Joint Head Selection and Airtime Allocation for Data Dissemination in Mobile Social Networks

by   Zhifei Mao, et al.

Mobile social networks (MSNs) enable people with similar interests to interact without Internet access. By forming a temporary group, users can disseminate their data to other interested users in proximity with short-range communication technologies. However, due to user mobility, airtime available for users in the same group to disseminate data is limited. In addition, for practical consideration, a star network topology among users in the group is expected. For the former, unfair airtime allocation among the users will undermine their willingness to participate in MSNs. For the latter, a group head is required to connect other users. These two problems have to be properly addressed to enable real implementation and adoption of MSNs. To this aim, we propose a Nash bargaining-based joint head selection and airtime allocation scheme for data dissemination within the group. Specifically, the bargaining game of joint head selection and airtime allocation is first formulated. Then, Nash bargaining solution (NBS) based optimization problems are proposed for a homogeneous case and a more general heterogeneous case. For both cases, the existence of solution to the optimization problem is proved, which guarantees Pareto optimality and proportional fairness. Next, an algorithm, allowing distributed implementation, for join head selection and airtime allocation is introduced. Finally, numerical results are presented to evaluate the performance, validate intuitions and derive insights of the proposed scheme.


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1 Introduction

Mobile social networks (MSNs) enable people to share content and communicate without Internet access by exploiting short-range wireless communication technologies such as WiFi Direct and Bluetooth [1, 2, 3]. Due to the nature of intermittent connectivity, MSNs are often regarded as a special type of delay tolerant network that utilizes opportunistic contacts among mobile users to deliver data [4, 5, 6]. Nowadays, people are becoming increasingly inseparable from their portable smart devices such as smartphones. This brings numerous opportunities for people to form temporary groups to exchange information when their portable devices are within each other’s transmission range. In particular, MSNs are promising communication systems for people in areas where Internet access is unavailable or too costly. For example, when disasters strike, Internet infrastructure such as cellular networks are among the first pieces of critical infrastructure to fail, leaving individuals disconnected from one another and from vital information sources [7]. In such scenarios, MSNs will be one of the fastest and most handy ways to provide digital connection among individuals.

Over the past years, significant MSN research effort has been conducted. However, the main focus has been on routing, data dissemination and community detection, leaving a fundamental MSN problem nearly completely untouched, which is local resource management [3]

. This surprising phenomenon is probably due to that for various types of wireless networks, local resource management has been extensively studied, and consequently one could expect that their existing solutions might be directly applied or easily extended to MSNs. Unfortunately, this expectation ignores a fundamental difference between MSNs and other popular types of wireless or mobile networks. The difference is that, in the latter cases, there exist base stations (BSs) or access points (APs) for mobile users to get connected to and through the Internet, and in such cases, local resource management, e.g. scheduling the use of airtime among users, typically implicitly assumes that the BSs or APs are always willing to serve, and hence considers only user devices in making the decision.

However, in MSNs, that seemingly unquestionable assumption may not hold, particularly when a user needs to use her/his smart device to store-carry-forward data for the others [8]. This is because, a smart device normally has limited capacities in terms of e.g. energy, storage, processing and communication. In consequence, local resource management in MSNs not only should consider the devices with data to send or receive, but also must not forget the helpers that contribute additionally in terms of local resources such as energy and communication to act like BSs or APs to help the others. This partly explains why research on MSNs has been progressing for more than one decade but real implementations and adoptions of MSNs in the public are rarely seen today: Overlooking the additional costs incurred to the helper essentially discourages anyone to be helper, which is a foundation for MSNs to work. Nevertheless, the concept of MSNs has drawn huge attention of industry besides academy. Recently, Mozilla and the U.S. National Science Foundation have been running a contest seeking innovative solutions to connect people who are disconnected from the Internet due to disaster or insufficient connectivity [7]. This provides a great opportunity to work on the missing pieces in MSN research towards public adoption of MSNs, and motives the work of this paper.

