From ocial Trust Assisted eciprocity (TA) to Utility-Optimal Mobile Crowdsensin Xiaowen Gon, Xu Chen, Junshan Zhan, H. Vincent Poor chool of Electrical, Computer and Enery Enineerin Arizona tate University, Tempe, AZ 8587 Email: {xon9, xchen79, junshan.zhan}@asu.edu Department of Electrical Enineerin Princeton University, Princeton, NJ 08544 Email: poor@princeton.edu Abstract Crowdsensin has been widely reconized as a promisin paradim for numerous applications in mobile networks. To realize the full benefit of crowdsensin, one fundamental challene is to incentivize users to participate. In this paper, we leverae social trust assisted reciprocity (TA), a syneristic marriae of social trust and reciprocity, to develop an incentive mechanism in order to stimulate users participation. We investiate thorouhly the efficacy of TA for satisfyin users sensin requests, for a iven social tie structure amon users. pecifically, we first show that all requests can be satisfied if and only if sufficient social credit can be transferred from users who request more sensin services than what they can provide to users who can provide more than what they request. Then we investiate utility maximization for sensin services, and show that it boils down to maximizin the utility of a circulation flow in the combined raph of the social raph and request raph. Accordinly, we develop an alorithm that iteratively cancels cycles of positive weihts in the residual raph, and thereby finds the optimal solution efficiently. I. INTODUCTION Mobile crowdsensin has recently emered as a promisin paradim for a variety of applications, thanks to the pervasive penetration of mobile devices in people s lives []. Althouh the benefit of crowdsensin is pronounced, a user would not participate in sensin without receivin adequate incentive. Therefore, effective incentive desin is essential for realizin the benefit of crowdsensin. ecent studies on incentive desin for crowdsensin (e.., [] [4]) mostly use monetary reward to stimulate users participation, which rely on a lobal (virtual) currency system that typically incurs sinificant implementation overhead. Therefore, it is appealin to desin a crowdsensin system that can motivate users to participate without usin a lobal currency, which is a subject of this study. Mobile users behaviors are increasinly influenced by their social relationships, mainly due to the rapid rowth of online social networkin services [5]. As an important aspect of social relationship, social trust can be exploited to stimulate crowdsensin: if Alice has social trust in Bob, then Alice is willin to help Bob since Alice can trust Bob, in the belief that Bob would help Alice in the future to return the favor. (a) (b) (c) (d) 4 4 (e) (f) Fi.. Examples of social trust assisted reciprocity cycles. (a)-(d) are special cases: (a) direct reciprocity cycle; (b) indirect reciprocity cycle; (c) direct social trust based cycle; (d) indirect social trust based cycle. olid edes are social edes. Dashed edes are request edes. In this paper, we devise an incentive mechanism to stimulate users participation in crowdsensin, by usin ocial Trust Assisted eciprocity (TA) - a syneristic marriae of social trust and reciprocity. The basic idea of TA is that Alice is willin to help Bob if someone who trusts Bob can help someone trusted by Alice (as illustrated in Fi. ). This is because that the overhead of Alice for helpin Bob is compensated, as the one trusted by Alice will help Alice in the future to return the favor. By takin advantae of reciprocity ( synchronous exchane ) with the assist of social trust ( asynchronous exchane ), TA can efficiently encourae users participation in crowdsensin. Furthermore, compared to traditional currency-based schemes, TA can incur a much lower implementation overhead due to the use of the existin social trust. The main thrust of this study is devoted to characterizin the fundamental performance of TA, particularly for satisfyin users sensin requests iven the social trust structure amon them. We are interested in answerin two important questions: What are the conditions under which all requests can be satisfied? What is the maximum utility that can be achieved by provided service? These two questions are similar in spirit to admission control and network utility maximization, respectively. We summarize the main contributions of this paper as follows. We desin TA, an incentive mechanism which stimulates users participation by usin social trust assisted reciprocity (TA). We investiate thorouhly the efficacy of TA for satisfyin users sensin requests, for a iven social trust structure amon
