Joint Coverage and Power Control in Highly Dynamic and Massive UAV Networks: An Aggregative Game-theoretic Learning Approach

TL;DR

This paper introduces an aggregative game-theoretic learning approach with a novel algorithm for UAV networks, improving efficiency and speed in highly dynamic, large-scale post-disaster scenarios.

Contribution

It develops a new aggregative game model and proposes the SPBLLA algorithm, enhancing learning speed and reducing information exchange in UAV network management.

Findings

SPBLLA outperforms revised BLLA in learning speed.

The model effectively handles large-scale, dynamic UAV networks.

The approach reduces communication overhead and improves deployment efficiency.

Abstract

Unmanned aerial vehicles (UAV) ad-hoc network is a significant contingency plan for communication after a natural disaster, such as typhoon and earthquake. To achieve efficient and rapid networks deployment, we employ noncooperative game theory and amended binary log-linear algorithm (BLLA) seeking for the Nash equilibrium which achieves the optimal network performance. We not only take channel overlap and power control into account but also consider coverage and the complexity of interference. However, extensive UAV game theoretical models show limitations in post-disaster scenarios which require large-scale UAV network deployments. Besides, the highly dynamic post-disaster scenarios cause strategies updating constraint and strategy-deciding error on UAV ad-hoc networks. To handle these problems, we employ aggregative game which could capture and cover those characteristics. Moreover, we propose a novel synchronous payoff-based binary log-linear learning algorithm (SPBLLA) to lessen information exchange and reduce time consumption. Ultimately, the experiments indicate that, under the same strategy-deciding error rate, SPBLLA's learning rate is manifestly faster than that of the revised BLLA. Hence, the new model and algorithm are more suitable and promising for large-scale highly dynamic scenarios.

Figures

Equations

$${\sigma }_{ip}(s_{-i})=(\sigma -P_i)\otimes C_i,$$

$${\tilde{D}}_i=\pi (h_i\tan \theta {)}^{2\cdot \beta },$$

$${\bar{D}}_i=D_i-\kappa \sum _{j\ne i}{D}_j,$$

$$U_D=\frac{{\tilde{D}}_i}{D}.$$

$$U_E=\frac{E}{\sum _{1\le n\le N}{P}_{in}}.$$

$$SNR=\frac{P_{in}}{{\sum }_{j\ne i}{P}_{jn}+N{s}_n}.$$

$$C=B{\log }_2(1+SNR),$$

$$U_{SNR}=\mu \sum _{i\in M}[P_i-\gamma {\sigma }_{ip}(s_{-i})]\otimes C_i-\alpha {\bar{D}}_i,$$

$$\begin{array}{rl}& U_i(s_i,{\sigma }_{ip}(s_{-i}))=A{U}_E+B{U}_{SNR}+C{U}_D\\ & =\frac{AE}{\sum _{1\le n\le N}{P}_{in}}+B\{\mu \sum [P_i-\gamma {\sigma }_{ip}(s_{-i})]\otimes C_i-\alpha {\bar{D}}_i\}+\frac{C{\tilde{D}}_i}{D},\end{array}$$

$$U=\sum _{i\in M}{U}_i(s_i,{\sigma }_{ip}(s_{-i})),$$

$$U_i(s)=u_i(s_i,{\sigma }_{i1}(s_{-i}),{\sigma }_{i2}(s_{-i}),...,{\sigma }_{iK}(s_{-i})),$$

$$g_k(s)=g({\sigma }_{ik}(s_i,s_{-i})).$$

$${\sigma }_{i2}(s_{-i})={\sigma }_{ia}(s_{-i})=\sum _{j\ne i}{D}_j.$$

$$g_1(s{)}_{(n)}=({\sigma }_{ip}(s_{-i})+P_i{)}_{(n)},$$

$$g_2(s)={\sigma }_{ia}(s_{-i})+D_i,$$

$$U_i(s_i,s_{-i}^{\ast })\le U_i(s_i^{\ast },s_{-i}^{\ast }),i\in M,s_i\ne s_i^{\ast }.$$

$$U_i(s_i^{\prime },s_{-i})-U_i(s_i,s_{-i})=\varphi (s_i^{\prime },s_{-i})-\varphi (s_i,s_{-i}),$$

$$s_i^2\in C_i(s_i^1)\iff s_i^1\in C_i(s_i^2).$$

$$0<\underset{ϵ\to 0+}{\lim }\frac{P_{ϵ}(s^n,s^m)}{{ϵ}^{R(s^n\to s^m)}}<\infty .$$

$$\underset{ϵ\to 0+}{\lim }{\mu }_{ϵ}(s)>0.$$

$${\mu }_{ϵ}(s)=\frac{{ϵ}^{-\varphi (s)}}{{\sum }_{\tilde{s}\in S}{ϵ}^{-\varphi (\tilde{s})}}.$$

$$R(L)=\sum _{k=1}^{m}R(s^{k-1}\to s^k).$$

$$R(T)=\sum _{s^{{}^{\prime }}\to s^{{}^{\prime \prime }}\in T}R(s^{{}^{\prime }}\to s^{{}^{\prime \prime }}).$$

$$\gamma (s)=\underset{T\in T(s)}{\min }R(T).$$

$$R(T_{min})=\underset{s\in S}{\min }\gamma (s).$$

$$m>2\Delta ,$$

$$\begin{array}{rl}\Delta =& AE\frac{\Delta P}{\sum C_i\cdot P_1\cdot (P_1+\Delta P)}+B\sum C_i\cdot \Delta P\\ & +B\gamma \Delta P\cdot \sum (\sum _{j\ne i}{C}_j\otimes C_i)\\ & +B\alpha \pi ta{n}^2\theta (2h_{nh}\Delta h-\Delta h^2)\\ & +B\alpha \kappa (M-1)\pi ta{n}^2\theta (2h_{nh}\Delta h-\Delta h^2)\\ & +\frac{C}{D}\pi ta{n}^2\theta [h_{nh}^{2\beta }-(h_{nh}-\Delta h{)}^{2\beta }],\end{array}$$

$$\varphi (s)=\sum _{i\in M}\{A\frac{E}{\sum _{1\le n\le N}{P}_{in}}+B[\sum _{1\le n\le N}{P}_{in}-\alpha {\bar{D}}_i]+C\frac{{\tilde{D}}_i}{D}\}.$$