In this paper, we focus on a fundamental problem in local resource management, which is airtime allocation among users in a temporary group formed on the move, since available airtime is typically limited in MSNs due to user mobility and short transmission range. The airtime required for disseminating a piece of data from its source to all the interested group members depends on its communication network topology. In this work, we consider that each group uses a star topology to communicate where one user is selected as group head to serve like a personal hot-spot open to the group and manage the group while other users connect to the head as peripherals111Group head and peripherals are called group owner and clients in Wi-Fi Direct terminology [9], and called master and slaves in Bluetooth terminology [10].. Such a star form is simple yet practical222Theoretically, mesh topology is also possible for connecting MSN users in proximity. However, it requires additional functionalities such as multi-hop routing and topology management implemented on users’ portable devices. Similarly, while wireless channels are broadcast in nature, using it for applications usually also requires changes to the applications and the various layers below, to be able to make use of this feature. because it is natively supported by the most popular off-the-shelf short-range communication technologies on portable devices, including WiFi Direct [9] and Bluetooth [10], making the underlying network functionalities transparent to application development.

With star topology, a group head must be selected among the users. The head needs to forward data for the peripheral users, and thus spends more battery power than them. Therefore, it is important to encourage users to be the head. In addition, since users may have different battery levels and link capacities, head selection is critical in that it impacts users’ utilities and the amount of data that can be disseminated with the limited airtime. In the literature, various fair airtime (or rate) allocation schemes have been proposed for traditional WLANs and cellular access networks [11, 12, 13, 14]. However, they all implicitly assume that the airtime is long enough so that all the data transmission can be finally completed. This assumption does not generally hold in MSNs where the contact duration among users is limited due to user mobility and short transmission range. In addition, the utility function they use, which is typically where is the allocated airtime (or rate), cannot characterize the specifics of users in MSNs, including data dissemination need, preference on other users’ data and battery level. Furthermore, unlike previous work, the group head, counterpart of AP in WLANs [14] and BS in cellular networks [13] respectively, must also be a target of the airtime allocation, equivalent to the other users connecting to it. These add more difficulty in designing a fair airtime allocation scheme for local data dissemination in MSNs.

In this paper, we address airtime allocation jointly with head selection among a group of users in an MSN, which, to the best of our knowledge, has never been considered previously. Since anyone in the group may or may not (want to) be the head, a game-theoretic approach is naturally adopted. Specifically, we formulate the problem of joint head selection and airtime allocation as a Nash bargaining game. An advantage of using Nash bargaining is that the solution, if it exists, is known to be Pareto optimal, proportionally fair, and acceptable by all users. Motivated by this, we prove the existence of optimal solution to the joint head selection and airtime allocation Nash bargaining game using decomposition. In addition, we propose a distributed algorithm for joint head selection and airtime allocation, based on the decomposition idea. Moreover, numerical results are presented to provide an overview of the performance, validate intuitions and derive insights of the proposed joint head selection and airtime allocation scheme.

The rest of this paper is organized as follows. In Section 2, we introduce the system model including network model, dissemination model, incentive scheme, and user utility function. The Nash bargaining solution (NBS)-based head selection and airtime allocation scheme is proposed and studied in Section 3. In Section 4, we show the numerical results. Section 5 presents related work. Finally, we conclude in Section 6.

2 System Model

Since there are many notations in this paper, we summarize them in Table I for reader’s convenience.

Notation Description
User group
Number of users in
Set of all directed links
Link that sends data from node to node
Rate of link
Set of data that node intends to disseminate
Unit reward
Amount of data node forwards for other nodes
Utility of node
Valuation function
Cost function
Amount of data disseminates
Amount of data of interests receives
Total energy consumption of node
Head indicator
unit energy consumption for sending data
unit energy consumption for receiving data
Amount of data node sends
Amount of data node receives
Energy budget of node
Node ’s sensitivity to battery power consumption
Airtime for the dissemination of node ’s all data
Bargaining power of node
Available airtime
Airtime for disseminating node ’s data

TABLE I: Notations.

