Incentive Mechanisms for Crowdblocking Rumors in Mobile ...

9220 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 68, NO. 9, SEPTEMBER 2019

Incentive Mechanisms for CrowdblockingRumors in Mobile Social Networks

Yaguang Lin , Zhipeng Cai , Senior Member, IEEE, Xiaoming Wang , and Fei Hao

Abstract—Mobile social networks (MSNs) have become an indis-pensable way for people to access information, express emotions,and communicate with each other. However, the advent and ex-tensive use of MSNs has also created fertile soil for the breedingand rapid spread of rumors. Therefore, blocking the spread ofrumors in MSNs has always been a hot topic in this field. Withthe idea of crowdsourcing, we propose a novel rumor controlframework, called Crowdblocking, in which users can implementcontrol schemes in a collaborative and distributed way, so thatthe rumors can be controlled more effectively. With the proposedframework, the main problem that arises is how to motivate moreusers to actively participate in rumor blocking activities. To thisend, we design two effective incentive mechanisms in this paper.First, we propose an incentive mechanism based on the Stackelberggame for homogeneous control tasks. We theoretically analyze theStackelberg equilibrium to maximize the utility of the networkmanager and users involved in blocking rumor tasks, and ensurethat no user can improve its own utility by unilaterally changingthe current strategy. Second, for heterogeneous control tasks, wedesign a real-time reverse auction incentive mechanism, whichallows users to have more autonomy and freely customize theirown plans to participate in control tasks. Also, we prove thatthe mechanism possesses the desired properties of task timeliness,computational efficiency, user rationality, manager profitability,and price truthfulness. Finally, we validate the efficiency of theproposed mechanisms through extensive simulation experimentson the real datasets.

Index Terms—Mobile social networks, rumor blocking, incen-tive, auction, Stackelberg game.

I. INTRODUCTION

THANKS to the rapid proliferation of mobile intelligentdevices and the quick development of instant messaging

services, Mobile Social Networks (MSNs) have become themost important way for people to access information, express

Manuscript received March 17, 2019; revised June 6, 2019; accepted July16, 2019. Date of publication July 24, 2019; date of current version September17, 2019. This work was supported in part by the National Natural ScienceFoundation of China under Grants 61872228, 61877037, 61702317, 61601273,and 61602289, in part by the National Science Foundation under Grant 1704287,in part by the Fundamental Research Funds for the Central Universities of Chinaunder Grants 2017TS046 and GK201803081, in part by the Scholarship of ChinaScholarship Council under Grant 201806870045, and in part by the NaturalScience Basic Research Plan in Shaanxi Province of China under Grant 2019JM-379. The review of this paper was coordinated by Prof. L. Guo. (Correspondingauthor: Xiaoming Wang.)

Y. Lin, X. Wang, and F. Hao are with the Key Laboratory of Modern TeachingTechnology, Ministry of Education, Xi’an 710062, China, and also with theSchool of Computer Science, Shaanxi Normal University, Xi’an 710119, China(e-mail: [email protected]; [email protected]; [email protected]).

Z. Cai is with the Department of Computer Science, Georgia State University,Atlanta, GA 30303 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TVT.2019.2930667

emotions and communicate with each other [1], [2]. Many pop-ular mobile social networking platforms, such as Twitter, Face-book, WeChat and Sina Weibo, demonstrate the unprecedentedpower of MSNs in information dissemination and sharing [3],[4]. Compared to the traditional way of information interaction,MSNs provide certain benefits such as low transmission delay,high real-time performance and no space limitation, etc. In recentyears, MSNs have attracted increasing attention in academia andindustry due to their potential value [5], [6].

The development of MSNs provides a continuous impetus forthe flow of information. Because of its ubiquitous and accessiblecharacteristics, MSNs not only improve the scope of informationsharing, but also speed up the information dissemination [7], [8].However, it may also become a fertile breeding ground for gen-erating and disseminating malicious rumors or misinformation[9]. When people use MSNs to enjoy convenient and high-speedinformation services, some unauthenticated information or ru-mors will flow into it and may spread widely. This poses a seriousthreat to normal social network activities, such as harming theinterests of others, disturbing public order and threatening socialstability [10], [11]. Next, let’s introduce a motivating scenario.In September 2018, a powerful hurricane “Hurricane Florence”caused severe damage in the Carolinas of America. However,there were rumors on MSNs that “the Brunswick nuclear powerplant is in danger due to nearby flooding”, “the environmentalprotection agency does not have the budget to respond to naturaldisasters”, etc [12]. These have led to serious panic among thepublic and made the rescue even more difficult.

Rumors spreading in MSNs have been a critical threat toour society. Therefore, how to effectively block the spread ofrumors has become a research hotspot. In the recent years,many researchers have devoted themselves to this problem andproposed a variety of control methods [9], [13]–[17]. Thesemethods can be divided into two categories. The first is toidentify or infer the source nodes of rumor diffusion based onthe network topology and the information flow records, andblock or isolate the source nodes to prevent them from furtherspreading rumors to the network [10], [18], [19]. Although thistype of method can uncover the source of rumors and implementcorresponding control measures, it ignores other users who arebeing infected by rumors, leading to the emergence of newsources. Thus, rumors cannot be completely eradicated from thenetwork. The second method is to inject correct information intothe network to fight rumors, so as to block the spread of rumors[13]–[15]. For example, some information can be released byauthoritative departments or managers. Then this authoritative

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LIN et al.: INCENTIVE MECHANISMS FOR CROWDBLOCKING RUMORS IN MOBILE SOCIAL NETWORKS 9221

information can be spread to the whole network to maximize itsinfluence, so that the users who may be misled will no longerbelieve the rumors that come to them. Meanwhile, there aretwo ways to spread authoritative information: continuance andpulse [11], [20]. The continuance method is to continuously sendauthoritative information to users, while the pulse method is todisseminate authoritative information to users at intervals. In re-cent years, inspired by the optimal control theory, some optimalcontrol strategies for blocking rumors based on continuance andpulse forms have been proposed [15], [21], [22]. That is, calcu-lating the optimal control intensity (sending different quantitiesor different contents of authoritative information against rumors)at each time or period under cost constraints, so as to achievethe optimal control effect.

However, some new problems arise in the implementation ofthe control methods mentioned above, especially in a wide-areanetwork scenario. It is not difficult to find that most of the currentcontrol measures are implemented by managers in a centralizedway. The direct result of this is that the managers take on allthe workloads of control tasks. The workloads are so heavy thatthe managers can’t finish their work. Following that, based on theconcept of maximizing influence by selecting Top-k seed nodesin social networks [23], some scholars proposed an improvedscheme to assist managers to perform control tasks by selectingsome specific users (e.g., users with a high degree of sociality)[13], [24], [25]. Although this relieves the pressure of executingusers to some extent, it still does not fundamentally solve theproblem of overload. At the same time, a method was proposedfor all users to perform tasks in turn, that is, to select differentusers as executors of tasks at different times [26], [27]. Theadvantage of this method is to further balance the workloadsamong users. However, in the network with diverse users andcomplex structure, the efficiency and quality of tasks are not highdue to the difference of social attributes among users. In addition,some users are reluctant to perform control tasks because ofselfishness or resource consumption [28], [29].