$$\begin{split}&U_{i}(s^{\prime}_{i},s_{-i})-U_{i}(s_{i},s_{-i})\\ &=\Bigg{(}A\frac{E}{\sum\limits_{1\leq n\leq N}P^{\prime}_{in}}+B\{\sum[P^{\prime}_{i}-\gamma\sigma_{ip}(s_{-i})]\otimes C_{i}-\alpha\bar{D}^{\prime}_{i}\}+C\frac{\tilde{D}^{\prime}_{i}}{D}\Bigg{)}\\ &\quad-\Bigg{(}A\frac{E}{\sum\limits_{1\leq n\leq N}P_{in}}+B\{\sum[P_{i}-\gamma\sigma_{ip}(s_{-i})]\otimes C_{i}-\alpha\bar{D}_{i}\}+C\frac{\tilde{D}_{i}}{D}\Bigg{)}\\ &=A(\frac{E}{\sum\limits_{1\leq n\leq N}P^{\prime}_{in}}-\frac{E}{\sum\limits_{1\leq n\leq N}P_{in}})+B(\sum(P^{\prime}_{i}-P_{i})\otimes C_{i})\\ &\quad+B\alpha[(D^{\prime}_{i}-\kappa\sum_{j\neq i}D_{j})-(D_{i}-\kappa\sum_{j\neq i}D_{j})]+C(\frac{\tilde{D}^{\prime}_{i}}{D}-\frac{\tilde{D}_{i}}{D})\\ &=A(\frac{E}{\sum\limits_{1\leq n\leq N}P^{\prime}_{in}}-\frac{E}{\sum\limits_{1\leq n\leq N}P_{in}})+B\sum\limits_{1\leq n\leq N}(P^{\prime}_{in}-P_{in})\\ &\quad+B\alpha(D^{\prime}_{i}-D_{i})+C(\frac{\tilde{D}^{\prime}_{i}}{D}-\frac{\tilde{D}_{i}}{D}).\end{split}$$

$$\begin{array}{rl}& \varphi (s_i^{\prime },s_{-i})-\varphi (s_i,s_{-i})\\ & =\sum _{i\in M}\{A\frac{E}{\sum _{1\le n\le N}{P}_{in}^{\prime }}+B[\sum _{1\le n\le N}{P}_{in}^{\prime }-\alpha {\bar{D}}_i^{\prime }]+C\frac{{\tilde{D}}_i^{\prime }}{D}\}\\ & -\sum _{i\in M}\{A\frac{E}{\sum _{1\le n\le N}{P}_{in}}+B[\sum _{1\le n\le N}{P}_{in}-\alpha {\bar{D}}_i]+C\frac{{\tilde{D}}_i}{D}\}\\ & =A(\frac{E}{\sum _{1\le n\le N}{P}_{in}^{\prime }}-\frac{E}{\sum _{1\le n\le N}{P}_{in}})+B\sum _{1\le n\le N}(P_{in}^{\prime }-P_{in})\\ & +B\alpha (D_i^{\prime }-D_i)+C(\frac{{\tilde{D}}_i^{\prime }}{D}-\frac{{\tilde{D}}_i}{D}).\end{array}$$

Full text

I Introduction

I-A Background and Motivation

Catastrophic natural and man-made disasters, such as earthquakes, typhoons, and wars, usually involve great loss of life and/or properties, historical interests in vast areas. Though sometimes unavoidable, the loss of life and property can be effectively reduced if proper post-disaster management approach has been taken. Because telecommunication infrastructure can be vital for search and rescue missions after disasters [1]. When these infrastructures are damaged by natural disasters, rapid network deployment to provide wireless coverage for ground users is quite essential for disaster management [2]. Under such circumstances, it is highly demanded to deploy post-disaster emergency networks (PENs) to restore wireless communication connectivity in accident and/or incident areas. Essentially, considering that repairing communication infrastructures takes a long time, building vehicle relay networks was a preferable solution during the critical first 72 hours [3]. In such relay networks, vehicles are distributed in the post-disaster areas and act as communication infrastructures, supporting mission-critical and emergency communications. However, vehicles are likely barricaded by damaged roads, torrential rivers, and precipices, etc, in the vast areas. Faced with those challenges, unmanned aerial vehicles (UAV) ad-hoc networks become a good alternative for emergency response due to its numerous advantages such as quick deployment and resilience in large-scale harsh networking conditions [4][5].

As we know, a number of techniques have been developed for extensive coverage, high capacity, and high energy efficiency in fifth generation (5G)/beyond-5G wireless networks. Accrodingly, energy consumption, signal to noise ratio (SNR) and coverage size would be the key factors to determine the performance of a UAV-enabled PEN [6]. In this paper, we consider joint power and coverage control for the highly dynamic and massive UAV networks, in which the power control decides the energy consumption and SNR of UAV, while the coverage decides the number of covering users that a UAV can support and the minimum value of SNR. Since the higher altitude a UAV is, the wider range it can cover, and the higher SNR it requires to support more users. Power and coverage controls are two essential factors with extensive researches that focus on their impacts on various networks. For example, the work in [9] constructed a coverage model which considered each agent’s coverage size on a network graph. However, such model considered only one characteristic of networks but ignored systems’ multiplicity, which would bring great loss in practice. Since a lot of energy will be consumed to improve UAV’s SNR or coverage size. The work in [7] made a progress by studying multi-factors of coverage and charging strategies, but the authors overlooked UAV’s power control which might waste lots of energy. To sum up, power and coverage control which determine energy consumption, SNR, and coverage size, are pivotal factors in improving network performance and ought to be considered jointly [10].

In post-disaster scenarios, due to environmental factors such as violent winds [11] and tempestuous rainstorms, the strategy set of UAVs is constrained. In other words, UAV’s action set has a restriction that cannot switch rapidly between extreme high power or elevate altitude to low ones, but only levels adjacent to them [12]. For instance, the power can change from $1$mW to $1.5$mW in the first time slot and from $1.5$mW to $2$mW in the next one, but it cannot alter it directly jumping from $1$mW to $2$mW, which results in the constrained strategy sets. To tackle these problems, we propose the aggregative game model into such scenarios and permit UAVs to learn in the constrained strategy sets. Because the aggregative game can measure and integrate the impact of all other UAVs (e.g., the aggregated interfering power level) on one UAV, which reduces the computational complexity from polynomial time to linear time. Such an advantage is more obvious when the number of UAVs is extremely large since figuring out each others’ strategies is unrealistic [8]. Therefore, the aggregative game with constrained strategy sets is an ideal model for the UAV deployment in post-disaster networking scenarios.

In this setting, efficient learning algorithms seeking for the UAVs’ Nash equilibrium are necessary. In existing algorithms, UAVs learn by prediction [8, 9] and know current strategies with corresponding payoffs. However, under highly dynamic scenarios, UAV merely knows the current and next strategies to calculate their expected payoffs, but can only learn from previous experiences [11]. Therefore, a novel algorithm which can learn from previous experiences is required, and the algorithm with faster learning speed is more desirable. In this case, if the two strategies are different, UAV chooses the strategy with the larger payoff as the current strategy in the next iteration. Otherwise, if the two strategies are the same, a new strategy is randomly selected as the current strategy of the next iteration. To sum up, existing algorithm calculates payoff for a new strategy (equivalent to prediction) and chooses it based on prediction, while an improved algorithm can be available when UAV only knows strategy’s payoff if strategy has been selected, and then UAV can decide whether to stick to the current strategy or return to the past strategy by comparing their payoffs. Therefore, an algorithm which can learn from previous experiences is required.

However, the learning rate of the existing algorithm is less than satisfied [13]. The algorithm called binary log-linear learning algorithm (BLLA) has been proposed by [14]. However, each UAV is allowed to change strategy one by one in each iteration based on current game state, which means that UAVs are not permitted to update strategies at the same time. Besides, to determine which UAV to update strategy, the coordinating process will occupy plenty of channel capacities and require more time between two iterations [15]. If the algorithm can learn synchronously, more than one UAV can update strategies based on the current game state in one iteration. Thus, the algorithm can be more efficient. In a word, synchronous update algorithms which can learn from previous experiences are desirable, but only a few research has investigated on it.