2.1 Network Model

Consider a group333We assume nearby groups use different channels for data dissemination and data transmission on each channel is independent from the other channels, e.g. in WiFi-Direct. of users (or nodes) in an MSN, which come into contact by opportunity and would like to disseminate their data to other interested nodes in this group. The nodes can communicate with each other by forming a star network where is the set of all directed links. One of the nodes is selected as the head of the group while other nodes, referred to as peripheral nodes, connect to each other through the head. Denote the rate of link that sends data from node to node . We assume the links in may have different rates.

2.2 Dissemination Model

Denote the set of data that node intends to share to other interested nodes in the group during the contact (There may be some nodes that do not have any data to disseminate but are interested in other nodes’ data.). Given that the data of a peripheral node can interest multiple nodes in the group, the head can intentionally store the data (or part of the data) once receiving it from the source node for the first time and then forward it to the rest recipients, so that the limited airtime can be utilized more efficiently than directly sending multiple times from the source node to each recipient.

2.3 Incentive Scheme

Forwarding data for the peripheral nodes will incur a high cost to the energy, storage, etc., therefore, rational nodes are not willing to be the head and forward data for others unconditionally. To encourage nodes to become the head, we assume there is an incentive scheme such that the forwarding behavior is rewarded by the system. Note that the peripheral users do not have to pay to the head for forwarding their data: In practice, such a reward could be in various forms such as popularity and/or reputation in the MSN. For simplicity of analysis, in this paper, we do not restrict the form of implementing the reward and use a linear abstract form of rewarding function, i.e., the node will receive a reward of if it forwards an amount of data for others, where is the unit reward.

2.4 User Model

Nodes are effectively autonomous agents, since there is no network-wide control authority. Each node can decide, on its own will, whether to join the group and contribute resources to facilitate data dissemination. In addition, the node selected as the head contributes more resources than client nodes. Therefore, it is reasonable to assume that each node seeks to maximize its utility from data dissemination over a contact. Denote the utility of node , it is given by the valuation of the data it disseminates and the data of interests it receives, minus the energy cost for sending/receiving data, plus the reward for forwarding data for others if is the head:


where is the valuation function, is the cost function, is the amount of data disseminates, is the amount of data of interests receives, is the amount of data forwards for other nodes if it is the head, is the total energy consumption for sending and receiving data, and is the head indicator. For any node , means it is selected as the group head while means it is a peripheral node. Since there will be only one head, we have . Denote and the unit energy consumption for sending and receiving data, respectively. Then we have where is the amount of data it sends and is the amount of data it receives. To clarify the difference between and , and the difference between and , an example is illustrated in Fig. 1.

Fig. 1: Differences between and , and between and . Node and are interested in node ’s data A of MB, but node is not. Since they cannot communicate directly, node will send A to the head, i.e., node first, then node will forward data A to node and . As we can see, the amount of data node disseminates is MB, while the amount of data it directly sends is MB. For node , the amount of data it receives is MB, while the amount of data of interests it receives is MB since it is not interested in data A.

For the valuation function , we assume it is a strictly concave, positive, and increasing function of , and if . Function satisfies the above assumptions. Such logarithmic function has been often used in the literature (e.g., [15, 16, 17]) to model a network user’s satisfaction or evaluation over certain network resources. For the energy cost function , we assume it is a strictly convex, positive, and increasing function of , and if . In addition, each node has an energy budget of that can be spent during the contact period. Clearly, we need to have . Function satisfies the above assumptions, where is a normalization parameter that indicates user ’s sensitivity to battery power consumption444This cost function is a modified version of that used in [17] which does not satisfy .. For example, a user may have high sensitivity when battery charging is inconvenient. As a rational node will not participate in the group if it will become worse off, it requires for all .

