be able to communicate with microscale machines using electromagnetic waves as well.
2.2 Modulation
where s(t) is a series of ‘01’ symbols. Note that n can be regarded as the molecule emission power.
2.3 Channel model
3 Interference analysis
3.1 Intersymbol interference
3.2 Interuser interference
The IUI defined in traditional wireless communication systems represents the interference power from one transmitter to the other. Similarly, in molecular communications, when there are two or more transmitters releasing molecules into the medium, they would interfere each other at the receiver side, as shown in the right part of Figure 4. Let us consider the twotransmitter case as shown in Figure 1, where the receiver is located d _{1} away from transmitter (TX) I and d _{2} away from TX II. The emission power of TX I is n _{1} and that of TX II is n _{2}, i.e., releasing n _{1} or n _{2} molecules to represent bit ‘1’. Similar to the aforementioned ISI analysis, the IUI for TX I is caused by the molecules which leaked from TX II, and vice versa. For the symbol of TX II transmitted at time slot m, the molecules leaked to it should be considered from both the previous symbol at time slot m−1 and the current symbol of TX I. There are four cases: ‘00’, ‘01’, ‘10’, and ‘11’, where ‘10’, for example, means TX I transmitted ‘1’ in the previous time slot and transmits ‘0’ in the current time slot.
where P _{2c }=P _{ a }(d _{2},t _{ s }) and P _{2p }=P _{ a }(d _{2},2t _{ s }) can be calculated by (3). Note that multiplexing techniques can be adopted to alleviate the IUI, which is not the focus of this paper. The techniques from traditional wireless communications may be used to improve the multiaccess performance. For examples, the timedivision multiplexing techniques can be easily applied in molecular communications, where the molecular transmissions are scheduled by time slots and by utilizing the Neural Delay Boxes (NDBs) connecting different molecular transmitters to the shared medium [35]. Moreover, the space multiplexing techniques can also be applied in molecular multiple access, e.g., MIMO communication based on molecular diffusion [36]. The IUI analysis in this paper can be easily extended to the multitransmitter scenario. When there are K transmitters, one transmitter may be interfered by other K−1 transmitters. Since each transmitter is with four cases: ‘00’, ‘01’, ‘10’, and ‘11’, there are totally 4(K−1) cases. Since the interference is additive, there are totally 4(K−1) summations in (11). Therefore, the complexity would be linear with the number of transmitters, i.e., \(\mathcal {O}(K)\).
3.3 Bit error rate performance
BER is defined as the number of bit errors divided by the total number of transmitted bits during an evaluated time interval, which is usually approximated by the bit error probability, i.e., the probability that bit ‘0’ or ‘1’ is wrongly decoded by the receiver. In the amplitude modulationbased system, the receiver compares the number of molecules absorbed in one time slot with some predefined threshold τ to decode the bit. In such a case, the BER performance is heavily related with the threshold τ. Therefore, we need to first derive the optimal τ that minimizes the BER of each transmitter.
4 Molecule emission control game
In this section, we discuss how to control the molecule emission for both transmitters in Figure 1, i.e., how to control n _{1} for TX I and how to control n _{2} for TX II. As discussed in the introduction, the centralized method can globally optimize all the transmitters’ molecule emission by minimizing the total BER P eI+P eII derived in (26) and (27). However, the centralized method needs a nanoscale server/coordinator to collect each transmitter’s location information, and each coordinator responds with the optimal emission control scheme, which inevitably incurs huge communication and energy cost for the nanoscale molecular machines. Moreover, minimizing the total BER P eI+P eII may lead to unfairness phenomenon, e.g., one transmitter is with extremely low BER and the other is with extremely high BER. In the following, we first discuss a noncooperative game formulation of the emission control problem and then consider a cooperative game formulation based on Nash bargaining to improve the system efficiency and ensure the fairness between TX I and TX II. The Nash bargaining solution shows its merits of low computational cost, low information exchange cost, and proportional fairness.