I-B Contribution

We establish a multi-factor model for large-scale UAV networks in highly dynamic post-disaster emergency communication network scenarios. Considering the limitations in existing algorithms, we devise a novel algorithm which is capable of updating strategies simultaneously to fit the highly dynamic environments. The main contributions of this paper are as follows:

• •

We desgin a novel framework that jointly considers power and coverage control in multi-channel scenario, which are key network influence factors to ensures the reliability of large-scale highly dynamic UAV networks.

• •

We propose a novel PEN model with the aggregative game which is compatible with the large-scale highly dynamic environments, in which several influences are coupled together. In the aggregative game, the interference from other UAVs can be regarded as the integral influence, which makes the model more practical and efficient.

• •

We devise a novel payoff-based binary log-linear learning algorithm (PBLLA). PBLLA outperforms BLLA in the sense that it is capable of learning previous utilities and strategies to fit for post-disaster scenarios.

• •

We further propose the synchronous payoff-based binary log-linear learning algorithm (SPBLLA) which has the following properties: 1) SPBLLA can learn with restricted information; 2) In certain conditions, SPBLLA approaches NE with constrained strategies sets; 3) SPBLLA allows UAVs to update strategies synchronously, which significantly speeds up the learning rate; 4) SPBLLA avoids system disorder and ensures synchronous learning, which is a main breakthrough.

We organize this paper as follows. The related works will be introduced in section II. In section III, we first introduce the UAV’s power control in the multi-channel communication and coverage problems, then form a system model in highly dynamic scenarios. Moreover, in section IV, we formulate the problem as an aggregative game and prove the existence of the NE. In section V, we propose the two algorithms for approaching the NE. Section VI presents the simulation results and discussions. Ultimately, section VII gives a conclusion.

II Related Work

The literature mainly lies in four areas, we will present the studies of UAV communication network, channel reuse in the communication network, game theoretical approaches, and the implementation algorithms sequentially.

II-A UAV Communication Network, Post-Disaster Wireless Commnications, and PEN

With the rapid commercialization of UAVs, a lot of researches have emerged into this field [16]. To efficiently deploy UAVs, studies have been made to find out UAV distribution on network graph [9] and a graphical model has been proposed for channels reuse [17]. The resource allocation of channel and time is also a hot area to study [8]. As mentioned previously, those models are impractical for post-disaster scenarios since they fail to jointly consider power control and coverage which are coupled and dominant factors. In the context of post-disaster wireless commnications, a booming number of works have been seen, which use difference wireless technologies, sush as, LTE/LTE-A [18], WiFi [19], IoT [20], D2D/MANET [21], Satellite [22]. Due of their flexibility and fast deployments, adpoting UAVs to constrcut the PENs highly dynamic post-disaster circumstances is now becoming a pivotal choice. Only recently, we have seen works studied the coverage problem by using geometry based mathematical framework [23], the joint bandwidth-power (resource) allocation scheme in known composite fading channel [24], task offloading [26], joint channel allocation and data delivery [27], delay-tolerant blockchain-based secure offline transactions [28] and topological aspects of PENs [25]. In comparision, our study of the joint power-and-coverage control is make a first step for the intersection of the wilress resource and network topological managments in PENs, which has actually taken the mobility of UAVs and tough communication circumstances into account.

II-B Channel Reuse

Typical wireless protocol 802.11b/g only provides limited channels for users, which is far more than enough for high-quality communication services [29]. To reduce the load in central system, making use of distributed available resources in networks turns out to be an ideal solution. Underlay Device-to-Device (D2D) communication is considered as one of the crucial technologies for cellular spectrum reuse in communication networks [30]. The advantage of D2D communication that allows end users to operate on licensed channels through power control sheds light on how interference management would work in PENs [34].

II-C Game Theoretical Approaches

Game theory provides an efficient tool for the non/cooperation through resource allocation and sharing [31, 32, 33]. A computation offloading game has been designed in order to balance the UAV’s tradeoff between execution time and energy consumption [37]. A sub-modular game is adopted in the scheduling of beaconing periods for the purpose of reducing energy consumption [35]. Sedjelmaci et al. applied the Bayesian game-theoretic methodology in UAV’s intrusion detection and attacker ejection [36]. However, most existing models focus on common scenarios with less number of UAVs, which are not compatible with large-scale scenarios with massive numbers of UAVs [38]. Aggregative game is a characteristic game model which treats other agents’ strategies as a whole influence, thus avoids overwhelming strategies information from every single agent [39][40]. Inspired by this, our model is built upon the aggregative game theory which suits for large-scale scenarios.

II-D Cooperative Algorithms

Various algorithms have been employed in the UAV networks in search of the optimal channel selection [41][43], such as stochastic learning algorithm [42]. The most widely seen log-linear learning algorithm (LLA) is an ideal method for approaching NE [9][44]. The BLLA has been employed by [45], which is modified from LLA to update strategies in each iteration to converge to the NE. However, only a single agent is allowed to alter strategies in one iteration. In large-scale scenarios, more iterations are required, which makes BLLA inefficient. It is obvious that more UAVs altering strategies in one iteration would be more efficient. To achieve it, the works in [46] and [47] have provided a novel synchronous algorithm. However, there exist superabundant restrictions that make the algorithm impractical in most scenarios. Compared with the formers, the SPBLLA proposed by us has fewer constraints and can achieve synchronous operation, which can significantly improve the computational efficiency.

In summary, our work differs significantly from each of the above-mentioned works, and other literatures in PENs. As far as we know, our proposed algorithm is capable of learning previous utilities and strategies, achieve NE with restricted information and constrained strategies sets, and update strategies synchronously, which significantly speed up the learning rate.

III System model and Problem formulation

III-A System Model

In our PEN, we construct a PEN in a post-disaster scenario with $M$ identical UAVs being randomly deployed, in which $M$ could be a huge number compared with normal multi-UAV system. All the UAVs have the same volume of battery $E$ and communication capability. The topological structure of Multi-UAV network is shown in Fig. 1 (a). We study the altitude of UAV in one time slot, and UAV can move horizontally in different time intervals. We study UAV changes in each time slot. Random selection of location in modeling shows that the optimal solution can be found in any location.

To support the communication mission, all UAVs are required to cooperate and support the user communication in need. UAVs work above the post-disaster area $D$. If a user (${\rm User}_{1}$) needs to communicate with another user (${\rm User}_{2}$) far away from him/her, the communication link can be set either directly between ${\rm User}_{2}$ (${\rm User}_{1}\rightarrow{\rm User}_{2}$), or use UAVs above in the sky as relay (${\rm User}_{1}\rightarrow{\rm UAV}_{1}\rightarrow{\rm UAV}_{2}\rightarrow\cdots\rightarrow{\rm UAV}_{n}\rightarrow{\rm User}_{2}$), as shown in Fig. 1 (b). Since the users are far away from each other, the link through UAVs provides better quality of service than the direct communication. Thus users will choose Link$2$ if they are covered by UAVs. Notice that users may not be covered by UAVs due to the high uncertainty in post-disaster areas.