3 Nash Bargaining to Head Selection And Airtime Allocation

When a number of users come into each other’s proximity, they create a contact opportunity to form a group to exchange data with interested ones. Before they can do that, they have to make a proper decision on head selection and airtime allocation. Since the users are autonomous and rational, each of them would like to benefit from the contact by disseminating its data, receiving data of interests, or obtaining reward. However, the airtime can be very limited due to their mobility so that it would be impossible that everyone gains as much as he/she wants. Therefore, the final decision of head selection and airtime allocation should be acceptable to everyone in order to resolve conflicts of interest. Otherwise, there would be no guarantee that the group will be formed. For such bargaining problems where players not only have incentive to cooperate but also have incentive to oppose each other, Nash bargaining solution (NBS) is an axiomatic approach that can uniquely identify an outcome by its four axioms. In this section, we use Nash bargaining to formulate the problem of airtime allocation jointly with head selection, and analyze the existence of its optimal solution. First of all, we review the basics of NBS in the following section.

3.1 Basics of Nash Bargaining Solution

In this section, we briefly review the concepts and results related to NBS. Consider a bargaining game of players who bargain or compete for a share of a limited resource (airtime in our case). Throughout the game, the players either reach an agreement on an allocation of the resource or come into disagreement. Let be the share of the resource that player gets, is called a feasible allocation. For each player , it has a utility function where is the set of all possible allocations. Denote the utility of player when the players come into disagreement, is called the disagreement point. Then a bargaining game can be formally given by the pair where

is the set of all feasible utility vectors


Let a bargaining solution that assigns to the bargaining game an element of . is said to be an NBS if the following axioms are satisfied:

  • PAR (Pareto optimality). For any , if , then .

  • ILT (Independence of Linear Transformations). Suppose that the game

    is obtained from by the transformations , for all , then for all .

  • SYM (Symmetry). If is invariant under the exchanges of player and player and , then , for all possible .

  • IIA (Independence of Irrelevant Alternatives). If and are two bargaining games with and , then .

PAR ensures no wastage in the resource. ILT states that the bargaining solution is invariant with respect to linear utility transformations. SYM means that if any two players have the same utility function and disagreement utility, they will have the same utility in the bargaining solution. IIA says that if the feasible utility set shrinks, but the bargaining solution remains feasible in the smaller set, then the bargaining solution to the game with the smaller utility set should be the same. The latter three axioms (i.e., ILT, SYM and IIA) are often regarded as axioms of fairness [18, 19], as they allow NBS to select a fair allocation among the set of all Pareto optimal allocations. More details and interpretations of NBS can be found in [20].

Assuming the utility set is compact convex and there is at least one such that for all , then there exists a unique bargaining solution fulfilling the above four axioms, which maximizes the following Nash product (or Nash welfare) [18]:


Though no explicit fairness is defined within the four axioms, NBS shows strong fairness property. It is well-known that when for all , NBS guarantees proportional fairness (PF) in utility. An allocation that satisfies PF should be that, moving away from the PF allocation or NBS to any other feasible allocation will not increase the aggregate of proportional changes in utilities [21, 22, 23]. In mathematical terms, where is the PF allocation and is any other feasible allocation. Due to such relationship, NBS is often regarded as a generalization of proportional fairness. By relaxing the axiom of SYM [24, 18], the so-called generalized (or asymmetric) NBS can be obtained by maximizing


where is the bargaining power and . Generalized NBS satisfies the axioms of PAR, ILT and IIA and guarantees weighted proportional fairness which satisfies [25].

In the following, we will first elaborate the utility function of users in the cases of homogeneous user preference and heterogeneous user preference, model the head selection and airtime allocation using generalized NBS, and then discuss the existence of optimal solution for both cases. The intention of using generalized NBS instead of standard NBS is to see whether bargaining power allows the head to be selected to gain higher utility than other users, which motivates the users to become the head willingly.