Different from the existing control methods, we propose anovel control framework inspired by the idea of crowdsourcing[30]–[32], called Crowdblocking, that is, users can participate inthe activities of control rumors more flexibly and autonomously,and all users can cooperate to implement the whole controlscheme. Furthermore, on the basis of the Crowdblocking rumorcontrol framework, if more capable users in the network canbe motivated to spontaneously participate in the Crowdblockingrumors and forward the authoritative information, it will be fasterto block the spread of rumor information.

Toward this end, we further design incentive mechanisms toencourage more users in MSNs to take an active part in blockingthe spread of rumors, so as to achieve distributed and cooperativeimplementation of all control tasks among more users, andto control rumors faster and more efficiently. Considering theattributes of control tasks, we establish homogeneous controltask model and heterogeneous control task model. Then, basedon the Stackelberg game theory [33] and the reverse auctiontheory [34], we establish efficient incentive mechanisms for theabove models, respectively. The multi-fold contributions of thispaper are summarized as follows:

� We put forward a new Crowdblocking framework for con-trol rumors in MSNs by using crowd intelligence. Besides,we propose incentive mechanisms to encourage more usersto actively participate in rumor blocking activities. Moreimportantly, to the best of our knowledge, this paper pio-neer the concept of Crowdblocking and incentive mecha-nisms for control rumors in MSNs.

� For homogeneous control tasks, we design an incentivemechanism based on the Stackelberg game, called HTIM.We solve the Stackelberg equilibrium, and theoreticallyanalyze the optimal strategies of users which can maximizetheir utilities when other users’ strategies are fixed, andensure that no user can improve its utility by unilaterallychanging the strategy. In addition, we propose a method forcalculating the optimal budget of the manager to maximizeits utility.

� For heterogeneous control tasks, we propose an incentivemechanism based on the real-time online reverse auctioncalled RTA to assign tasks to active participants at the earli-est. Meanwhile, we theoretically prove that RTA possessesthe desired properties of task timeliness, computationalefficiency, user rationality, manager profitability, and pricetruthfulness.

� We evaluate the performance of the proposed incentivemechanisms on the real dataset. The effect of the HTIM isvery close to that of the theoretical optimal strategy, whichcan create maximum utility for the manager and users. Inaddition, compared to existing mechanisms, the proposedRTA can speed up the task allocation by 20%.

The remainder of the paper is organized as follows. In SectionII, some previous works are reviewed. We present the systemmodel and problem formulation in Section III. Then, in SectionIV and V, we propose HTIM and RTA for homogeneous taskmodel and heterogeneous task model, respectively. After evalu-ating the performance our proposed models and mechanisms inSection VI. We conclude this paper in Section VII.

II. RELATED WORK

Rumor blocking has become a research hotspot in the fieldof MSNs in the recent years [25]. Scholars have realized the ur-gency of blocking and controlling rumors, and proposed variouscontrol schemes from the perspective of network managers [10],[11], [13]–[21].

One of the representative methods is to find the source ofrumors and block it. In [18], Luo et al. studied the problemof exploring the source of rumors. According to the cascaderelationship of rumor propagation, they established a compu-tational model to estimate the probabilities of users who aremost likely to be the source of rumors. Then the problemis transformed into a mixed integer quadratic programmingproblem with quadratic constraints. The optimal solution ofthe above problem is obtained by using standard optimizationtoolboxes. In addition, considering the constraints of algorithmcomplexity, they also proposed an improved algorithm basedon the heuristic optimization to estimate the sources (users) ofrumors more quickly. Jiang et al. [10] proposed that identifying



the sources of rumors and isolating them in time play a vitalrole in reducing the harm caused by rumors in social networks.They used a back-propagation strategy to identify suspects whospread rumors, and established a microscopic rumor spreadingmodel to calculate the maximum likelihood for each suspect.Their work can more accurately identify the sources of rumors.However, adopting the method of blocking some users to controlthe spread of rumors will inevitably affect the experience of someusers and the quality of service of the network, and to someextent disturb the normal communication order of the network.Wang et al. [14] realized this realistic problem and proposed amodel of dynamic rumor influence minimization with the userexperience. They regarded user experience as a constraint ofnetwork services and set a threshold of maximum blocking timefor each user. Once the maximum blocking time was exceeded,the normal communication activities of the blocked user wererestored immediately.

Another mainstream approach to effectively control thespread of rumors in MSNs is that sending positive informationcorresponding to rumors. Wang et al. [15] explored the impact ofuser mobility on information diffusion, and proposed a computa-tional model to describe the process of rumor propagation. Then,a double pulse control strategy is proposed to dispel rumors bysending two kinds of positive information regularly in the net-work. Besides, considering cost constraints, an optimal controlmethod for epidemic information diffusion in social networksis proposed [21]. They modeled real-time control measures astime-varying dynamic functions, and solved the optimal distri-bution of dynamic functions through dynamic programming,thus minimizing the control cost in the network. He et al. [11]paid attention to the heterogeneity of users in social network,and designed a control method which combines the real-timecontrol with pulse control to suppress the spread of rumors.They also calculated the implementation intensity of controlstrategies for different victims to ensure cost effectiveness. Insummary, all of the above works have designed effective controlmeasures, providing a feasible solution for rumor control inMSNs. However, none of these efforts addressed the issue ofhow to implement control strategies.

Inspired by the problem of how to select Top-k seed nodesto maximize the impact of information in social networks,Tong et al. [13] explored how to select k seed users under agiven budget to maximize the number of unaffected users bydisseminating positive information. They proposed a samplingmethod based on the reverse tuple to select seed users to per-form tasks, and proved that the proposed method has low timecomplexity. In order to solve the problem of user selection inrumor blocking, Zhu et al. [24] studied how to use the leastcost to select the least positive seed user, so that the effect ofpositive seed can reach the preset threshold. At the same time,considering the disturbance of rumor information to seeds andthe time-effectiveness of information, they proposed a scalablegreedy algorithm to determine the positive seed nodes. Bothmethods can find the users set of task execution under a certainbudget, but the premise is that all users are willing to participatein the task execution. So, they only focus on the selection ofusers, not the design of incentives.

With the increasing scale of MSNs, the types of users inthe network become more and more complex. Scholars haveproposed some incentive mechanisms to encourage users toparticipate in the task execution independently [30], [35], [36].These incentives are mostly common in the field of crowdsensingand so on. Referring to the auction theory in economics, Gaoet al. [35] proposed a Lyapunov-based Vickrey Clarke grovesauction mechanism to motivate users to perform tasks, whichgreatly improves the proportion of user participation. Similarly,an incentive mechanism called LSB is proposed to motivateusers to participate in the crowdsourcing tasks [30]. The authorsdesigned a reverse auction mechanism, proved the submoduleproperty of its objective function, and proposed a greedy strategyto select the user who performs the task. However, the LSBmechanism has been proved to be inconsistent with the truthful-ness property.