In the PNE (i.e., PEN), there are $N$ channels available for UAVs to communicate, which are labeled as $C=[C_{1},...,C_{n},...,C_{N}]$. Each channel holds at most $C_{max}$ UAVs due to interference constraint. Following the water-filling allocation scheme [48], UAVs select one channel $C_{n}$ in the beginning of the process and will not change their decision afterwards. Assuming that ${\rm UAV}_{i}$’s decision is represented by the vector $C_{i}=[C_{i1},...,C_{in},...C_{iN}]$, where $C_{in}=1$ if UAVi selects channel $C_{n}$, otherwise $C_{in}=0$. Since the channel of each UAV is fixed, UAVs cannot switch to other channel with less noise to improve SNR, but only capable of raising SNR by coordinating their own power. The power for ${\rm UAV}_{i}$ in each channel is denoted by the vector $P_{i}=[P_{i1},...,P_{in},...,P_{iN}]$, $P_{in}>0$ when ${\rm UAV}_{i}$ selects the channel $C_{n}$, otherwise, $P_{in}=0$. The intrinsic noise in each channel is $Noise=[NS_{1},...,NS_{n},...,NS_{N}]$.

UAVs have several power levels and altitude levels. In the midst of extreme environments, UAVs cannot change its voltage dramatically but merely change to the adjacent power level [12]. Similarly, the altitude changing also has a limitation that only adjacent altitude level conversion is permitted in each move. We denote power set and altitude set to be $P=\{P_{1},...,P_{k},...,P_{np}\}$ and $h=\{h_{1},...,h_{k},...,h_{nh}\}$, respectively, where $np$ is the number of power levels and $nh$ is the number of altitude levels. We assume that the gap between different levels of power and altitude are equal. Let $\Delta P$ and $\Delta h$ denote the distance value of adjacent power levels and altitude levels, respectively.

When UAVs communicate, and the signal to noise rate (SNR) mainly determines the quality of service. UAVs’ power and inherent noise are interferences for each other. Since there are hundreds of UAVs in the system, each UAV is unable to sense all the other UAVs’ power explicitly, but only the aggregative interference and treat it as an integral influence. Though increasing power can improve SNR, excessively large power causes more energy consumption and results in less running time. Therefore, proper power control for UAVs needs to be carefully designed.

Coverage is another factor which determines the performance of each UAV. As presented in Fig. 1 (c), the altitude of UAV plays an important role in coverage adjusting. The higher altitude it is, the larger coverage size a UAV has. A large coverage size means a substantial opportunity of supporting more users, but a higher SNR will be needed. Furthermore, the turbulence of upper air disrupts the stability of UAVs with more energy consumption. Thus, a suitable height is essential to determine the coverage area.

III-B Problem Formulation

Defining a PEN game $\Gamma=(U_{i},S_{i})_{i\in M}$, where $U_{i}$ is the utility function of ${\rm UAV}_{i}$ and $S_{i}$ is the strategy set of ${\rm UAV}_{i}$. Take $s_{i}\in S_{i}$ as one strategy of ${\rm UAV}_{i}$, then $s_{i}=[C_{i},P_{i},h_{i}]$. For simplicity, we denote $S_{-i}\in\prod_{j\neq i}S_{j}$ as the strategy set of UAVs except ${\rm UAV}_{i}$ and $s_{-i}\in S_{-i}$. $S=\prod_{i}S_{i}$, and $s\in S$ is the strategy profile of the game.

As is described previously, UAVs can only sense the aggregative influence in each channel they select. Defining such influence as a power interaction term, which of ${\rm UAV}_{i}$ is written as:

[TABLE]

where $\sigma=\sum_{i\in M}P_{i}+Noise$ and $\otimes$ is the operator that multiplies corresponding elements in two vectors. For the convenience of notation, denoting the $nth$ element in $\sigma_{ip}(s_{-i})$ to be $\sigma_{ip}(s_{-i})_{(n)}$, which is the power interaction in $C_{n}$. It should be noticed that, when ${\rm UAV}_{i}$ does not choose $C_{n}$, $\sigma_{ip}(s_{-i})_{(n)}=0$.

Suppose that a UAV covers a round area below it with a field angle $\theta$ as shown in Fig. 1 (b). Thus the coverage of ${\rm UAV}_{i}$ is $D_{i}=\pi(h_{i}{\rm tan}\theta)^{2}$. Considering that the higher the altitude is, the severer the air turbulence a UAV suffers, the utility of a coverage area is

[TABLE]

where $\beta$ is the air turbulence index, which decreases as the altitude increases.

When there are numbers of UAVs in the network, it is possible for the coverage areas of different UAVs to overlap. When a UAV overlaps with another, they will not support all users but share the mission. The users in the overlaps will be served randomly with equal probability by each UAV. Fig. 2 presents the overlaps between two UAVs, i.e., ${\rm UAV}_{1}$ and ${\rm UAV}_{2}$ will support half of the users in the overlap area. In this condition, the true coverage $\bar{D}_{i}$ is smaller than $D_{i}$, and it is written as

[TABLE]

where $\kappa$ is the index which decides the influence of overlap. Since $\bar{D}_{i}$ must satisfy that $\bar{D}_{i}>0$, $\kappa$ is a tinny index. In a large-scale network, $D_{i}<<\sum_{j\neq i}D_{j}$ and $\kappa<<1$.

In order to support as many users as possible, UAVs are required to enlarge coverage size, which is equal to enlarge the coverage proportion in the mission area. Higher altitude indicates larger coverage size as shown in Fig. 1 (c). The utility of coverage size is denoted as

[TABLE]

Now we define the utility functions of energy, SNR and coverage to calculate UAVs’ payoffs. For power selection of ${\rm UAV}_{i}$, a large power does not necessarily result in high utility due to the large interference comes with it. Taking energy saving and longer lifetime into consideration, choosing the right amount of power that balances the tradeoffs between power and interference, brings the highest utility. We define the energy utility function $U_{E}$ as:

[TABLE]

It should be noted that a high power provides UAVs with higher communication quality and larger coverage requires higher power.

As for power control, the SNR in channel $C_{n}$ for ${\rm UAV}_{i}$ is

[TABLE]

And it can also be written as $\mu C_{in}[P_{in}-\gamma\sigma_{ip}(s_{-i})_{(n)}]$ [8], where $\gamma$ is the SNR balance index and $\mu$ is the SNR index. If given the Shannon capacity

[TABLE]

which decides the communication quality to be improved, SNR is supposed to be enlarged. The summation SNR of ${\rm UAV}_{i}$ is $\mu\sum_{i\in M}[P_{i}-\gamma\sigma_{ip}(s_{-i})]\otimes C_{i}$, where we broaden the meaning of $\sum$ and the formula is the summation of all elements in the vector.

According to the previous definitions, we use an equation and an index to balance the tradeoff between power control and coverage, and we formulate the SNR utility function as:

[TABLE]

where $\alpha$ is the index that balances the tradeoff between SNR and coverage size.

Finally, the optimal utility for each UAV is where there is a balance among power limitation, SNR and coverage size. The utility function $U_{i}$ of ${\rm UAV}_{i}$ is defined as:

[TABLE]

where $A$, $B$ and $C$ are balance indices that balance three utilities on the basis of post-disaster scenario. The ultimate goal for enlarging the utility of the networks is to maximize the summation of utility function (9) of each UAV, and we define the global utility function as the objective function:

[TABLE]

which presents the performance of the PEN.