3.2 Homogeneous User Preference


Assume the nodes have homogeneous preference on the data, i.e., they are interested in any data that any other nodes would like to disseminate. Define a dissemination of the data of any node the set of transmissions (or links) that send ’s data from one node to the other. For a peripheral node, its dissemination includes the transmission from itself to the head and transmissions from the head to other nodes in the group. For the head, its dissemination consists of transmissions from itself to all the peripheral nodes. Denote the airtime for the dissemination of node ’s data, then the airtime constraint is given by


where is the available airtime. For each node , we have


where is the size of all the data in and is the set of all the links that disseminate ’s data. The constraint shown in (5) means that the airtime allocated to ’s dissemination should not exceed what it needs.

Within the dissemination of any node’s data, we also aim a fair data distribution among all the transmissions. Ideally, the progress of all the transmissions of a given dissemination, defined as the amount of data transmitted, should be equal when the dissemination stops. Mathematically, we have


where denotes the amount of data transmitted by every link in and is the airtime for link in to disseminate node ’s data. Now we can express the airtime for sending ’s data via each link in in terms of :




the amount of ’s data transmitted by each link can be given by


Then, the amount of data disseminates within is given by


where is the number of transmissions in the dissemination of ’s data in . if the head has not stored ’s data, and if the head has. The amount of data of interests receives within can be given by


where is the set of users in except . If will be selected as the head, the amount of data forwards for other nodes is


where means the head has not stored node ’s data and needs to send the data to the head, otherwise . For a peripheral node , it only sends its data to the head, therefore . However, for a head , it not only sends its own data but also others’ data to all the interested nodes, therefore we have . Using a unified expression, we have


In the case of homogeneous preference, we have , i.e., the amount of data of interests receives equals the amount of data receives. Finally, the total consumed energy of is


Replacing , , , and in (1) by (10), (11), (12) and (3.2), we obtain the utility of any user shown in (3.2) at the bottom of this page.


From (3.2), we can see that is a function of and where and . We assume there is at least one makes otherwise nodes have no motivation to join the group and disseminate their data. Formally, the generalized NBS for the problem of head selection and airtime allocation for the case of homogeneous user preference can be obtained by maximizing the following generalized Nash product:

s.t. (17)

Under the generalized NBS framework, a higher generalized Nash product means a better decision of head selection and airtime allocation. In our case, users’ utilities are zero at the disagreement point since they will get nothing if no group is formed. Eq. (17) represents the airtime constraint for all the group members. Eq. (18) states that the airtime allocated to the dissemination of any user’s data should be nonnegative and not be longer than the maximum airtime required. Eq. (19) ensures individual rationality. Eq. (20) limits the energy consumption of each user to its energy budget. Finally, Eq. (21) and (22) indicate that only one of the users would be selected as the group head.

The problem (16) has at least one optimal solution. The proof of this statement is skipped, because in Section 3.3 below, a more general case, the heterogeneous case, is studied. For this more general case, it will be proved with details that the same statement holds, as shown in Theorem 1, for the more generalized problem (32) that corresponds to the problem (16) here.

3.3 Extension To Heterogeneous User Preference

The above model applies to MSN systems where users are interested in the same data. However, in some MSN systems (e.g., publish-subscribe systems), users may be interested in different data. In this section, we extend the above model to cases with heterogeneous user preferences.

Consider that there could be multiple data in . Let be the set of links that disseminate node ’s data . Denote the airtime for disseminating ’s data . Then the total airtime for disseminating ’s data is where is the number of data in . Then the airtime constraint is given by


For each node and its data , we have


where is the size of data . The total amount of data disseminates is


where is the number of transmissions in the dissemination of ’s data . Denote the set of nodes that disseminate data to (or equivalently the set of nodes that is interested in their data). The amount of data of interests receives is


where is the set of ’s data sent to . If will be the head after selection, the amount of data it forwards is


where means the head has not stored ’s data and needs to send to the head, otherwise . The amount of data sends is given by