In addition, participating in sensing tasks may expose users’privacy information, in order to motivate privacy-sensitive usersto perform sensing tasks. Koh et al. [37] proposed an incentivemechanism based on the Stackelberg game, which can meetthe privacy requirements of users, and increase the range ofsensors and the diversity of data. Then the unique solution ofStackelberg equilibrium is derived and the stability of the mech-anism is proved. Nevertheless, aforementioned mechanisms areall offline, and cannot be applied to real-time task allocation envi-ronment. Lin et al. [38] designed an online incentive mechanismcalled SOS, which can make assignment decision immediatelywhen a user or task arrives, and proved that it can overcome theSybil attack in the network. However, they assume that only oneuser can be assigned to tasks in a period of time. When multipletasks or users come at the same time, the mechanism allocatestasks slowly.

The design of incentive mechanisms has also been studied inmany other fields, such as cooperative communication, spectrumtrading, network routing, etc [39], [40]. These incentive mecha-nisms provide a theoretical basis on how to make users activelyparticipate in the execution of tasks, and the proposed methodsare worthy of our reference. However, because of the differentobjectives and different nature of the scenarios, the proposedincentive mechanism cannot be directly applied to the scenarioof blocking rumor spreading in MSNs.

For all this, we will design new incentive mechanisms toencourage users to actively participate the task of controllingthe spread of rumors. Considering different types of tasks, wewill propose incentive mechanisms for homogeneous tasks andheterogeneous tasks, respectively.

III. SYSTEM MODELS AND PROBLEM FORMULATION

Firstly, we introduce the Crowdblocking rumor control frame-work. Secondly, considering the attributes of control tasks, weestablish the homogeneous control task model and the heteroge-neous task model. In the homogeneous task model, the managerpublishes the same control tasks. For example, the managerwants to refute rumors by sending the same correct informationto users. On the contrary, in the heterogeneous task model, thecontent of control tasks released by the manager is different. For



Fig. 1. The diagram of Crowdblocking framework.

instance, the manager needs to send specific control informationto prevent or correct users according to the degree of rumorinfection. In addition, in the homogeneous task model, usersreceive the same payment for each task because all tasks are thesame, whereas, in the heterogeneous task model, the paymentof each task is different as the content and value of tasks aredifferent.

Therefore, the above two models are different but complemen-tary. The design of incentive mechanism in the homogeneoustask model can provide task assignment scheme for homoge-neous networks or scenarios with the same control measures.On the contrary, the design of incentive mechanism in hetero-geneous task models can be applied to heterogeneous networksand scenarios where more flexible control measures are adopted.In addition, the two models proposed can also be applied toinformation maximization scenarios with minor modifications.

A. Crowdblocking Framework

We consider the Crowdblocking rumor control framework tohave a manager M and n network users S = {1, 2, . . . , n−1, n}, where n ≥ 2. As shown in Fig. 1, all users can com-municate with the manager in the cloud. Firstly, the managerbroadcasts some tasks for rumor control to uninfected neighborusers according to the rumor blocking plan (step 1©). Afterconsidering the factors of cost and energy consumption, etc.,the users receiving broadcast messages may select some tasksfrom M to make their own execution plans and carry out thecorresponding tasks (step 2©). After users complete the selectedtasks, the execution results are submitted to the manager (step3©). Then, the manager checks the results of users and pays them

a certain amount of rewards (we call payments) to compensatefor their loss in energy consumption and so on (step 4©), and thetask assignment of this round is completed. It is worth noting thatthe manager can identify whether the users in the network areinfected by rumors or not, and it will not assign tasks to users whohave been infected by rumors. In fact, some effective methodsfor identifying the nodes (users) infected by rumors through theanalysis of user activity records, have been proposed in recentyears [10], [11], [15], [18]. Note that although the method ofidentifying infected nodes is beyond the scope of this work,the proposed Crowdblocking framework is compatible with anyexisting method.

In addition, we assume that the manager and users in thenetwork are selfish and rational. The goal of the manager is toassign tasks to network users in a timely manner with minimalcost. For network users, the goal is to earn benefits to themselvesby participating in the control tasks. Undoubtedly, users will not

actively participate in control tasks without guaranteed profits.Therefore, while completing the control tasks, it must be guar-anteed that a user will be paid no less than its cost, otherwisethe user will refuse to participate in the control tasks. Similarly,in each round of task assignment, the manager must ensure thatthe payment allocated to all users is less than the budget, that isto say, its own income is positive.

B. Homogeneous Task Model

In the homogeneous task model, the manager publishes somehomogeneous control tasks and announces that the total budgetfor the tasks is R, where R > 0. If the user i is not infectedwith rumors and it decides to participate in the execution oftasks, we use ni to represent the number of tasks that user idecides to perform, where ni is a positive integer, i.e., ni ∈ N+.Undoubtedly, a user needs to spend a certain amount of cost toexecute tasks. Here, we define the total cost of user i to executeni tasks as the following non-linear function:

ci = α+ γini + κn2i . (1)

Where, α > 0 is the cost of receiving messages from the man-ager. As all tasks are the same in the homogeneous task model,each user only needs to receive messages once. γi > 0 is theunit cost caused by channel occupancy, energy consumptionand other factors when user i forwards a message. κn2

i is usedto represent the additional cost to users due to bad phenomenasuch as message retransmissions caused by signal interference orpoor channel quality, where κ > 0. Note that, we do not imposea strong assumption on the ci since any other form of non-linearfunction could also be used here to derive the solution. Further,we assume that R � ci, and when users finish their selectedtasks, the manager will divide R proportionally according to thenumber of tasks each user performs as its payment. If the user iperforms ni tasks, the payment of user i is pi = R∑

j∈S njni, and

the utility of user i is:

ui =R

∑j∈S nj

ni −(α+ γini + κn2

i

). (2)

It is worth noting that each user in the network is selfish, andusers make decisions before participating in the tasks to ensureui > 0. Then, if R∑

j∈S nj

≤α+γini+κn2i

ni(i.e., the payment for each

task is less than the cost of each task performed by the useri), user i will not participate in the ni-th task. For the managerwith the budgetR, the more control tasks completed by users, thegreater the benefits created for it. Here, we represent the numberof tasks completed by all users as m0, then m0 =

∑j∈S nj , and

we define the utility of the manager as follows:

U = f(m0)−R, (3)

where f is the manager’s valuation function of the number oftasks completed by all users. We assume that f is a strictlyconvex function and increases monotonously with the increaseof m0. It is worth noting that this assumption is feasible andrealistic, and has been accepted and adopted in the previousworks [30], [35], [37], [40].

Under the above model, the problems and challenges we facedin designing incentive mechanism are:



Problem 1: max ui (ni), that is, how to determine the strat-egy of user i so as to maximize its utility ui when the budget Rand other users’ strategies are fixed.

Problem 2: max U (R), that is, how to determine the budgetR to maximize the manager’s utility U .