4.1 Noncooperative game formulation

Player: The players of \(\mathcal {G}_{c}\) are TX I and TX II in the system, i.e.,$$ \mathcal{P}_{c}=\{\text{TX I, TX II}\}. $$(28)

Strategy: The strategy for each transmitter is defined as its molecules emission power: n _{1} for TX I and n _{2} for TX II, i.e., how many molecules are released to represent bit ‘1’. We can see that the strategy space of each transmitter is finite and discrete. Suppose the molecule emission constraints for TX I and TX II are \(N_{\text {max}}^{\text {I}}\) and \(N_{\text {max}}^{\text {II}}\). Thus, the strategy spaces of TX I and TX II, denoted by \(\mathcal {S}_{1}\) and \(\mathcal {S}_{2}\), respectively, are as follows:$$ \mathbf S_{c}=\{\mathcal{S}_{1},\mathcal{S}_{2}\},\quad \text{where} \left\{\begin{array}{l} \mathcal{S}_{1}=\left\{0, 1, \dots, N_{\text{max}}^{\text{I}}\right\},\vspace{2mm}\\ \mathcal{S}_{2}=\left\{0, 1, \dots, N_{\text{max}}^{\text{II}}\right\}. \end{array}\right. $$(29)
Moreover, the pure strategy profile of game \(\mathcal {G}_{c}\), which is defined as a set of strategies including each player’s specified action, can be written as (n _{1},n _{2}) [38].

Utility function: The utility function should be defined proportional to the player’s benefit. Since each transmitter expects a lower BER, the utility should be defined inversely proportional to the BER performance. Considering that the worst case of BER is \(\frac {1}{2}\) (when each transmitter adopts strategy n _{1}=n _{2}=0 and the receiver cannot distinguish between bit ‘1’ and ‘0’), we can define the utility function as the difference between \(\frac {1}{2}\) and BER to ensure a positive utility. Let U _{1} and U _{2} denote the utility functions of TX I and TX II, respectively, which can be written as follows:$$ \begin{aligned} &\mathbf U_{c}=\{ U_{1},U_{2}\},\\ & \text{where} \left\{\begin{array}{l} U_{1}=\frac{1}{2}P^{\text{I}}_{e}=\frac{1}{2}\bigg(Q\left(\frac{\tau_{1}\mu_{1,\text{I}}}{\sigma_{1,\text{I}}}\right)Q\left(\frac{\tau_{1}\mu_{0,\text{I}}}{\sigma_{0,\text{I}}}\right)\bigg),\vspace{3mm}\\ U_{2}=\frac{1}{2}P^{\text{II}}_{e}=\frac{1}{2}\bigg(Q\left(\frac{\tau_{2}\mu_{1,\text{II}}}{\sigma_{1,\text{II}}}\right)Q\left(\frac{\tau_{2}\mu_{0,\text{II}}}{\sigma_{0,\text{II}}}\right)\bigg). \end{array}\right. \end{aligned} $$(30)
According to the utility function above, we can see that U _{1} is an increasing function in terms of n _{1} and a decreasing function in terms of n _{2}. On the other hand, U _{2} is a decreasing function in terms of n _{1} but an increasing function in terms of n _{2}.
Based on the game model of \(\mathcal {G}_{c}=\langle \mathcal {P}_{c}, \mathbf S_{c},\mathbf U_{c}\rangle \) above, we can further find its NE. Since the utility functions of \(\mathcal {G}_{c}\) are monotonic in terms of both transmitters’ strategies, the NE can be guaranteed to exist and is unique. Apparently, the NE is that each transmitter uses the highest molecule emission power, which would lead to low system efficiency compared with the centralized scheme. Therefore, to improve the system efficiency, we formulate the emission control problem as a cooperative game using the Nash bargaining solution in next subsection.
4.2 Cooperative game formulation based on Nash bargaining
In game theory, a cooperative game studies how a group of players should cooperate with each other when noncooperation leads to an unfavorable outcome for each player, e.g., in the noncooperative molecule emission control game discussed above, each transmitter adopts the highest molecule emission power and heavily interferes with each other. Under such circumstances, the players have the incentive to cooperatively enhance the system efficiency in conjunction with fairness. The bargaining game is one branch of the cooperative game, where two players try to reach an agreement on trading/sharing a limited amount of resources. These two individual players have a choice to bargain with each other so that both of them can gain benefit higher than that without cooperation. In a bargaining game, since there might be an infinite number of social optimal agreement points (i.e., bargaining solutions), three kinds of bargaining solutions  egalitarian solution, KalaiSmorodinsky solution, and Nash bargaining solution [38]  were proposed to refine the multiple bargaining solutions, among which the Nash bargaining solution is mostly widely adopted with the emphasis of fairness and social optimality [39]. In the following, we formulate the emission control game between two transmitters as a bargaining game and find the Nash bargaining solution for them.