users. pecifically, we first show that all requests can be satisfied if and only if users who request more sensin service than what they can provide can transfer sufficient social credit to those users who can provide more than what they request. Then we investiate utility maximization for sensin service, and show that this problem is equivalent to maximizin the utility of a circulation flow in the combined social raph and request raph. Based on this observation, we develop an alorithm that iteratively cancels the cycles of positive weihts in the residual raph, and thereby finds the optimal solution efficiently. The rest of this paper is oranized as follows. In ection II, we desin an incentive mechanism based on social trust assisted reciprocity (TA). Based on TA, ection III investiates the conditions for satisfyin all sensin requests and the utility maximization for sensin service. The paper is concluded in ection IV. II. TA: OCIAL TUT AITED ECIPOCITY BAED INCENTIVE MECHANIM A. ystem Model We consider a crowdsensin system consistin of a set of mobile users V = {,, N}. Each user can request a certain amount of sensin service from another user. We model users sensin requests by a request raph G (V, E ), where user i and user j are connected by a directed request ede e ij E if user j requests sensin service from user i. The capacity ij > 0 of each request ede e ij represents the amount of service requested by user j from user i. The flow fij > 0 on the request ede e ij represents the amount of service provided by user i to user j. A user obtains utility from its requested sensin service, which depends on the amount of service it receives from each user requested by her. We assume that user j obtains a utility of U ij for each unit amount of service provided by user i. We further assume that a user s utility is equal to the total utility of the service provided to her. More complex forms of utility will be studied in future work. We model the social trust structure amon users by a social raph G (V, E ), where user i and user j are connected by a directed social ede e ij E if user j has social trust in user i. The capacity ij > 0 of each social ede e ij represents the social credit limit, which specifies the maximum amount of social credit that can be transferred from user i to user j. The flow fij on the social ede e ij represents the amount of social credit transferred between user i and user j. Note that fij = f ji holds for each pair of social edes between two users, where fij > 0 or f ji > 0 indicates that credit of amount fij or f ji is transferred from user i to user j or from user j to user i, respectively. For brevity, we use sensin service and service interchaneably throuhout the paper. (a) (c) (b) Fi.. An example of (a) the social raph; (b) the request raph; (c) the combined social and request raph; (d) two TA cycles in the socialrequest raph. B. Desin Description The basic incentive structure of the TA mechanism is a social trust assisted reciprocity cycle (TA) in which a set of users have incentive to provide service. It is defined in the combined social and request (social-request) raph G (V, E E ) (as illustrated in Fi. ). Definition : A social trust assisted reciprocity cycle is a directed cycle in the social-request raph G. In a TA cycle, a user is willin to provide service since the overhead is compensated by receivin credit or service from another user in the cycle. For each user in a TA cycle, the amount of service or credit it receives should be equal to that of service or credit it provides or spends, respectively. Let f c denote a balanced flow alon a TA cycle c, which has the same flow value on each ede in c. The flow on a social or request ede in the areate flow f of a set of balanced flows {f c, c C} alon cycles C is iven by X X X fij = f c f c, fij = f c c C:e ij c c C:e ji c c C:e ij c respectively. Note that the credit transferred from user i to j (i.e., the flow on e ij E ) in the balanced flow alon a TA cycle can be partly or completely canceled by that from user j to i in another TA cycle. Users can participate in a set of balanced flows alon TA cycles if and only if the areate flow satisfies the capacity constraints on request and social edes. Definition : A set of balanced flows alon TA cycles is feasible if the areate flow satisfies the followin capacity constraints: ji fij ij, fji = fij, e ij E () 0 f ij ij, e ij E. () Under the TA mechanism, all users are willin to participate in any feasible set of balanced flows alon TA cycles. III. EXPLOITING TA TO ATIFY ENING EQUET A. atisfyin All ensin equests Based on TA cycles, we first show that it suffices to focus on circulation flows in the social-request raph (d)
