Social Group Utility Maximization Game with Applications in Mobile Social Networks

1. Social Group Utility Maximization Game with Applications in Mobile Social Networks Xiaowen Gong, Xu Chen, Junshan Zhang Arizona State University Allerton Conference 2013 Oct. 4th, 2013

2. Outline • Introduction • Social Group Utility Maximization Framework • Random Access Control Game under SGUM • Power Control Game under SGUM • Conclusion

3. Non-cooperative Game v.s. Network Utility Maximization • Non-cooperative game (NCG) • Each user is selfish, aiming to maximize its individual utility • Widely applied in networking field to model strategic interaction among autonomous network entities • Network utility maximization (NUM) • Users are altruistic, with the same objective of maximizing the total utility of all users • Extensively studied for network resource allocation • NCG v.s. NUM are two extreme cases: socially oblivious v.s. fully social-ware Question: What is between these two extremes?

4. Mobile Social Network • Mobile social network • Hand-held mobile devices are operated by human beings • People have diverse social relationships and care about their social neighbors at different levels (e.g., family, friends, acquaintances) • New framework between NCG and NUM is needed • Social network overlaying mobile network • Physical domain: physical coupling based on physical relationships • Social domain: social coupling due to social ties among users

5. Social Group Utility Maximization Framework • Social graph model • Two users are connected by a directed edge if one has social tie towards the other • : strength of the social tie from user to user with • The social tie strength of user to itself is , • Social group utility maximization game (SGUM) • User areplayers • : user ’s strategy, : all users’ strategies except user ’s • : individual utility of user • : social group utility of user • Each user aims to maximize its social group utility

6. Social Group Utility Maximization Framework • Social-aware Nash equilibrium (SNE) • is a SNE if no user can improve its social group utility by unilaterally changing its strategy • NCG and NUM are captured under SGUM as special cases • If no social tie exists(i.e., ), SGUM degenerates to NCG as • is a Nash equilibrium (NE) if no user can improve its individual utility by unilaterally changing its strategy • If all social ties have the maximum strength(i.e., ), SGUM degenerates to NUM as • is network optimal (NO) if it maximizes the network utility

7. Related Work • Explore social aspects in networking • Exploit social contact pattern for efficient data forwarding [Costa et al, 2008] [Gao et al, 2009], leverage social trust and reciprocity to improve D2D communication [Chen et al, 2013] • Little attention paid to the continuum space between NCG and NUM • Routing game among altruistic users [Chen et al, 2008] [Hoefer et al, 2009], random access game between twosymmetricallyaltruistic users [Kesidis et al, 2010] • SGUM is different from cooperative game (CG) • Each user in a CG only cares individual utility, although it is achieved through cooperation with other users • A user in a CG can only participate in one coalition, while it can be in multiple social groups under SGUM coalitions in a CG: {1,2,3}, {4,5} social groups under SGUM: {1,2,3},{3,4,5}

8. Random Access Control Model • Protocol interference model • Each user is a link consisting of a transmitter and a receiver • causes interference to if is in the interference range of • : set of the receivers that causes interference to • : set of the transmitters that causes interference to • Random access control model • Each user decides accessprobability tocontend for data transmission • If multiple users contend, a collision occurs and no user can grab the transmission opportunity

9. Random Access Control Game under SGUM • Random access control game under SGUM: • : the successful contention probability of user • User ’s individual utility • : user ’s efficiency of utilizing the transmission opportunity (e.g., transmission rate) • The log function is widely used to model utility of wireless users THEOREM 1: There exists a unique SNE in the random access control game under SGUM, and . • Remark: each user’s SNE strategy is a dominant strategy

10. Random Access Control Game under SGUM LEMMA 1: The SNE strategy is decreasing in . • Remark • Each user decreases the successful contention probability of any user within its interference range if it increases its access probability user decreases when the social tie increases LEMMA 2: The network utility at the SNE is increasing in , and is optimal when . • Remark • Users’ individual utilities are equally weighted in the network utility each user ’s SNE strategy becomes closer to the network optimal one when other users’ individual utilities weigh more in user ’s social group utility

11. Random Access Control Game under SGUM • Remark • As the social tie strengths increase from 0s to 1s, the SNE strategy of each player migrates from the NE strategy of a NCG to the NO strategy for NUMSGUM spans the continuum space between NCG and NUM • An example of two-user game with

12. Power Control Model • Physical interference model • Each user is a link consisting of a transmitter and a receiver • : transmission channel gain of link • : interference channel gain from to • : noise at • Power control model • Each user decides transmit power of • : signal-to-interference-plus-noise ratio (SINR)

13. Power Control Game under SGUM • Power control game under SGUM: • User ’s individual utility • : user ’s cost per unit power consumption • can be a good approximation of the channel capacity THEOREM 2: The power control game under SGUM is a supermodular game, and hence it has at least one SNE. • Remark • The game is supermodular if • Since the game is supermodular, each user can update its strategy with best responsefrom , such that it will monotonically converge to the SNE

14. Power Control Game under SGUM • We focus on two-user power control game under SGUM • Provide useful insight into the impact of social ties • The game with more users is difficult for analysis THEOREM 3: There exists a unique SNE in the two-user power control game under SGUM, and ,, where , , , . • Remark: each user’s SNE strategy is a dominant strategy LEMMA 3: The SNE strategy is decreasing in and is decreasing in . • Remark • Each user decreases the SINR of another user if it increases its transmit power user decreases when the social tie increases

15. Power Control Game under SGUM LEMMA 4: The network utility at the SNE is increasing in and , and is optimal when . • Remark • Similar to the random access control game under SGUM, the network utility improves when the other user’s individual utility weighs more in a user’s social group utility • SGUM spans the continuum space between NCG and NUM • An example of two-user game with

16. Conclusion • Contribution • Developed social group utility maximization (SGUM) framework that bridges the gap between non-cooperative game and network utility maximization, two traditionally disjoint paradigms • Showed that there exists a unique social-aware Nash equilirium (SNE) in the random access control game under SGUM, and investigated the impact of social ties on the SNE strategy and network utility • Showed that the power control game under SGUM is a supermodular game and hence has at least one SNE, and investigated the impact of social ties for the two-user case • Future work • SGUMprovides rich modeling flexibility by spanning the continuum space between NCG and NUM • Study SGUM game for more applications (e.g., spectrum access) and investigate the impact of social ties on different performance metrics (e.g., fairness)