For a peripheral node , we still have . However, for the head, it does not hold, since it may receive some data of no interest and only for forwarding. Then the amount of data receives is


Finally, the total consumed energy of is


Replacing , , , and in (1) by (25), (26), (27), and (3.3), we obtain the utility of any user shown in (3.2) at the bottom of this page. Formally, the generalized NBS for the problem of head selection and airtime allocation for the case of heterogeneous user preference can be obtained by maximizing the following optimization problem

s.t. (33)

Constraints (33) to (38) have the same meaning with constraints (17) to (22), respectively. Assuming there is at least one makes , we have the following theorem.

Theorem 1.

There exists at least one optimal solution to optimization problem (32) for joint head selection and airtime allocation.


In fact, the optimization problem (32) has two levels of optimization. At the lower level, each user in the group solves a sub-problem (a local generalized NBS problem) that finds optimal airtime allocation among all the users when user is the head. At the higher level, we have a master problem that chooses the best to be the head, which gives the highest generalized Nash product.

Mathematically, the sub-problem for each user is given by


where is fixed and is only a function of . Since and are strictly concave functions, by the concavity preserving rules in [26], we can see that is a strictly concave function in . Since the function of is concave and monotonic, the objective function of problem (3.3) is equivalent to [18]


It is easy to see that (40) is strictly concave and the constraints in (3.3) are convex. Additionally, we have assumed that there is at least one feasible point, meaning the constraint set is non-empty, therefore there exists a unique optimal solution to problem (40) and equivalently to problem (3.3) [26].

At the higher level, the master problem is

s.t. (41)

where is the optimal objective value of problem (3.3) for a given (namely, a given user being the head). Since there will be only one equals and the rest are , the objective of the master problem (3.3) is essentially finding the largest within real numbers, which always exists. The pair(s) resulting in the largest number will be the optimal solution(s)555Strictly speaking, there might be multiple maximum in real numbers. Therefore, we do not claim uniqueness of the optimal solution. to the whole problem. ∎

Remark 1: In the above proof, we show that there is a unique optimal solution to each local generalized NBS problem (3.3), meaning the axioms of PAR, ILT and IIA presented in Section 3.1 are satisfied in maximizing each local generalized NBS problem. Note that the master problem is just finding the user giving the highest generalized Nash product among all users. Suppose user is finally selected as the head, then the final optimal airtime allocation will be the optimal airtime allocation to the local generalized NBS problem of user , which satisfies the axioms of PAR, ILT and IIA. Therefore, no matter which user is selected as the head, the axioms of PAR, ILT and IIA are always satisfied.

Remark 2: Letting and , we can see that it reduces to the model for the homogeneous case. In other words, the homogeneous case is a special instance of the heterogeneous case. Therefore, Theorem 1 holds for the homogeneous case as well.

3.4 Algorithm

Based on the idea of decomposition in the proof of Theorem (1), we present an algorithm that can find a unique optimal solution in a distributed fashion. First of all, each node solves problem (3.3) with . Since problem (3.3) is a convex optimization problem with inequality constraints, its optimal solution can be found by interior point methods[26]. After solving the problem, node sends the optimal results and , i.e., the optimal airtime allocation and generalized Nash product given is the head, to all other nodes in . Once receiving the optimal results from all other nodes, each node checks which node being the head will result in the largest generalized Nash product. If node happens to have the largest generalized Nash product, it will become the group head and will be the final optimal airtime allocation. It is possible that multiple nodes have the largest generalized Nash product and any such node can be the head. In such cases, the node with the lowest index will be selected as the head without loss of generality666Certainly, there are other approaches to determine the final group head, such as selecting the node with the highest battery power. Actually, in the numerical results, it will be shown that user with high energy budget is preferred to be the head if we do not specifically select them.. The above steps is summarized in Algorithm 1.