C. Heterogeneous Task Model

In heterogeneous task model, the manager publishes m dif-ferent control tasksΓ = {τ1, τ2, . . . , τm−1, τm}within time stept ∈ [1, · · · , T ]. Because of the heterogeneity of tasks, the costsfor completing different tasks are different. Moreover, due todifferent network structures, the cost for performing the sametask varies with different users. We use cji to represent the costfor user i to complete task τj . In addition, by implementingcontrol tasks, users are no longer misled by rumors. Therefore,for managers, performing each task can create a certain value.We define the value created by completing the task τj as vτj .Besides, each user has its own working time [ti, t

′i], in which

ti, t′i ∈ [1, · · · , T ] and ti < t′i. Users can only perform tasks

during their working time, and when t /∈ [ti, t′i], the cost of user

i performing any task is +∞. Users can select tasks they wantto perform during their working time independently and formtheir own tasks set. Assuming that user i selects a set of tasksΓi, where Γi ⊆ Γ.

Then, the user i will submit a bidding information Bi =(ti, t

′i,Γi, bi) to the manager, where bi is the bid for the user

i to perform the tasks set Γi, and bi ≥∑

τj∈Γicji . We call Bi

the time-task-bid, to express its willingness and requirements toparticipate in the control tasks. After receiving users’ time-task-bid pair, the manager will select some users as the executors toform a set of winners S, where S ⊆ S. If i ∈ S, the managerwill pay the user i a certain amount of payment pi after allthe selected tasks have been performed, where pi ≥ bi. By thisstage, the utility ui of user i by participating in these tasks is asfollows:

ui =

{pi −

∑τj∈Γi

cji , if i ∈ S;

0, otherwise.(4)

After all winners complete their tasks, the utility that the man-ager will gain is shown as follows:

U =∑

τj∈∪i∈SΓi

vτj−∑

i∈Spi (5)

Another problem and challenge we face are:Problem 3: How to design an effective incentive mechanism

to motivate more users to participate in heterogeneous controltasks more autonomously, while satisfying the following condi-tions:� Task timeliness: the mechanism can assign tasks to users

at the earliest time.� Computational efficiency: the mechanism can determine

winners and calculate payment in polynomial time.� User rationality: the mechanism can guarantee that no

user’s utility is negative.� Manager profitability: the mechanism can ensure that the

utility of the manager is non-negative.

TABLE INOTATIONS IN HOMOGENEOUS TASK MODEL

� Price truthfulness: None of the users can get more paymentby raising the bid that is different from its cost.

IV. INCENTIVE MECHANISM FOR HOMOGENEOUS TASKS

From the homogeneous task model we established inSub section III-B, there is a obvious conflict of interest betweenthe manager and users (that is, users want the budget as muchas possible, while the manager wants the budget as little aspossible). From the perspective of Game Theory [33], we buildthe strategic interaction between the manager and users as anon-cooperative game. Each user’s strategy is to determine thenumber of tasks ni, it has to perform so as to maximize its utilitywhen the other users’ strategies are fixed. The manager’s strategyis to determine the budget R to maximize its own utility. Sincethe Stackelberg game can solve the conflict of interests betweenusers and manager, and find out their optimal strategy [33], [40],we model the Homogeneous Tasks Incentive Mechanism as aStackelberg game, called HTIM. Both users and manager areplayers in HTIM, and the manager is the leader and the usersare the followers, where both users and manager only seek tomaximize their own utilities. We use S to represent the set ofall users in the network, and S−i to represent the set of all usersexcluding user i, we have S = S−i ∪ i. Then, we use N andN−i to denote the set of all users’ strategies and the set of allusers’ strategies excluding user i, respectively. So, we also haveN = N−i ∪ ni. For clarity, we list the frequently used notationsin homogeneous task model and their respective descriptions inTable I.

A. Theoretical Optimal Strategies of Users

First, we calculate the optimal strategy n∗i to maximize the

utility of user i in case of a fixed budget R and a strategiesset N−i, where n∗

i ≥ 0 and n∗i ∈ N . We calculate the first and

second derivatives of the utility function Eq. (2), respectively:

∂ui

∂ni=

R∑

j∈S−inj

(∑j∈S nj

)2 − γi − 2κni. (6)

∂2ui

∂ni2= −2R

∑j∈S−i

nj(∑

j∈S nj

)3 − 2κ. (7)



For each user j ∈ S, we have nj ≥ 0. Besides, we have R >0 and κ > 0. So, the second-order derivative of ui is alwaysnegative. Thus, the utility function ui is strictly convex, whichmeans that if the optimal strategy n∗

i of user i exists, then n∗i is

unique. Next, by setting Eq. (6) to 0, we have:

R∑

j∈S nj− Rni

(∑j∈S nj

)2 − γi − 2κni = 0, i ∈ S. (8)

If user i participates in the tasks, then i is winner and ni > 0,otherwise ni = 0. We define the set of winners as S, and S ={j ∈ S |nj > 0}. Let k0 be the number of winners, and k0 =∣∣S

∣∣. Then, considering that

∑j∈S nj =

∑q∈S nj , we have:

R∑

q∈S nq− Rnp

(∑q∈S nq

)2 − γp − 2κnp = 0, p ∈ S. (9)

For above equation, we sum all the elements in S and get:

k0R∑

q∈Snq

− R∑

q∈S nq−∑

p∈Sγp − 2κ

∑

q∈Snq = 0. (10)

Further, we solve the∑

q∈S nq . By discarding the meaning-less negative value, we can get the following result:

∑

q∈Snq =

√(∑p∈S γp

)2+ 8Rκ (k0 − 1)−∑

p∈S γp

4κ. (11)

Let θ =∑

q∈S nq , and bring θ into Eq. (9), we have:

n∗p =

θ (R− γpθ)

R+ 2κθ2, p ∈ S. (12)

Hence, the optimal strategy of user i can be obtained as:

n∗i =

{θ(R−γiθ)R+2κθ2 , if i ∈ S,

0 otherwise.(13)

It is worth noting that the calculated n∗i is a positive number,

not a positive integer, that is, n∗i ∈ N and n∗

i /∈ N+.

B. Nash Equilibrium

In this subsection, we explore whether a stable state can beachieved in HTIM when all users adopt the optimal strategies,so that no user can improve its utility by unilaterally changingthe current strategy in this stable state. As we all know, this stateis the Nash equilibrium in the non-cooperative game [41]. Next,we define Nash equilibrium for HTIM.

Definition 1: All users’ strategies (nne1 , nne

2 , . . . , nnen−1,

nnen ) in HTIM are the Nash equilibrium state. Then for any user

i, the strategy nnei satisfies:

ui

(nnei , Nne

−i

) ≥ ui

(ni, N

ne−i

).

Given a budgetR, if the Nash equilibrium exists and is uniquein HTIM, we can determine the unique optimal strategy foreach user to maximize its utility when other users’ strategies arefixed. Next, we analyze the existence and uniqueness of Nashequilibrium in HTIM. We have:

Theorem 1: For a given budget R, if the set of strategiesN ∗ =

(n∗

1, n∗2, . . . , n

∗n−1, n

∗n

)for all users is the unique Nash

equilibrium of HTIM, then it satisfies the following conditions:i) let S = {i ∈ S |n∗

i > 0}, then∣∣S

∣∣ ≥ 2;

ii) n∗i = {

θ(R−γiθ)R+2κθ2

0if i∈S

otherwise;

iii) if γp ≤ maxj∈S {γj}, then p ∈ S;iv) The cost of user i to perform a homogeneous task is

c1i = α+ γi + κ. We rank the costs of n users in a non-

decreasing sequence γ1, γ2, . . . , γn−1, γn. Let h be thelargest positive integer less than n and satisfy γh < R

θ <γh+1, then S = {1, 2, . . . , h− 1, h}.