s P P P P 4 Fi.. The extended social raph constructed from the social raph in Fi. where P > 0, P = 0, P < 0, P 4 < 0, P 5 = 0, P > 0. The notation next to an ede is its capacity. defined as follows. Definition : A flow f in the social-request raph G is a circulation if f satisfies the capacity constraints (), (), and the flow Xconservation X constraints X fij + fij = fji, i V. () j:e ij E j:e ij E j:e ji E It is clear that the areate flow of any feasible set of balanced flows alon TA cycles is a circulation flow in G. The followin lemma shows that the converse is also true. Lemma : Any circulation flow in the social-request raph amounts to the areate flow of a feasible set of balanced flows alon TA cycles. Due to space limitation, all the proofs of this paper are iven in our online technical report []. We define P i as the total amount of service requested by user i minus the amount that user i can X provide: X P i ji ij. j:e ji E j:e ij E Then we construct an extended social raph G + from the social raph G by addin a directed ede with capacity P i from a virtual source node s to each node i with P i > 0, and addin a directed ede with capacity P i from each node i with P i < 0 to a virtual destination node t (as illustrated in Fi. ). X Let P be defined X as P P i = P i. i:p i >0 i:p i <0 Theorem : All sensin requests can be satisfied under TA if and only if P is equal to the maximum flow value from s to t in the extended social raph G +. emark : Theorem provides a useful insiht: all requests can be satisfied if and only if users who request more service than what they can provide can transfer sufficient social credit to users who can provide more than what they request, to compensate their imbalance in requests. Intuitively speakin, the social raph serves as a buffer to partially or completely absorb the mismatch amon users requests. It is worth notin that the maximum amount of service provided under TA is in eneral not equal to the maximum flow value from s to t in G +. emark : We note that an important difference between [7] and our study is that the results in [7] is based on the assumption that all users are connected in the social network, whereas our model here does not have this assumption. This is essentially because that reciprocity is t used in TA but not in [7]. B. Utility Maximization for ensin ervice Due to the mismatch of sensin service requests and social credit limits, it is possible that not all requests can be satisfied. In this case, a natural objective is to maximize the total utility of provided service. The next result follows from Lemma. Theorem : The maximum utility of sensin service provided under TA is equal to the maximum utility of a circulation flow in the social-request raph. Note that the flow on a social ede does not enerate any utility. By Theorem, Xour problem can be written as maximize U ij f fij,f ij ij (4) i,j:e ij E subject to constraints (), (), (). Note that we can maximize the total amount of service provided under TA by solvin problem (4) with the utility U ij set to for each request ede e ij. In the followin, we will solve problem (4) usin an alorithm inspired by the cycle-cancelin alorithm for solvin the minimum cost flow problem [8]. We should note that problem (4) is very different from a typical network flow problem in that two nodes can be connected by multiple edes (request edes and social edes). Furthermore, request edes and social edes carry different types of flows: the flows on all request edes are non-neative and independent (as in constraint ()), while the flows on social edes can be neative and must be inverse between a pair of users (as in constraint ()). We start with constructin a residual raph G f (V, Ef E f ) of the social-request raph G for a iven flow f. pecifically, for each request ede e ij E, we construct a forward ede e ij E f and a backward ede e ji Ef with capacity ij = ij fij, ij = fij respectively. For each pair of social edes e ij, e ji E, we construct a pair of edes e ij, e ji E f with capacity ij = ij fij, ji = ji fji respectively. We do not construct an ede in the residual raph if its capacity is zero. Then we set the weiht of each forward ede e ij E f and each backward ede e ji Ef as W ij = U ij, W ij = U ij respectively. The weihts of each pair of edes e ij, e ji E f are set to W ij = W ji = 0. The followin lemma establishes the optimality condition for solvin problem (4). Lemma : A flow f is optimal for problem (4) if and only if there exists no cycle of positive weiht in the residual raph G f.