IV Multi-aggregator Aggregative Game

IV-A Multi-aggregator Aggregative Game Model

We formulate the PEN game in large-scale post-disaster area as a multi-aggregator aggregative game [39], where we calibrate its definition in our UAV network model and put it as follows.

Definition 1

(Multi-aggregator Aggregative Game) The game $\Gamma=(U_{i},S_{i})_{i\in M}$ is called a multi-aggregator aggregative game with two or more aggregators $g={g_{1},g_{2},...,g_{K}}$, $g_{k}:S\rightarrow R^{n}$ ($n$ is the dimension of aggregators), and $S$ is bounded, if there exists two or more interaction functions $\sigma_{i}=\{\sigma_{i1},\sigma_{i2},...,\sigma_{iK}\}$, and $\sigma_{ik}:S_{-i}\rightarrow X_{-i}\subseteq R^{n},i\in M$ such that each of the payoff functions $i\in M$ can be written as:

[TABLE]

where $u_{i}:S_{i}\times X_{-i}\times X_{-i}\times...\times X_{-i}\rightarrow R$, and for any fixed strategies profile $s$:

[TABLE]

In the PEN game, let $\sigma_{i1}(s_{-i})$ represents the power interaction term $\sigma_{ip}(s_{-i})$, which is the aggregative power influence from other UAVs for ${\rm UAV}_{i}$. Another interaction term $\sigma_{i2}(s_{-i})$ is named as the area interaction term $\sigma_{ia}(s_{-i})$:

[TABLE]

Besides, the aggregators $g_{1}(s)$ and $g_{2}(s)$ of the PEN game are as follows:

[TABLE]

where $g_{1}(s)_{(n)}$ is the $n$th element of $g_{1}(s)$. $g_{1}(s)_{(n)}$ is the summation of power and noise in channel $n$, which has been chosen by ${\rm UAV}_{i}$. If the strategy profile $s$ is fixed, the summation of power and noise in all channels are fixed. Therefore, for any value of $i$, $g_{1}(s)_{(n)}$ is the same, if the stratefy profile $s$ is fixed. Let $g_{2}(s)$ stand for the summation of each UAV’s coverage area.

[TABLE]

where $\sigma_{ia}(s_{-i})$ is the area covered by UAVs except of ${\rm UAV}_{i}$. Therefore, $g_{1}(s)$ is the summation of all UAV’s coverage area. For any value of $i$, $g_{2}(s)$ is the same, if the strategy profile $s$ is fixed.

The utility function $U_{i}(S_{i},\sigma_{ip}(S_{-i}))$ in the PEN game conforms to the definition of payoff function $U_{i}(S)$. Therefore, the PEN game is a multi-aggregator aggregative game.

IV-B Analysis of Nash Equilibrium

In game theory, Nash Equilibrium (NE) is a special state that no UAV can gain more payoff by changing its strategy. Thus, NE is an ideal solution for all UAVs in multi-UAV relay mission game [8]. The potential game is usually used in analyzing the existence of NE. We first define the Pure Strategy Nash Equilibrium (PSNE) with several assumptions, then prove that the PEN game has a NE.

Definition 2

(Pure Strategy Nash Equilibrium) A strategy profile $s^{*}=\{s_{1}^{*},s_{2}^{*},...,s_{m}^{*}\}$ is a pure strategy Nash equilibrium if and only if any UAV is unable to gain more payoff by altering their strategies when no other UAVs change their strategies, i.e.,

[TABLE]

Definition 3

(Potential Game) A game $\Gamma=(\phi_{i},S_{i})_{i\in M}$ such that $S_{i}$ is the strategy of ${\rm UAV}_{i}$, $s_{i}\in S_{i}$. We broaden the meaning of $\prod$ and write the set of all strategies as $S=\prod S_{i}$. Labeling the set of UAVs except ${\rm UAV}_{i}$ as $S_{-i}=\prod_{j\neq i}S_{j}$,$s_{-i}\in S_{-i}$. There exists a potential function $\phi_{i}(s_{-i}):S_{-i}\rightarrow S_{i}$. If there exists a function $P:S\rightarrow R$ for $\forall i\in M$ such that

[TABLE]

then this game is a potential game.

Definition 3 indicates that the change of utility function is the same amount of the change of potential function, which gives an ideal property to the potential game.

Theorem 1

Any potential game with finite strategy has at least one PSNE.

Proof:

Refer to the proof of Corollary 2.2 in [49]. ∎

Theorem 2

The PEN game is a potential game and has at least one PSNE.

Proof:

Please refer to Appendix A. ∎

Since the PEN game is a special type of potential game, we can apply the properties of the potential game in the later analysis. Some algorithms that have been applied in the potential game can also be employed in the PEN game. In the next section, we investigate the existing algorithm with its learning rate in large-scale post-disaster scenarios and propose a new algorithm which is more suitable for the PEN in such scenarios.

V Learning Algorithm

V-A Payoff-based Binary Log-Linear Learning Algorithm

In the literatures, most works search PSNE by using the Binary Log-linear Learning Algorithm (BLLA). However, there are limitations of this algorithm. In BLLA, each UAV can calculate and predict its utility for any $s_{i}\in S_{i}$ in the complete strategy set. While in PENs, UAVs are accessible only to the constrained strategy set and corresponding utilities in the last two decision periods. Thus, conventional BLLA is no more suitable for the scenario we considered here. Therefore, we propose a revised algorithm based on BLLA, called Payoff-based Binary Log-linear Learning Algorithm (PBLLA) to resolve the issue.

In this section, we will present how PBLLA works in Algorithm 1 and the convergence of PBLLA.

Theorem 3

Let denote $\tau$ as the dynamic degree of the scenarios. The harsher environment the networks suffers, the higher $\tau$ it is. In the highly dynamic scenarios, we suppose that $\tau\geq 0.01$. With proper $\tau$, PBLLA asymptotically converges and leads the PEN game approaching to the PSNE.

Proof:

The proof of the convergence of PBLLA is similar to that of BLLA in [50]. According to [50], we only need to illustrate that PEN game conforms to the following two assumptions.

Assumption 1

For $\forall i\in M$ and all strategy profile pairs $s_{i}^{0},s_{i}^{n}\in A$, there exist a series of strategies $s_{i}^{0}\rightarrow s_{i}^{1}\rightarrow...\rightarrow s_{i}^{n}$ satisfying that $s_{i}^{k}\in C_{i}(s_{i}^{k-1})$ for $\forall k\in\{1,2,...,n\}$, where $C_{i}(s_{i}^{k})$ is the constrained strategies set of strategy profile $s_{i}^{k}$.