The emission control bargaining game for molecular communications can be described as follows. Let \(\mathcal {P}_{c}=\{\text {TX I, TX II}\}\) denote the set of two players, i.e., two nanoscale transmitters. Let \(\mathbf W=\{(U_{1},U_{2})\} \subset \mathbb R^{2}\) represent the set of feasible payoff allocations that each transmitter can get if they can reach an agreement to cooperate. Let \(\mathbf U_{\text {min}}=\left (U^{\text {I}}_{\text {min}},U^{\text {II}}_{\text {min}}\right)\), where \(U^{\text {I}}_{\text {min}}\) and \(U^{\text {II}}_{\text {min}}\) represent the minimal payoffs that TX I and TX II can be satisfied if both of them cooperate with each other. One way is to set the \(U^{\text {I}}_{\text {min}}\) and \(U^{\text {II}}_{\text {min}}\) to be the NE of the noncooperative molecule emission control game analyzed in the previous subsection. This means that both of the transmitters agree to cooperate only if their respective utility cannot be less than that of the noncooperative scenario. However, in such a case, fairness cannot be guaranteed since the TX, which can obtain a much higher utility in the noncooperative scenario, can still obtain a much higher utility than the other in this cooperative scenario, which may lead to the failure of cooperation. Therefore, as in most cases, to ensure fairness among transmitters, we set UminI=UminII=0, which means that as long as the utility is larger than 0, both transmitters would be cooperative. In such a case, all the possible solutions in W satisfy the individual rationality, i.e., \(\left \{(U_{1},U_{2})U_{1}\ge U^{\text {I}}_{\text {min}},U_{2}\ge U^{\text {II}}_{\text {min}}\right \}\subset \mathbf W\) is the set of all the possible bargaining solutions. This is also practical since the NE of the noncooperative game usually corresponds to the scenario that each transmitter has little utility, i.e., \(U^{\text {I}}_{\text {min}}\rightarrow 0,\ U^{\text {II}}_{\text {min}}\rightarrow 0\). Thus, the pair (W,U _{min}) can be defined as the emission control bargaining game. We can see that there are numerous possible operating points for TX I and TX II in W. The Nash bargaining solution (NBS), defined as follows, provides a unique, fair and efficient operating point in W with the idea that after the minimal requirements are satisfied for both transmitters, the rest of the resources are allocated proportionally to each of them according to their conditions.
 1.
Pareto optimality: There does not exist a point \(\mathbf U^{\prime }=\left (U^{\prime }_{1},U^{\prime }_{2}\right)\) other than U ^{∗} in W, satisfying that \(U^{\prime }_{1}\ge U^{\ast }_{1}\) and \(U^{\prime }_{2}\ge U^{\ast }_{2}\).
 2.
Symmetry: If the feasible set W is symmetric in terms of two transmitters, for example, when TX I and TX II are homogenous, then U ^{∗} is also symmetric, i.e., \(U^{\ast }_{1}=U^{\ast }_{2}\).
 3.
Independence of irrelevance alternatives: If U ^{∗}∈W ^{′}⊂W is an NBS for (W,U _{min}), then it is also an NBS for (W ^{′},U _{min}).
 4.