() (0) () (0) (0) (0) (0) (0) (0) (0) 4 4 () () 4() () ( ) 4(0) (a) 4 () ( ) () ( ) (c) (0) (0) 4() ( ) ( ) 4(0) (b) 4 ( ) ( ) () () (d) (0) (0) Fi. 4. An example of runnin Alorithm. (a) Initial social-request raph with the empty flow; (b) esidual raph after aumentin with a flow of value alon cycle 4 ; (c) esidual raph after aumentin with a flow of value alon cycle ; (d) esidual raph after aumentin with a flow of value alon cycle 4. For each ede, the number before () is its capacity; the number in () is its weiht. Usin Lemma, we can develop an alorithm as described in Alorithm to solve problem (4). The alorithm starts with the empty flow in the network. It iteratively searches for a cycle of positive weiht in the residual raph and cancels the cycle by aumentin the current flow in the raph with a balanced flow alon the cycle, until no cycle of positive weiht exists. In each iteration, the value of the flow to aument with is set to be the residual capacity of the cycle, which is the minimum capacity of all edes in that cycle. We show how Alorithm works by an illustrative example in Fi. 4. As for step in Alorithm, we can use an alorithm similar to the Bellman-Ford alorithm [9] to find a cycle of positive weiht in the residual raph, if there exists one. For ease of exposition, we will focus on problem (4) with rational parameters: the utilities and capacities of all social and request edes are rational numbers. This settin is of important interest in eneral, since the parameters of most practical problems are rational numbers. Then problem (4) with rational parameters can be equivalently converted to one with interal parameters by multiplyin with a suitably lare inteer K. The solution of the oriinal problem (with rational parameters) is equal to the solution of the new problem (with interal parameters) divided by K. For problem (4) with rational parameters, let U and denote the maximum utility and maximum capacity of a request ede, respectively (i.e., U = max eij E U ij, = max eij E ij). The followin theorem shows that Alorithm is correct and efficient. Theorem : For problem (4) with divisible sensin service and rational parameters, Alorithm finds the optimal flow and has runnin time O( V E ( E + Alorithm : Find the optimal flow for problem (4) in social-request raph G input : ocial-request raph G output: The optimal flow for problem (4) Initialize an empty flow f in G; while There exists a cycle of positive weiht in the residual raph G f of flow f do Find a cycle c of positive weiht in G f ; 4 Compute the residual capacity r c of cycle c; 5 Aument flow f with a balanced flow of value r c alon cycle c; end 7 return Flow f; E )UK ). emark : It is worth notin that, when sensin service is indivisible such that its amount has to be an inteer, problem (4) is essentially an inteer linear proram, which is NP-hard to solve in eneral. In this case, we can still use Alorithm to find the optimal flow for problem (4). In other words, usin a network flow approach, we can capture and exploit the specific combinatorial structure of problem (4), based on which a polynomial-time alorithm can be developed to solve it. IV. CONCLUION In this paper, we have desined TA, an incentive mechanism usin a syneristic marriae of social trust and reciprocity. Based on the TA mechanism, we have established the conditions under which all sensin requests can be satisfied. We have also developed an efficient alorithm to maximize the utility of sensin service provided under TA. An interestin research direction underway is to leverae the social tie structure amon mobile users to stimulate their altruistic behaviors for cooperative networkin. EFEENCE []. K. Ganti, Y. Fan, and L. Hui, Mobile crowdsensin: current state and future challenes, IEEE Communications Maazine, pp. 9, 0. [] D. Yan, G. Xue, X. Fan, and J. Tan, Crowdsourcin to smartphones: incentive mechanism desin for mobile phone sensin, in ACM MOBICOM 0. [] T. Luo and C.-K. Tham, Fairness and social welfare in incentivizin participatory sensin, in IEEE ECON 0. [4] I. Koutsopoulos, Optimal incentive-driven desin of participatory sensin systems, in IEEE INFOCOM 0. [5] emarketer: ocial networkin reaches nearly one in four around the world. [Online]. Available: http://www.emarketer.com/article/ocial- Networkin-eaches-Nearly-One-Four-Around-World/00997 [] X. Gon, X. Chen, J. Zhan, and H. V. Poor, From social trust assisted reciprocity (TA) to utility-optimal crowdsensin in mobile networks, Technical eport. [Online]. Available: http://informationnet.asu.edu/pub/ta-lobalsip4-t.pdf [7] Z. Liu, H. Hu, Y. Liu, K. W. oss, Y. Wan, and M. Mobius, PP tradin in social networks: The value of stayin connected, in IEEE INFOCOM 00. [8]. Ahuja, T. Mananti, and J. Orlin, Network flows: Theory, alorithms, and applications. Prentice Hall, 99. [9] X. Huan, Neative-weiht cycle alorithms, in 00 International Conference on Foundations of Computer cience.