Assumption 2

For $\forall i\in M$ and all strategy profile pairs $s_{1}^{0},s_{2}^{n}\in A$,

[TABLE]

In the PEN game, UAVs are only permitted to select the adjacent power and altitude levels. It is evident that for any strategy profile pair $s_{i}^{0},s_{i}^{n}\in A$, $s_{i}^{0}$ can change power and altitude step by step to reach $s_{i}^{n}$, vice versa. Thus, PEN game satisfies Assumption 1 and Assumption 2. The remaining proof can follow Theorem 5.1 in [50]. ∎

The essence of PBLLA is selecting an alternative UAV randomly in one iteration and improving its utility by altering power and altitude with a certain probability, which is determined by the utilities of two strategies and $\tau$. UAV prefers to select the power and altitude which provide higher utility. Nevertheless, highly dynamic scenarios will cause UAVs to make mistakes and pick the worse strategy. The dynamic degree index $\tau$ determines the dynamic degree of the situation and UAV’s performance. Small $\tau$ means less dynamic scenarios and fewer mistakes when UAVs are making decisions. When $\tau\rightarrow 0$ which equals to stabilization, UAV will always select the power and altitude with higher utility; when $\tau\rightarrow\infty$ where exists severe dynamics, UAV will choose them randomly. However, PBLLA has its limitations that PBLLA is only one single UAV is allowed for altering strategies in one iteration. We will propose a new algorithm in the next section to overcome the restrictions.

V-B Synchronous Payoff-based Binary Log-linear Learning

Since PBLLA only allows one single UAV to alter strategies in one iteration, such defect would cause computation time to grow exponentially in large-scale UAVs systems. In terms of large-scale UAVs ad-hoc networks with a number of UAVs denoted as $M$, $M^{2}$ times of exchange messages will be needed to coordinate and guarantee that only one UAV changes strategy in each iteration. Such a process not only consumes large energy but also prolongs the convergence time. Hence, algorithms that can improve the learning rate and reduce messages exchange is urgently needed. Thus, we propose the Synchronous Payoff-based Binary Log-linear Learning Algorithm (SPBLLA) in Algorithm 2, which permits each UAV altering their strategies synchronously and learning with no message exchange.

As shown in Algorithm 2, the process of SPBLLA let UAVs free from message exchange. Therefore, there is no waste of energy or time consumption between two iterations, which significantly improves learning efficiency. All UAVs are altering strategies with a certain probability of $\omega$, which is determined by $\tau$ and $m$. $\tau$ also presents the dynamic degree of scenarios. The chance of UAVs to make mistakes when altering strategies is determined by the dynamic degree as in PBLLA.

To prove the convergence of SPBLLA, we first provide some conceptions.

Definition 4

*(Regular Perturbed Markov Process) Denote $P$ as the transaction matrix of a Markov Process which has a finite state space $S$. This Markov Process is called regular perturbed markov process with noise $\epsilon$ if the following conditions are met.

1. $P_{\epsilon}$ is aperiodic and irreducible when $\epsilon>0$.

2. $\lim_{\epsilon\rightarrow 0}P_{\epsilon}=P_{0}$, where $P_{0}$ is an unperturbed process.

3. For any $s^{n},s^{m}\in S$, when $P_{\epsilon}(s^{n},s^{m})>0$, there exists a function $R(s^{n}\rightarrow s^{m})\geq 0$, which is called the resistance of changing strategy from $s^{n}$ to $s^{m}$, such that*

[TABLE]

Definition 5

(Stochastically Stable Strategy) Denote $P_{\epsilon}$ as the transaction probability of a regular perturbed Markov process in a state space $S$, and $\mu_{\epsilon}(s)$ is the probability that the state transforms to $s$. The state is a stochastically stable strategy if

[TABLE]

In our model, when UAVs of repeated PEN game adheres to regular perturbed Markov process, the probability of being in $s$ is

[TABLE]

Let $L$ denote a series of adjacent strategies, then $L=\{s^{0}\rightarrow s^{1}\rightarrow...\rightarrow a^{n}\}$. In any adjacent strategies pairs $(s^{k-1},s^{k})$, the resistance of changing strategy from $s^{k-1}$ to $s^{k}$ is $R(s^{k-1},s^{k})$. The resistance of path $L$ is the sum of each move

[TABLE]

According to $L$, we give the definition of resistance tree.

Definition 6

(Resistance Tree) In a strategy profile space $S$, strategy profiles are linked. A tree $T$, rooted at strategy profile $s$, is a set of directed edges that any other strategy profile has only one directed path, which consists of several directed edges, leads to $s$. The resistance of $T$ is the summation of resistance of all directed edges,

[TABLE]

Denoting $T(s)$ as all the trees that rooted at strategy profile $s$, then the stochastic potential of strategy profile $s$ can be written as

[TABLE]

The minimum resistance tree of strategy space $S$ is the tree that has the minimum stochastic potential,

[TABLE]

For example, Fig. 3 shows different tree rooted at $S^{3}$. The branches are linked in different ways.

With these definitions in the hand, we can prove the convergence theorem of SPBLLA as follows.

Theorem 4

Let us consider several UAVs with PEN game with potential function $\phi:S\rightarrow R$. When all UAVs adhere to SPBLLA, if $m$ is large enough, the stochastically stable strategies are maximizers of the potential function, which are PSNEs.

Proof:

Please refer to Appendix B. ∎

Remark 1

According to Appendix B, to make the SPBLLA converge, $m$ should be twice larger than the most massive altering amount of each UAV’s utility function, which can be summarized in the following theorem.

Theorem 5

$m$* is an index that indirectly influences the learning rate. If $m$ satisfies*

[TABLE]

*where *

[TABLE]

then SPBLLA converges.

Proof:

Please refer to Appendix C. ∎

Remark 2

Let each UAV alter strategy as large as possible to make utility function change significantly. Calculating the most significant difference that a utility function can make in an iteration, and we are capable of learning the range of $m$.

VI Simulation Result and Discussion

In this section, we will investigate how the key parameters in the PEN affect the performance of PBLLA and SPBLLA. In the simulation, besides the quantity of UAVs and channel, other parameters are fixed as constant values. We set up the post-disaster sqaure area as $D=4000km^{2}$. $M=100$ UAVs are scattered in the area. There are $N=30$ channels and each channel can hold $C_{max}=25$ UAVs at most, and each UAV selects $N_{C}=5$ channels set $np=40$ levels of power are accessible $P=\{0.025W,0.05W,0.75W,...,0.975W,1W\}$, and the gap between two levels is $\Delta P=0.025W$. Noise is randomly distributed from $0.025W$ to $1W$. $nh=46$ levels of altitude are accessible $h=\{1km,1.2km,1.4km,...,9.8km,10km\}$, and the distinction between two levels is $\Delta h=0.2km$. The battery capacity of UAV for communication is $E=5mAh$. Field angle which determines the coverage size is $\theta=30^{o}$. The balance indices vary from one to another in the diverse scenarios. Here we choose $A=0.002$, $B=0.005$, $C=0.03$, and $\alpha=0.002$, $\gamma=0.002$, $\kappa=10^{-4}$, $\mu=10$. $\beta$ decreases as $h$ increases. According to Theorem 5 in the last section, when $m>2\Delta$, SPBLLA is feasible.