2:for  do
3:     Solve problem (3.3) with to get and
4:     Send to all other nodes in
5:     while receiving from all  do
6:         if  then
7:              set and
8:         else
9:              set
10:         end if
11:     end while
12:end for
Algorithm 1 Joint Head Selection And Airtime Allocation

3.5 Handling Dynamics

Once the group members are known to all, Algorithm 1 will be applied to select a proper head and allocate airtime to each user. In practice, node mobility would introduce dynamics to a formed group, such as node leaving and joining. To cope with such dynamics, we presume a group management function is implemented at each node, which allows the head on duty to update group information periodically. Once an event (e.g., a new node joins) is detected, the head on duty will inform other users to start a new round of head selection and airtime allocation by applying Algorithm 1.

4 Numerical Results

In this section, we demonstrate how the NBS-based approach to the joint head selection and airtime allocation problem performs. Particularly, we show the impacts of different parameters such as energy budget, unit reward and link capacity on the behavior of the proposed approach. The tools used in solving the optimization problems and obtaining the results are AMPL777A powerful tool that can solve high-complexity optimization problems. ( jointly with MATLAB. In the utility function of users, we use as the valuation function and as the cost function.

Candidate head User 1 User 2 User 3 User 4 User 1 User 2 User 3 User 4

GNP is short for generalized Nash product. Each colored value is the highest GNP among all four values and implies that the corresponding user is selected as the head.

TABLE II: Utilities of the users with different users being the head when user ’s budget is Joules.

4.1 Setup

We consider a set of nodes that are in proximity with each other888Though the number of users is few, the most fundamentals are revealed. Evaluation with more than four users has also been conducted, with the same observations.. We assume that the communication technology they use is Wi-Fi Direct which supports star network topology [9]. Since it is built on traditional Wi-Fi infrastructure mode, Wi-Fi Direct can achieve typical Wi-Fi speeds. In a recent experimental study we conducted, it is found that the network is able to provide an average capacity of more than MB/s (equivalent to Mb/s) for local data dissemination [27]. We consider an average energy consumption of Joule/MB for both sending and receiving via Wi-Fi Direct [17]. At the beginning of the contact, each user has one data to share with others and the size of each data is MB (e.g., a short video clip or high-definition photo). The available airtime is seconds.

4.2 Energy Budget and Sensitivity to Energy Consumption

Hereafter, we use budget and sensitivity to represent energy budget and sensitivity to energy consumption respectively. We set user ’s budget to Joules and vary user ’s budget in Joules to see how head selection and airtime allocation are affected. We also consider two types of sensitivity, i.e., where all the users have the highest sensitivity and where user is insensitive because e.g., it has a power bank. In addition, the users have the same bargaining power and unit reward is set to .

Fig. 2: Results of airtime allocation and head selection when .

Fig. 2 shows the selected heads and airtime allocated to the dissemination of each user’s data with varying budget of user and . In the figure, "H" above a bar is short for ‘head’ and implies that the corresponding user is selected as the head. When the budget of user is Joules, only s out of s is allocated. The reason is that the NBS-based approach guarantees fairness in utility. As can be seen from Fig. 2, the utilities of all the users are rather close except that the utility of the head is slightly larger than others (the reason will be explained in Section 4.3). Fig. 2 shows that the budget of user will be totally utilized when it is Joules. Therefore, if all the s is allocated, only the utilities of user will increase while user ’s utility will not, which is not fair. However, this does not necessarily mean that the NBS-based approach is inefficient. On the contrary, it provides fairness in utility within each execution, the rest available time will be allocated among user in the next execution and fairness is guaranteed among them.

Fig. 2 also shows that the user with the highest budget is always selected as the head, provided that their sensitivities are the same. This seems reasonable since the head consumes significantly more energy than others, as shown in Fig 2. However, if the sensitivity of user , i.e. , is zero, user will become the head when its budget is Joules which is smaller than user ’s budget. This is because the cost in its utility function is always zero no matter how much energy is consumed. Table II shows that, when user is the head, the utility of user and that of user is respectively larger than that of user and that of user when user is the head. As a result, user being the head has larger generalized Nash product than user being the head does. However, when user ’s budget is or Joules, it is not selected as the head simply because its budget does not support to utilize the whole s and therefore leads to low utility for all the users.