Proof of Lemma APPENDIX Consider a non-empty circulation flow f. We can find a node v with a positive flow on an outoin ede from v and trace alon this ede to another node v. Due to the flow conservation constraint, we can find an outoin ede from v with a positive flow and trace alon it to a node v. We continue this tracin process until we visit a node v j that has been visited before, i.e., v i = v j for some i < j, and hence we find a TA cycle v i v i+ v j. Then we subtract flow f by a balanced flow alon this cycle with value equal to the minimum flow value on an ede in that cycle. Thus the remainin flow is still a circulation flow in which the number of edes with non-zero flows is reduced. We can repeat this arument to subtract the remainin flow by a balanced flow alon a cycle until it is empty. This implies that flow f is the areate flow of the subtracted balanced flows alon the cycles, which is also feasible. Proof of Theorem By Lemma, all requests can be satisfied if and only if there is a circulation flow f in the social-request raph G that saturates all request edes (i.e., fij = ij, e ij E ). We first show the if part. uppose is equal to the value of the maximum flow f + from s to t in G +. Let f be the flow comprised of the flows on the social edes E in f + (i.e., not includin the edes from s and to t in G + ). Let f be the flow in the request raph G that saturates all request edes. Then we aument flow f in the social-request raph G with flow f to obtain a flow f in G. Accordin to the construction of G +, we have P j:e ij E f ij = P i for each node i V, while we also have P j:e ji E f ji P j:e ij E f ij = P i. This shows that f is a circulation flow. Next we show the only if part. uppose f is a circulation flow in G that saturates all request edes. Let f be the flow comprised of the flows on the social edes E in f. Then we aument flow f with saturated flows on the edes from s and to t in G + to obtain a flow f + in G +. Accordin to the construction of G +, f + is a flow in G + satisfyin the capacity and flow conservation constraints, with a flow value of P from s to t. Proof of Lemma The only if part is easy to show: If there exists a cycle of positive weiht in G f, then we can aument the flow f with a balanced flow of value ϵ > 0 alon that cycle to construct a circulation flow with larer utility. Next we show the if part. uppose there exists no cycle of positive weiht in G f but there exists a circulation flow f in G with larer utility than f. imilar to the residual raph G f, we construct a raph G (V, E E ) from G by constructin e ij, e ij E for each e ij E and e ij, e ji E for each pair of e ij, e ji E, and settin their weihts the same as those in G f. The difference between G f and G is that all the edes are constructed in G (an ede is not constructed in G f if its capacity is 0) and have unlimited capacities. Therefore, the edes in G f is a subset of the edes in G. Then we can define a flow in G by definin the flows in on the edes of G as ij = max{0, f ij fij }, e ij E ij = max{0, fij f ij }, e ij E ij = f ij f ij, e ij E. It follows from the definition that ij ij = f ij fij, e ij E. Then Xthe net flow value at each node i V in flow is X ij + X ji + X ij X ij ji j: e ij E X j: e ji E j: e ij E j: e ij E j: e ji Š X Š X E Š = f ij fij f ji fji + f ij fij j:e ij E j:e ji E j:e ij E = X f ij j:e ij E X X Ž + f ij f ji j:e ij E j:e ji E X X X fij + fij fji = 0 j:e ij E j:e ij E j:e ji E where the last equality follows from that f and f are circulation flows in G. Therefore, is a circulation flow in G. We observe that the flow on any ede e E \ Ef is zero in because ) if e = e ij, then we have f ij = ij and hence ij = 0; ) if e = e ij, then we have f ij = 0 and hence ij = 0. We further observe that ij 0 for any ede e ij E \ Ef since we have f ij = ij. ince W ij = 0, e ij E, the weiht of flow in G is X W ij ij + W Š X Š ij ij = U ij f ij fij i,j: e ij E X i,j:e ij E X = U ij f ij U ij fij > 0 i,j:e ij E i,j:e ij E where the last inequality follows from the assumption that f has larer utility than f in G. ince only has positive flows on the edes in G f, usin a similar arument as in the proof of Lemma, is the areate flow of balanced flows alon cycles each comprised of edes in G f. Then the total weiht of these flows alon the cycles in G f is equal to the weiht of flow in G, which is reater than 0. This implies that there must exist a cycle of positive weiht in G f, which is a contradiction to the previous assumption. This completes the proof. Proof of Theorem As discussed earlier, we first equivalently convert the problem to one with interal parameters by multiplyin them by an inteer K. ince the capacities of all edes in the raph are interal Ž
and the initial empty flow is interal, the residual capacity of the cycle found in the first iteration of the alorithm is interal, and hence the flow after aumentation is interal. Thus, by induction, the updated flow after each iteration is also interal. This shows that the alorithm finds an interal flow when it terminates, which is optimal by Lemma. The utility of the initial empty flow is 0. The utility of any flow is upper bounded by the utility of the flow that saturates all request edes, which is E UK. ince the capacities of all edes are interal, the flow utility increases by an inteer no less than one at each iteration of Alorithm. Therefore, it takes the alorithm at most E UK iterations to terminate. ince each iteration has runnin time O( V ( E + E ), the desired result follows.