We prove the condition (i) as follows.Proof: Let’s first assume that

∣∣S

∣∣ = 0, and no user currently

involves in the control tasks. So, any user i in the networkcan unilaterally change its strategy from ni = 0 to ni > 0 toparticipate in the control tasks and gain utility. This is contraryto the definition of the Nash equilibrium, so

∣∣S

∣∣ �= 0.

Then, we assume that if∣∣S

∣∣ = 1, user i will involve in the

control tasks, and we consider it in the following two Cases:Case 1: If ni > 1, the utility that the user i gets from the

manager is ui = R− α− γini − κn2i . At this point, the user

can gain more utility by changing the strategy from ni to ni − 1,which is contrary to the definition of the Nash equilibrium.

Case 2: If ni = 1, because R � ci, any other user j except iin the network can unilaterally change its strategy from nj = 0to nj > 0 to participate in control tasks and gain more utility. Itstill contradicts the definition of the Nash equilibrium.

To sum up, we prove that∣∣N

∣∣ ≥ 2. �

Then, we prove the condition (ii) as follows.Proof: From Eq. (13), we can see that if i ∈ S, i participates

in the control tasks. At this time, the optimal strategy takenby user i is n∗

i =θ(R−γiθ)R+2κθ2 , which can ensure that i can get the

greatest utility when other users’ strategies are fixed. In addition,when i /∈ S, then i is not involved in the control task, there is nodoubt that the strategy of i is ni = 0.

Next, we prove the condition (iii) as follows.Proof: Undoubtedly, if i ∈ S, then ni > 0. From (ii) we

know that if ni > 0, there is R− γiθ > 0. So, we have:

γi <R

θ, i ∈ S. (14)

Further, we can draw the following conclusion:

maxi∈S

γi <R

θ. (15)

Assuming that γp ≤ maxj∈S {γj}, and p /∈ S. From (ii) it isknown that the strategy of p is np = 0. At this point, we take thestrategy np = 0 and substitute it in Eq. (6) and get the following:

R∑

j∈S nj− γp =

R

θ− γp > max

i∈Sγi − γp ≥ 0. (16)

Then, we can conclude that if γp ≤ maxj∈S {γj} and p /∈ S,user p can increase its utility to up > 0 by unilaterally changingthe strategy tonp > 0, which is contrary to the Nash equilibrium.Therefore, the condition (iii) is proved. �

Finally, we prove the condition (iv) as follows.



Proof: The n users are sorted in a cost non-decreasing orderγ1, γ2, . . . , γn−1, γn. Becauseh is the largest positive integer thatsatisfies γh < R

θ , it is known h ∈ S from Eq. (14). From (iii)we can get that if p is the largest positive integer that satisfiesγp ≤ maxj∈S {γj}, then p ∈ S and S = {1, 2, . . . , p− 1, p}.So, we have p ≤ h. Assume that p < h, then there is γp+1 < R

θand p+ 1 /∈ S. Similarly, we take the strategy np+1 = 0 of userp+ 1 into Eq. (6) and get:

R∑

j∈S nj− γp+1 =

R

θ− γp+1 > 0. (17)

The above equation shows that user p+ 1 can gain moreutility by increasing the number of tasks performed, whichis contrary to the Nash equilibrium. So, we have p = h, andS = {1, 2, . . . , h− 1, h}. �

C. Incentive Mechanism

In the above subsections, we calculate the optimal strategy setN ∗ =

(n∗

1, n∗2, . . . , n

∗n−1, n

∗n

)for all users with a given budget

and prove that the strategy setN ∗ is the unique Nash equilibriumfor HTIM.

Next, we design the detailed steps of HTIM, as shown inAlg. 1. It is worth noting that the theoretical optimal strategyn∗i is a decimal value. Here, we have to consider an important

scenario constraint that the number of tasksni that user i choosesto perform must be an integer value. In order to ensure user utilityui ≥ 0, the n∗

i is rounded down as the number tasks ni of useri in HTIM, that is, ni = n∗

i�.In Alg. 1, we initialize the winners set S, the winners’ task

number set {ni} and the winners’ payment set {pi} (line 1).Then we sort all the users belonging to S in a non-decreasingorder in accordance with the cost γi, and add the first two users tothe set S as winners (lines 2-3). Next, we add other eligible users(i.e., users get more payments than their costs for participatingin tasks) as winners to the set S (lines 4-6). We calculate the

number of tasks performed by each user (lines 8-11). Finally,we calculate the payment for all users according to the numberof tasks they performed (lines 13-16).

The time complexity of Alg. 1 is analyzed as follows. Sortingall users (lines 4-6) takes O (nlog2n), The time required todetermine the winners (lines 4-6), calculate the number of tasksfor all users and calculate the payment for all users is equal toO (n). Therefore, the overall time complexity is dominated bysorting, and is O (nlog2n).

D. Optimal Strategy of the Manager

In HTIM, both users and manager are players, in which themanager is the leader and the users are the followers. In thecase of any budget R, all users’ strategies have the unique Nashequilibrium state. So, the manager can determine the value ofR to maximize its own utility. We bring the optimal strategycalculated by Eq. (12) into Eq. (3) and conclude that:

U = f

(∑

i∈Sn∗i

)

−R, (18)

where

n∗i =

{θ(R−γiθ)R+2κθ2 , if i ∈ S.

0 otherwise.

Theorem 2: There is a unique optimal strategy R∗, whichconstitutes HTIM’s unique Stackelberg Equilibrium (R∗, N ∗),where N ∗ is the set of all users’ optimal strategies calculatedby Eq. (12) when the budget is R∗. And, under the condition of(R∗, N ∗), the manager’s utility is maximized.

Proof: As can be seen from Eq. (3), the manager’s utilityfunction U(ni) is a strictly concave function related to ni undera fixed R. From the Subsection IV-A, we can conclude thatHTIM can determine the unique optimal strategies set N ∗ forall users at any value of R. Therefore, we can conclude thatthe manager’s utility function U(R,N ∗) is still strictly convexat any R. There must be the unique R∗, so that the manager’sutility can be maximized under the condition of (R∗, N ∗), andthe unique R∗ can be calculated by using either bisection orNewton’s method [42]. �

V. INCENTIVE MECHANISM FOR HETEROGENEOUS TASKS

The auction-based mechanism is considered to be an effectivesolution to motivate users to participate in the execution oftasks. In this section, we propose a Real-Time reverse Auctionmechanism, called RTA with the aim of motivating users toparticipate in heterogeneous tasks. We consider the managerto be the auctioneer and the users as bidders. Table II lists thefrequently used notations and descriptions in heterogeneous taskmodel while the meanings of symbols n, S, S, pi are the sameas that in Section IV.