VI-A Effect of Scenarios’ Dynamic Degree $\tau$

In this subsection, we investigate the influence of environment dynamics on the network states. With different scenarios’ dynamic degree $\tau\in(0,\infty)$, PBLLA and SPBLLA will converge to the maximizer of objective function with different altering strategy probabilities. Fig. 6 presents the influence of the dynamics on PBLLA. We can find out that the fluctuation during converging is severe in both algorithms, which is different from other related works. It does not result from the bad performance of algorithms but from the highly dynamic scenarios. When the environment is highly dynamic with high values of $\tau$, which brings about more mistakes when selecting powers and altitudes. Thus, when UAVs have low probabilities to select the right strategy, it will result in non-optimal decisions. In the rest simulations, similar phenomena can also be observed.

As the dynamic degree index $\tau$ decreases from $0.03$ to $0.01$, the objective function’s values are increasing, which illustrates that lower values of $\tau$ approach to maximizer of the global utility function. When $\tau=0.03$, the value of $U$ does not increase much before converge. The fact that the severe interference from the environment seriously influences the UAV network making UAVs, making mistakes when altering strategies, can account for such a result. In Fig. 6, $\Delta U$ is the value that presents the difference between the value of each iteration and the average function value after convergence. Comparing the two lines of $\tau=0.01$ and $\tau=0.03$ in Fig. 6, we see that the fluctuation of $\tau=0.03$ is around 50% more than that of $\tau=0.01$, showing that the rough environments lead to unstable converge states. Fig. 9 and 9 show the performance of SPBLLA, which is similar to that of PBLLA. In the simulation of the SPBLLA, $m=0.03$, which follows the instruction of Theorem 5. Those mentioned figures show that when $\tau$s in PBLLA and SPBLLA are equal, the final optimum states and maximizer are similar.

VI-B Performance on SNR and Coverage

Since PBLLA is similar to SPBLLA, we skip the performance analysis for PBLLA, but focus on SPBLLA only. We set $m=0.03$ and $\tau$ from $0.01$ to $0.03$. Fig. 6 and Fig. 9 present the performance average SNR and the coverage size of the whole PEN. The interference from the environment causes fluctuation. When the scenario dynamic degree is getting lower, the network is capable of approaching to a better outcome, and the SNR and coverage size is getting larger. In the slower dynamic scenario ($\tau=0.01$), the PENs can cover 95% post-disaster area. Even though the interference from the environment is serious ($\tau=0.03$), the coverage proportion can be up to 80%.

VI-C Altering strategies probability $m$ on SPBLLA

The opportunities for UAVs changing strategies in each SPBLLA’s iteration are determined by the probability index $m$. The altering probability $\omega=(e^{-\frac{1}{\tau}})^{m}$ illustrates that higher $m$ provide less probability for UAVs altering strategies. Higher probability enables higher chances for more UAVs altering strategies at the same time. Thus lower $m$ ensures high frequencies of changing strategies and faster converge rates. According to the limitation of $m$ in equ. (26), we figure out that $m>0.028$.

Fig. 12 shows the effect of $m$ on the behavior of SPBLLA. Setting $\tau=0.01$ and $m>0.028$, we choose $5$ values from $0.03$ to $0.05$. As $m$ getting higher, SPBLLA needs more time for convergence. Since higher $m$ results in less opportunity for UAVs to alter strategies, fewer UAVs change strategies at the same time. For the sake of getting to the same optimum state, higher $m$ requires a longer time for convergence. It is also presented in Fig. 12 that $m$ does not influence the optimum state. The same value of $\tau$ and various values of $m$ converge to the state that shares the similar value of the global utility function, and a higher probability for altering strategies save PENs’ convergence time.

VI-D Power Control

Allocating appropriate power is one of the key factors that affect UAV’s utility, and the proper value of power will significantly improve the quality of service.

Fig. 12 presents the sketch diagram of a UAV’s utility with power altering. The altitudes of UAVs are fixed. When other UAVs’ power profiles are altering, the interference increases and the curve moves down. The high interference will reduce the utility of the UAV. Fig. 12 also shows that utility decreases and increases with power improving. Small and large power both provide high utilities, which is because small power will save energy and large power will increase SNR. The UAV might select the largest power to increase utility. However, The more power one UAV uses, the more interference other UAVs will receive and other UAVs’ utilities will reduce. For the sake of enlarging the global utility, the largest power is not the optimal strategies for the whole PEN. The best power will locate in some values that smaller than the largest power (The optimal value in the figure is a sketch value).

VI-E The number $N$ of UAVs

In the large-scale PENs, the number of UAVs is another feature that should be investigated. Since the demanding channel’s capacity should not be more than the channel’s size we provide, we limit the number of UAVs in the tolerance range which satisfies that each UAV’s channel selection is contented. In this scenario, there are $N=50$ channels, and the number of UAVs should be limited to $M\leq 250$.

Fig. 12 shows how the number of UAVs affects the computation complexity of SPBLLA. Since the total number of UAVs is diverse, the objective functions are different. The objective functions’ values in the optimum states increase with the growth in UAVs’ number. Since objective functions are the summation function of utility functions, more UAVs offer more utilities which result in higher potential function value. Moreover, more UAVs can cover more area and support more users, which also corresponds with more utilities. Fig. 12 also shows how many iterations that PEN needs to approach to convergence. With the number of UAVs improves, more iterations are required in this network.

VI-F Comparison of PBLLA and SPBLLA

In this subsection, we perform a comparison between PBLLA and SPBLLA to investigate the superiority of SPBLLA. We fix $\tau$ at several different values then compare the convergence rates of PBLLA and SPBLLA.

Fig. 15 presents the learning rate of PBLLA and SPBLLA when $\tau=0.01$. As $m$ increases the learning rate of SPBLLA decreases, which has been shown in Fig. 15. However, when $m$ is small, SPBLLA’s learning rate is about 3 times that of PBLLA showing the great advantage of synchronous learning. When $\tau=0.015$ and $\tau=0.02$ as shown in Fig. 15, such phenomenon also exists. Since PBLLA merely permits a single UAV to alter strategies in one iteration, SPBLLA’s synchronous learning rate will much larger than PBLLA. Moreover, in the large-scale UAV network with high dynamic, PBLLA needs information exchange to decide the update order, which would severely prolong the learning time. PBLLA’s learning time might be four times as long as that of SPBLLA. Thus we can make the conclusion that in the same condition (the same $\tau$ and other indexes), SPBLLA performs better and is more suitable for large-scale highly dynamic environment than PBLLA, and SPBLLA can improve the learning rate several times. With larger altering strategies probability, SPBLLA will be even more powerful.

However, we have to recognize that the altering strategies probability $\omega$ severely impacts on the efficiency of SPBLLA. If Theorem 5 limits $m$ to be a large value, the probability will decrease. When $m$ is too large, UAVs are hard to move, and the learning rate will decrease. To some points, the learning rate of SPBLLA will lower than that of PBLLA. In our PEN scenario, when $\tau=0.01$ and $m=0.03$, which is circled in Fig. 15, the probability of altering strategies $\omega<0.01$. The probability of altering strategies in SPBLLA is less than that of PBLLA, and the SPBLLA will spend more learning time.