4.3 Unit Reward

Fig. 2 and Table II have shown that the selected head can have higher utility than other users. Especially, it holds no matter which user is finally selected as the head in the examples shown in Table II. The form of utility function (1) implies that the forwarding reward to the head might be the cause. To study its impact on users’ utility, we vary the unit forwarding reward in . In addition, we let all the users have the same budget Joules and the same sensitivity .

Fig. 3 shows, on one hand, that the utility of the head (user ) increases with the unit reward. However, when there is no reward or the unit reward is too small, i.e. in , the utility of the head is lower than that of other users as shown in the small window in Fig. 3. In such cases, user may not be very willingly to be the head. When the unit reward is high enough, the head can gain a higher utility than other users. On the other hand, the utilities of the peripheral users (user ) also increase with the unit reward but slower than that of the head does and stop increasing when the unit reward is and higher. Fig. 4 illustrates the airtime allocated to the users’ disseminations. It can be seen that higher unit reward motivates the head to allocate more airtime to other users until the ‘others-first’ point where all the airtime is allocated to the other users and it get zero airtime for the dissemination of its own data. From that point where onward, increasing unit reward will not change the airtime allocation anymore, and the peripheral users will not be able to disseminate or receive more data. This explains why their utilities keep unchanged when and higher in Fig. 3.

Fig. 3: Utilities of users with varying unit reward.
Fig. 4: Airtime allocated to each user with varying unit reward.

In summary, a higher unit reward certainly motivates the users to be the head. However, a higher unit reward does not necessarily motivate the head to forward more data for others due to the existence of the others-first point. Therefore, a unit reward lying between zero and the others-first point is recommended in real application, which can not only make a trade-off between disseminating the head’s data and disseminating the peripheral users’ data, but can also control the gap between the users’ utilities.

4.4 Bargaining Power

It is expected that a user’s utility increases with its bargaining power. Fig. 5 shows the utilities of the users with increasing bargaining power of the head. In a loose sense, the expectation is reasonable. However, with a higher unit reward, the head seems less keen on obtaining longer airtime for its own data (as can be seen from Fig. 6), since a higher unit reward already allows it to achieve much higher utility than others. From another point of view, changing the bargaining power is less effective than changing the unit reward to control the gap between the users’ utilities. Especially when the unit reward is high (e.g., ), even the bargaining power of the head is much smaller than that of others, it still have much higher utility than others. Nevertheless, increasing the bargaining power of the head enables the head to obtain a higher utility than other users even when there is no reward for forwarding (i.e. ).

Fig. 5: Utilities of users with varying bargaining power.
Fig. 6: Airtime allocated to each user with varying bargaining power.

4.5 Data Load

Larger data load means that longer airtime is required to complete the dissemination. To investigate whether data load affects head selection and airtime allocation, we vary the data load of user from MB to MB and keep others’ data load fixed to MB. Table III shows the head selection under three different bargaining power settings. We can see that when the bargaining power of user is lower than or equal to that of others, the selected head does not change with user ’s data load. However, when the bargaining power of user is very high (i.e., ), the final head is changed from user to user when the data load of user becomes larger than that of others. To find the reason behind such a change, the airtime allocation with bargaining power is plotted in Fig. 7. It can be seen that if user is the head, the airtime allocated to itself increases linearly with its data load. As a result, the gap between the utility of user and that of other users increases with user ’s data load, as can be seen in Fig. 8. Larger utility gap gives smaller generalized Nash product and thus is regarded less fair by generalized NBS. That is why user is selected as the head by generalized NBS when user has a larger data load than others.

Bargaining power Data load of user (in MB)