A. Objectives of RTA Mechanism

The RTA must satisfy task timeliness, computational effi-ciency, user rationality, manager profitability, and price truth-fulness. Task timeliness is used to ensure that RTA can assign



TABLE IINOTATIONS IN HETEROGENEOUS TASK MODEL

Fig. 2. The schematic diagram of RTA.

tasks to users in a timely manner. Computational efficiency isused to ensure that RTA can be executed in polynomial time.User rationality is used to ensure that each user can benefit fromparticipating in tasks. Similarly, manager profitability is usedto ensure that the manager’s utility is non-negative. In addition,price truthfulness is used to ensure the fairness and robustness ofRTA. We use the following theorem to ensure that RTA possessesthe property of price truthfulness.

Theorem 3: [43] If RTA ensures price truthfulness, the fol-lowing two conditions must be satisfied.� The winner selection rule must be monotonous, that is, if

user i wins the auction by Bi = (ti, t′i,Γi, bi), user i can

also win it by submitting a lower bid bi or a broader workingtime ti, t

′i, where bi ≤ bi and ti ≤ ti, t

′i ≥ t′i;

� The payment pi to each winner must be the critical value,and if the user i changes the bid to bi > pi, the user i willnot win the auction.

B. Design of RTA Mechanism

We design the implementation steps of RTA, as shown inAlg. 2. It consists of two parts. The first part is the winnerselection (line 2). The second part is the calculation of eachwinner’s payment (lines 3-6). In addition, we show the functionsand relationships of the Algorithms in Fig. 2.

The detailed steps of winner selection are shown in Alg. 3.In each time step, we first calculate the active users set St at

t time step (lines 3-5), and then we assign tasks to users in agreedy manner according to the marginal value (we will showthe definition of marginal value in Theorem 5) created by usersand the bid of users (lines 6-12).

We show detailed steps of winner pricing in Alg. 4. For eachuser i, at each time step t in its working time, we first calculateSt−i, which is the set of active users in time step t excluding

user i (line 2). Then, we compute the maximum bid of user isuch that i can be selected instead of other users in St

−i in timestep t. Finally, we use the maximum bid of the user i during theworking time as the critical value of its payment (lines 3-10).

C. Theoretical Analysis of RTA Mechanism

In this subsection, we will theoretically prove that RTA has theproperties of task timeliness, computational efficiency, user ra-tionality, manager profitability, and price truthfulness. As shownin the following theorems:

Theorem 4: RTA achieves computational efficiency with atime complexity of O(nm3).

Proof: In winner selection stage, the worst case is that onlyone task is assigned in each while-loop (lines 2-14 of Alg. 3).Calculating the active users set St takes O(n) time (lines 3-5



of Alg. 3). It takes O(nm) to compute the user with the largestmarginal value. Because the number of tasks is m, each winnergets assigned at least one task, and the maximum number ofwinners is m. Thus, the time complexity of winner selectionstage is O(nm2). When calculating the payments for each useri, the worst case is to re-select the winners from all usersexcluding user i (this process is similar to Alg. 3), and it takesO(nm2) time. Since the maximum number of winners is m, thefor-loop (lines 3-6 of Alg. 2) takes O(nm3) time. To sum up,the time complexity of RTA is dominated by this for-loop, andis O(nm3). �

Theorem 5: RTA is rational for all users.Proof: From the line 8 of Alg. 3, if i can become winner,

there must be vi(S) ≥ bi, where vi(S) is the new added utilitycreated by user i for the manager on the basis of the selectedwinners set S, that is, vi(S) =

∑τj∈Γi\∪k∈SΓk

vτj . We call vi(S)themarginal value created by the user i to the manager. From line6 of Alg. 4, we can get pi ≥ min{vi − (vij − b

ij), vi}, where

user ij is the j-th winner after removing user i at t ∈ [ti, t′i]. At

this point, we consider two Cases:Case 1: If ii exists, then we have pi ≥ min{vi − (vii −

bii), vi}. Because the order of winners is determined by the

Greedy method, we have vi − bi ≥ vii − bii

. So, bi ≤ vi −(vii − b

ii). Then we can get bi ≤ min{vi − (vii − b

ii), vi}

≤ pi.Case 2: If ii does not exist, it means that the user who replaces

user i at the i-th position does not exist when user i is removed,that is, the condition of line 3 of Alg. 4 is not satisfied. Accordingto line 10 of Alg. 4, we have bi ≤ vi = pi. �

Therefore, RTA has been proved to be rational for users.Theorem 6: RTA can ensure manager profitability.Proof: As can be seen from Alg. 3, each user i ∈ S sat-

isfies bi ≤ vi. Therefore, the utility of the manager is U =∑i∈S (vi − pi). Next, we prove that pi ≤ vi is satisfied for any

user i ∈ S.Firstly, it can be concluded from line 6 of Alg. 4, that is:

pi = max

{

max1≤j≤k

(vi(j) −

(vij − bij

)), vi(k+1)

}

(19)

where ij is the j-th winner after removing user i in t ∈ [ti, t′i],

and k is the number of winners. vi(j) represents the marginalvalue of user i at j-th position. Then we consider the followingtwo cases:

Case 1: If ij exists, we assume that after removing user i, ther − th winner in t ∈ [ti, t

′i] satisfies:

r = arg maxj∈St

−i\Ctmin{vi(j) − (vij − b

ij), vi(j)} (20)

Further, we have:

pi = min{vi(r) − (vir − bir), vi(r)} (21)

From the lines 4-8 of Alg. 4, we have vir > bir

. Because theorder of winners is determined by the Greedy method, we alsohave vi(r) ≤ vi. Moreover, we can conclude that pi = vi(r) −(vir − b

ir) < vi(r) ≤ vi

Case 2: If ij does not exist, it means that the user who replacesuser i at the j-th position does not exist when user i is removed,

that is, the condition of line 3 of Alg. 4 is not satisfied. Accordingto line 10 of Alg. 4, we have vi = pi.

Thus, the manager profitability of RTA is guaranteed. �Theorem 7: RTA is truthful for all users.Proof: We show that any user i can’t make more utility by

submitting a false working time [ti, t′i] or a false bid bi.

We first demonstrate that the winner selection rule of RTA ismonotonic. If a false working time window [ti, t

′i] submitted by

the user i is narrower than the true working time window [ti, t′i],

the user i may lose the chance to be selected as the winner. Atthe same time, the marginal value generated by submitting [ti, t′i]will not be greater than that generated by submitting [ti, t

′i]. If

the [ti, t′i] submitted by the user i is wider than [ti, t

′i], when

user i is selected to perform the task in t ∈ [ti, t′i]\[ti, t′i], then

the cost is ci = ∞ and the payment is pi = 0. Then, we canconclude that by submitting a false working time [ti, t

′i], the

user i can neither be selected as a winner in advance nor increaseits payment. In addition, as we can see from Alg. 3 that in thesame time step t, the value of vj

(S)− bj for each user j will

increase monotonously with a decreased bid bj . Therefore, thewinner selection rule of RTA is monotonic.