Fig. 15 shows $\tau$ and $m$’s impact on the probability of altering strategies $\omega=(e^{\frac{1}{\tau}})^{m}$. The red circle in Fig. 15 matches with the red circle in Fig. 15, where SPBLLA is not efficient. The line $\omega=(e^{\frac{1}{\tau}})^{m}=0.01$ marks the same efficiency of the two algorithms, where SPBLLA alters one UAV in one iteration on average. When post-disaster scenarios are in the same degree of dynamic, lower $m$ permits higher changing strategies probability. If we want to increase altering probability, we can limit the utility change in each iteration to reduce $m$. Nevertheless, such a method reduces the updated amount which also causes a negative influence on the learning rate. Then how to balance the altering probability and update amount is a new topic that needs further investigation. Besides, how to reduce $m$ in Theorem 5 is another task that requires further research.

VII Conclusion

In this paper, we establish a PEN in the large-scale post-disaster area with the aggregative game. We propose a synchronous learning algorithm (SPBLLA) which expedites the learning rate comparing to the asynchronous learning algorithm (PBLLA), and shows desired behavior in highly dynamic scenarios. The learning rate of SPBLLA can be $10$ times larger than that of PBLLA. Even though there exists fluctuation during convergence, SNR has been improved and the network can cover over 95% post-disaster area. From our analysis, we see that more UAVs provide higher utilities in the network. Though our proposed algorithm fits a highly dynamic environment, the learning rate of SPBLLA decreases when $m$ becomes large. In further work, improved SPBLLA which supports a wide range of $m$ will be needed.

Appendix A Proof of the Theorem 2

Proof:

Formulate a function $\phi$:

[TABLE]

When $UAV_{i}$ changes its strategy from $s$ to $s^{\prime}$, its utility changes

[TABLE]

As for function $\phi$

[TABLE]

Therefore,

[TABLE]

and $\phi$ is the potential function of PEN game. ∎

Appendix B Proof of Theorem 4

Proof:

Define the state that UAV ad-hoc netwok has the latest two strategy profiles $s(t-1)$ and $s(t)$ to be tuple $a(t)=[s(t-1),s(t),x(t)]$, where $x(t)=[x_{1}(t),x_{2}(t),...,x_{M}(t)]$.

Let $\epsilon=e^{-\frac{1}{\tau}}$. When $\epsilon$ approaches to $0+$, $\omega=0$, then the unperturbed process equals to UAVs never altering strategies. It is obvious that the recurrent classes of PEN game with unperturbed process are $[s,s,\textbf{0}]$, where two strategy profiles are identical and any element in $x(t)$ is [math]. Any transition in the process of SPBLLA can be written as

[TABLE]

to be specific

[TABLE]

[TABLE]

The probability of transition from $a^{1}$ to $a^{2}$ is

[TABLE]

Denote

[TABLE]

[TABLE]

Multiplying the numerator and denominator of $P_{\epsilon z^{1}\rightarrow z^{2}}$ by $\prod_{i\in M}\epsilon^{V_{i}(s^{0},s^{1})}$, and

let

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Since when $\epsilon\rightarrow 0+$,

[TABLE]

Thus, $0<lim_{\epsilon\rightarrow 0}\frac{P_{\epsilon z^{1}\rightarrow z^{2}}}{\epsilon^{R(z^{1}\rightarrow z^{2})}}<\infty$ and $R(z^{1}\rightarrow z^{2})$ is the resistance of transition from $a^{1}$ to $a^{2}$. Then SPBLLA induces a regular perturbed markov process.

According to Lemma 1 in [51], we learn that the stochastically stable strategy profiles are strategy profiles with minimum stochastic potential, and a strategy profile is a stochastically stable strategy profile only if it is in a recurrent class of unperturbed process $P^{0}$. Then we only need to prove that the root of minimum trees of strategy profiles $S$ is the maximizer of $\phi$. Denote $3$ kinds of strategy profile $X$, $Y$ and $Z$ ,where $X$ is what cannot be written as $[s,s,\textbf{0}]$, $Y=[s,s,\textbf{0}]$ is what is not a maximizer of $\phi$, and $Z=[s,s,\textbf{0}]$ is what is a maximizer of $\phi$. Only $Y$ and $Z$ are the candidates for stochastically stable strategy profiles.

Due to $X$ cannot be the stochastically stable strategy profile, we focus on $Y$ and $Z$. Build a minimum resistance tree $T$ rooted at $s*$, and some edges have multiple UAVs alter strategies. At each edge, there are several UAVs such that $s_{i}^{k}\neq s_{i}^{k-1}$, which compose a group $G\subseteq M$. Then the probability of transition from $s$ to $s^{\prime}$ is

[TABLE]

the resistance of transition from $s$ to $s^{\prime}$ is

[TABLE]

Denote the upper bound of $V_{i}(s,s^{\prime})-U_{i}(s^{\prime})$ for any $i\in M$ to be $\Delta$, the resistance of the transition from $s$ to $s^{\prime}$ satisfies the inequality as follows:

[TABLE]

In the first condition, there is merely a single UAV altering strategy for each edge $[s\rightarrow\tilde{s}]\in T$, then the argument in [50] illustrates that $s*$ is a maximizer of the potential function.

In another condition, assuming that there exists an edge $[s\rightarrow\tilde{s}]\in T$ with multiple UAVs atlering strategies, Let $\tilde{G}$ consist of the group of these multiple UAVs. The resistance of this edge meets

[TABLE]

Consider a path $L=\{s=s^{0}\rightarrow s^{1}\rightarrow...\rightarrow s^{\ \tilde{G}|}\}$, where for any $k\in{1,...,|\tilde{G}|}$, there is only one UAV altering strategy and the rest UAVs keep the original strategies. equ. (35) shows that the resistance of these edges is less than

[TABLE]

and the resistance of $L$ is at most

[TABLE]

Build a new tree $T^{\prime}$ rooted at $s*$ as well but remove redundant edges of $L^{R}$ from $T$ and add edges of $L$ to $T$. The redundant edges consist of $[s\rightarrow\tilde{s}]$ and other directed edges rooted at strategy profiles in $\{s^{1},s^{2},...s^{|\tilde{G}|-1}\}$ but not in $L$. Fig. 16 presents how the new tree $T^{\prime}$ forms. We can deduce from equ. (36) that $R(s\rightarrow\tilde{s})\geq m|\tilde{G}|$ and the remanding edges in $L^{R}$ is at least $m$. Then, the resistance of $L^{R}$ satisfies

[TABLE]

The resistance of tree $T^{\prime}$ is

[TABLE]

Since $|\tilde{G}|\geq 2$, if $m>2\Delta$ then $R(T^{\prime})<R(T)$. The conflict implies that trees with edges, which has multiple UAVs changing strategies, are not the minimum resistance trees and this condition does not exist. Therefore, only maximizers of potential function are stochastically stable. ∎

Appendix C Proof of Theorem 5

Proof:

Notice that $\Delta_{max}=|U_{i}(s^{\prime})-U_{i}(s)|_{max}$, where the distinction between $s^{\prime}$ and $s$ is capable to be made in one iteration. Define $\delta(x)$ as the discrepancy of $x$ in an iteration. In one iteration

[TABLE]

[TABLE]

If each $\delta(x)$ approaches maximum, $U_{i}(s^{\prime})-U_{i}(s)$ gets the maximum. The maximum $\delta(x)$ are as follows.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Thus, $\Delta_{max}$ equals to the summation of the right value of Eqs. (43)–(47). When $m>\Delta_{max}$, $m$ satisfies the assumption that $m$ is large enough. ∎