Independence of linear transformations: For any linear transformation ψ, if U ^{∗} is an NBS for (W,U _{min}), then ψ(U ^{∗}) is an NBS for (ψ(W),ψ(U _{min})).
where T is the updating period, λ is an integer step size which can be set as 1, and [U(n _{1}(t)+1, n _{2}(t))−U(n _{1}(t), n _{2}(t))] and [U(n _{1}(t),n _{2}(t)+1)−U(n _{1}(t),n _{2}(t))] are the approximated partial derivatives of U(n _{1},n _{2}) over n _{1} and n _{2}, respectively. Note that in the implementation, U(n _{1},n _{2}) is not calculated using (30), which, instead, is evaluated by the transmitters. Therefore, they need to exchange their utility U _{1}(t) and U _{2}(t) with each other to find the NBS. In Algorithm 1, we summarize the distributed NBS algorithm. Note that the structure of the algorithm can also be applied into multiple TX and RX scenarios, where an extra time slot is required for multiple transmitters exchanging their utility information, i.e., the transmitters sequentially broadcast their utility in step 3 of Algorithm 1.
From the definition of NBS, the Pareto optimality quantitatively shows the high efficiency of NBS, while fairness is only illustrated in a qualitative manner. To further quantitatively verify the fairness property of NBS, we introduce the concept of ‘proportional fairness’, which is a widely used metric to evaluate fairness performance in wired networks [40]. Proportional fairness is a compromise between fairness and performance and based upon maintaining a balance between two competing interests, i.e., trying to maximize the social welfare while at the same time satisfying each user’s minimal level of requirement. According to the general definition of proportional fairness [40], the specific definition of proportional fairness in the emission control bargaining game for molecular communications can be described as follows.
From the definition, we can see that the physical meaning of proportional fairness is that at this point, if one transmitter intends to improve its performance with some increasing percentage, then the other transmitter would suffer from some degrading performance with some decreasing percentage, and the increasing percentage is no larger than the decreasing percentage. Therefore, the Pareto optimality is a special case of proportional fairness. In the following theorem, we will show that the NBS of the emission control bargaining game derived by (31) is a proportionally fair solution.
The first inequality means for all feasible point \(\mathbf U^{\prime }=\left (U^{\prime }_{1},U^{\prime }_{2}\right)\in \mathbf W\) that is different from NBS point U ^{∗}, the overall change of two transmitters’ utilities is negative according to the gradient.We can see that the second inequality is same as the definition of proportional fairness in (35). Therefore, we can conclude that the NBS of the emission control bargaining game is equivalent to the proportional fairness.
4.3 Simulation results

Centralized scheme solution (CSS): The centralized scheme is to maximize the social welfare of the two transmitters, i.e.,$$ \max\limits_{n_{1}\le N_{\text{max}}^{\text{I}},n_{2}\le N_{\text{max}}^{\text{II}}} U_{1}(n_{1},n_{2})+U_{2}(n_{1},n_{2}). $$(39)

Nash equilibrium solution (NES): Nash equilibrium means the solution of the noncooperative emission control game, i.e., \(n_{1}= N_{\text {max}}^{\text {I}},\ n_{2}= N_{\text {max}}^{\text {II}}\).

Nash bargaining solution (NBS): Nash bargaining means the solution of the cooperative emission control game, i.e.,$$ \max\limits_{n_{1}\le N_{\text{max}}^{\text{I}},n_{2}\le N_{\text{max}}^{\text{II}}} U_{1}(n_{1},n_{2})\cdot U_{2}(n_{1},n_{2}). $$(40)
5 Conclusions
In this paper, we provided a gametheoretic framework for nanoscale molecular communication systems. We analyzed the intersymbol and interuser interference, optimal decision threshold, and BER performance in the molecular communication system with amplitude modulation. Based on the interference analysis, we formulated the molecule emission control problem as a noncooperative emission control game, where the Nash equilibrium was found and proved to be unique. In order to improve the system efficiency while guaranteeing fairness, we further modeled the proposed molecule emission control problem using a cooperative game based on Nash bargaining solution, which was proved to be proportionally fair. Finally, a simulation was conducted to compare the performance among the centralized scheme, Nash equilibrium scheme, and Nash bargaining scheme. The results verified that the NBS can achieve a tradeoff between social welfare performance and user fairness.
Declarations
Acknowledgements
This work was funded by projects 61371079, 61273214, 61271267, and 91338203 supported by NSFC China, and the Postdoctoral Science Foundation funded project. The work was done during Chunxiao Jiang’s visit to the University of Maryland.
Authors’ Affiliations
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