Next, we show that the payment to each winner is the criticalvalue. User i is selected as the winner by submitting bid bi intime step ti, and the payment is pi. If user i submits a false bid biand bi > pi, then during the period of t ∈ [ti, t

′i), user i will not

be selected as winner. This is because the winner selection ruleof RTA is monotonic and vi − bi < vi − bi. From Eq. (19), thereis always a user who satisfies vi − bi < vij − b

ijin t ∈ [ti, t

′i].

Conversely, if bi ≤ pi, the user i will be a winner in t ∈ [ti, t′i].

Hence, the payment pi is a critical value for user i to be winner.To sum up, we can conclude that RTA can ensure truthfulness

for all users. �Theorem 8: RTA is timely for all tasks.Proof: From Theorem 7, we conclude that the winner selec-

tion rule of RTA is monotonic. In addition, from line 8 of Alg. 3,we can know that in each time step t, as long as the candidateuser meets the criteria vi ≥ bi, it can become a winner, and theselected tasks can be executed in time at t. That is to say, RTAguarantees task timeliness to the greatest extent on the basis ofplatform profitability. �

VI. PERFORMANCE EVALUATION

In this section, we conduct extensive simulations to validatethe performance of our proposed models and mechanisms.Firstly, for homogeneous task model, we compare the numberof winners, the number of tasks executed and the user utilitywhile using the optimal strategy and HTIM, respectively, andwe compare the approximation between the optimal strategy andHTIM mechanism. Secondly, for heterogeneous task model, weanalyze the speed of assigning tasks, the utility of user and theutility of manager in RTA. In addition, we compare RTA withLSB mechanism [30] and SMS mechanism [38] to illustratethe advantages of RTA. Finally, we validate the adaptabilityand effectiveness of the HTIM and RTA in large-scale networkscenarios.



TABLE IIITHE DEFAULT VALUES OF BASIC PARAMETERS

A. Experimental Scenario Setting

In order to get more objective and realistic experimentalresults, we carry out experiments on the real Twitter dataset.This dataset has the attributes of nodes and the relationshipsamong nodes in Twitter network [44]. In the original Twitterdataset, we selected a connected subset of 10,000 users andtheir relationships to conduct our experiments. It is worth notingthat the above users are not deliberately selected, hence theexperimental results are without loss of generosity.

In the homogeneous task model, we consider that the costfor sending messages is related to the degree of a user onthe social network graph (i.e. the number of neighbors in anundirected graph), and the higher the degree of users is, thelower the corresponding cost will be. So, let’s assume thatγi = b/ ln (1 + λi) + η, where λi is the degree of user i on thesocial network graph, the term ln (1 + λi) is used to smooththe degree differences among users. And b is a constant thatdetermines the magnitude of the γi, and η ∈ [0, 1] is a randomvalue to add a perturbation to the γi. Note that we do not imposea strong assumption on the γi, and any other forms of functionsthat can reflect the above relationship could also be used here toderive similar results.

In addition, in the heterogeneous task model, we have thefollowing settings: 1) The values of tasks are distributed evenlyin [vmin, vmax]. 2) The working time of users is distributedevenly in [tmin, tmax] and the lengths of users’ working timeare distributed evenly in [tshort, tlong]. 3) The number of tasksselected by users is evenly distributed in [1, kmax]. 4) Thecost of user i for executing the selected tasks set is ci =(1 − σlog(1+λmax) (1 + λi)

)∑τj∈Γi

vτj , in which vτj is thevalue of task τj , λi is the degree of user i on the social networkgraph, and λmax is the maximum degree of users. The termlog(1+λmax) (1 + λi) is used to smooth the user’s degree andget a normalized result, and σ ∈ [0, 1] is a constant used todetermine the magnitude of the costs of all users. Similarly, theform of ci we present here is not imposed by a strong assumptioneither. Besides, we set the default values of basic parameters asmentioned in Table III.

B. Simulation Results of Homogeneous Tasks

Number of winners: Fig. 3 shows the relationships betweenthe number of winners and the manager’s budget or parameterb under the optimal strategy and HTIM. As we can see, withthe increase in the budget, more and more users will become

Fig. 3. The number of winners vs. the total budget.

Fig. 4. The number of tasks performed vs. the total budget.

winners. In addition, as the value of b increases, the cost of usersperforming tasks increases, resulting in fewer winners. There-fore, we can conclude that the larger the total budget or the lowerthe cost of users performing tasks, the more likely users willbecome winners. It is worth pointing out that the uneven trendof the number of winners is due to the nonuniform distributionof users’ degree in the Twitter dataset. More importantly, bycomparing Fig. 3(a) with Fig. 3(b), we can see that the numberof winners determined by the optimal strategy and HTIM is veryclose in different scenarios. It is proved that the proposed HTIMand the optimal strategy have a high approximation.

Number of tasks performed: Fig. 4 shows the relationshipsbetween the number of tasks performed and the manager’sbudget or parameter b under the optimal strategy and HTIM. Wecan see that the number of tasks performed increases with anincrease in the budget or a decrease in the value of b. Therefore,if the manager’s budget is bigger or each user’s cost is lower, themore users are motivated to perform more tasks. By comparingFig. 4(a) with Fig. 4(b), it can be found that the number of tasksin HTIM is slightly lower than that in the optimal strategy. Thereason is that HTIM adopts rounded down method on the basisof the optimal strategy solution and abandons some tasks.

User utility: We randomly select three users (IDs are 8, 398and 2026) to explore the relationship between the utility of userand the manager’s budget under the optimal strategy and HTIM,as shown in Fig. 5. The relationship of the selected three usercosts is c8 < c398 < c2026. With the increase of the budget R, theutilities of three users keep increasing. In addition, we can alsodetect that the user’s utility in HTIM is slightly greater than thatin the optimal strategy, because a small part of tasks is abandonedin HTIM mechanism, resulting in fewer total number of tasksto be performed, so the payment for a single task will increase,and the user’s utility will increase accordingly.



Fig. 5. The user utility vs. the total budget.

Fig. 6. The utility of user vs. the number of tasks.

Fig. 7. The number of tasks assigned vs. time steps.

User strategy: We randomly select two users (IDs are 3121and 963) to analyze the relationship between user utility andthe number of tasks performed with a fixed budget and fixedother users’ strategies. Users 3121 and 963 are both winners.From Fig. 6, we can see that with the increase of the number oftasks performed, the utility of the corresponding user increasesfirst and then decreases. We can then verify that when thebudget and other users’ strategies are fixed, the user can getthe greatest utility by adopting the optimal strategy (45 and57, respectively). Changing the strategy, whether increasing orreducing the number of tasks, will lead to a decrease in the utility.In addition, the results in the HTIM are very close to those inthe optimal strategy, which proves that the optimal strategy weproposed can accurately calculate the optimal strategy for eachuser. The proposed HTIM can make the user’s utility very closeto the theoretical optimal value under constraints.

C. Simulation Results of Heterogeneous Tasks

Speed of assigning tasks: Fig. 7 shows the trend of the numberof tasks assigned over time steps in RTA and SOS. As can beseen, the number of tasks assigned in both mechanisms increasesover time steps. Obviously, the number of tasks assigned in RTAis growing faster than that in SOS. Therefore, we can conclude

Fig. 8. The bid of user vs. the utility of user.

Fig. 9. The utility of the manager vs. the number of tasks.

that RTA can assign tasks to executing users more quickly. Inaddition, we can also detect that the advantage of RTA is moreobvious in scenarios with more tasks. Because RTA allocates asmany tasks as possible to active users in each time step, whileSOS only chooses one user to allocate tasks in each time step.

Utility of user: We select two users (IDs are 105 and 271) toanalyze the changes of utilities obtained by submitting differentbids. The costs of user 105 and user 271 are 56 and 47, and thevalue created by performing all tasks is 82 and 79, respectively.From Fig. 8, we can see that with the increase of bids, the utilitiesof users in LSB increase linearly, which means that users canmake more utility by submitting a bid higher than the cost.However, in RTA and LSB, when the user increases the bid,the utility will remain unchanged until it is greater than a certaincritical value. Then, the user will lose the chance to become awinner, and the utility is 0. Thus, the critical values of user 105in RTA and SOS are 66 and 81. When the bid is below the criticalvalue, user will become the winner. RTA and SOS regard 66 and81 as the payment to user 105, and the corresponding utilitiesare 10 and 25, respectively. A similar conclusion can be drawnfor user 271. Therefore, RTA and SOS are truthful.

Utility of the manager: Fig. 9 shows the relationship betweenthe utility of the manager and the number of tasks. As the numberof tasks increases, the utilities of the manager are increasing inall three mechanisms with LSB having the greatest utility. This isbecause the payments to users in LSB are determined by the bidsof users, and LSB can create maximum utility for the manager ifusers’ bids equal to their costs. In RTA and SOS, the paymentsare given according to the calculated critical values, and thecritical value will not be lower than the bid, so the utilities ofthe manager in RTA and SOS are less than that in LSB. Thus,RTA and SOS can guarantee the price truthfulness at the cost ofa certain amount of manager’s utility.



Fig. 10. The performance of HTIM large-scale scenarios.

Fig. 11. The performance of RTA in large scale scenarios.

D. Simulation Results on Large-Scale Scenarios

In order to evaluate the performance of the proposed HTIMand RTA mechanisms in large-scale scenarios, we built a syn-thetic dataset as the large-scale network scenarios used in theexperiments. In the synthetic dataset, the number of users isdifferent while the other parameters are consistent with thesettings in the Subsection VI-A. When the number of networkusers varies from 10,000 to 200,000, we first investigate thenumber of winners and the number of tasks completed in HTIMmechanism, and then we simulate the number of tasks assignedby RTA mechanism. The simulation results are shown in Figs. 10and 11.

Performance of HTIM: Fig. 10(a) shows the relationshipsbetween the number of winners with the number of users andthe budget under HTIM. We can see that as the number of usersincreases, more and more users will be selected as winners. Also,an increase in the budget can lead to an increase in the numberof winners. In addition, comparing Fig. 10(a) with Fig. 3(b), wecan find that they show the same trend when the number of usersis fixed. Fig. 10(b) shows the relationships between the numberof tasks with the number of users and the budget under HTIM.Clearly, with the increase of the number of users or the budget,the number of tasks performed increases. In addition, we canstill observe that the trends in the number of tasks as budgetchanges in Fig. 10(b) and Fig. 4(b) are consistent. The aboveresults and conclusions show that our proposed HTIM can befully applicable to large-scale scenarios with satisfactory andstable performance.

Performance of RTA: Fig. 11 shows the changing trend of thenumber of tasks assigned with the number of users and time stepswhen the total number of tasks is 1,000 and 10,000, respectively.Similar to Fig. 7, the number of tasks successfully assigned toexecuting users in both scenarios increases over time steps. Inaddition, as the number of users increases, more and more usersparticipate in the task execution by submitting more bids, so the

task allocation and execution speed will be faster. Importantly,we can still conclude that RTA has robust performance in large-scale scenarios and can still assign tasks to users faster.

VII. CONCLUSION

In this paper, we propose two effective incentive mechanismsto encourage users to actively participate in the task of blockingthe spread of rumor information in MSNs. For homogeneouscontrol tasks, we focus on user fairness and propose an incentivemechanism based on the Stackelberg game. For heterogeneoustasks, we focus on user autonomy and timeliness of task alloca-tion, and propose an incentive mechanism based on the real-timeonline reverse auction. This paper can provide theoretical basisand feasible technical approaches for the user motivation or thetask allocation in rumor blocking scenarios. The proposed meth-ods can motivate more users to participate in the execution oftasks independently, so the advantage of these mechanisms willbe more prominent in large-scale networks. In addition, the pro-posed incentive mechanisms can be extended to other scenarios,such as advertising, cooperative communication, maximizinginfluence, etc. In future works, we plan on exploring other factorssuch as the users’ interests on the impact of incentive measures.

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Yaguang Lin received the B.S. degree in computerscience and technology from Xi’an TechnologicalUniversity, Xi’an, China, in 2013. He is workingtoward the Ph.D. degree with the School of ComputerScience, Shaanxi Normal University, Xi’an, China.He’s also a Visiting Ph.D. Student with the Depart-ment of Computer Science, Georgia State University,Atlanta, GA, USA, from 2018. His research interestsinclude information diffusion, rumor blocking, andmobile social networks.

Zhipeng Cai received the B.S. degree from the Bei-jing Institute of Technology, Beijing, China, and theM.S. and Ph.D. degrees in computing science fromthe University of Alberta, Edmonton, AB, Canada. Heis currently an Associate Professor with the Depart-ment of Computer Science, Georgia State University,Atlanta, GA, USA. His research interests includenetworking, privacy, and big data. He was the recip-ient of NSF CAREER Award. He is an Editor/GuestEditor for the IEEE TRANSACTIONS ON KNOWLEDGE

AND DATA ENGINEERING, the IEEE TRANSACTIONS

ON VEHICULAR TECHNOLOGY, the IEEE NETWORKING LETTERS, Algorith-mica, Theoretical Computer Science, the Journal of Combinatorial Optimiza-tion, and the IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND

BIOINFORMATICS.

Xiaoming Wang received the Ph.D. degree incomputer software and theory from Northwest Uni-versity, Xi’an, China, in 2005. He was a SeniorVisiting Scholar with the Department of ComputerScience, Georgia State University, USA, from 2007to 2008. He is currently a Professor and the Deanwith the School of Computer Science, Shaanxi Nor-mal University, Xi’an, China. He has served as areviewer/editor for many journals. He authored orcoauthored three books and more than 100 papers. Hismain research interests include information diffusion,

mobile social networks, social computing, and ubiquitous computing.

Fei Hao received the Ph.D. degree from Soonchun-hyang University, Asan, South Korea, in 2016. Heis currently an Associate Professor with the Schoolof Computer Science, Shaanxi Normal University,Xi’an, China. His research interests include socialcomputing, ubiquitous computing, and mobile cloudcomputing. He was the recipient of five best paperawards from KISM 2012, GreenCom 2013, MUE2015, UCAWSN 2015, and CUTE 2016. He is arecipient of the IEEE Outstanding Leadership Awardat CPSCom 2013 and the IEEE Outstanding Service

Awards at SmartData 2017 and DSS 2018.


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