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

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Outline

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

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Non-cooperative Game v.s. Network Utility Maximization

• Network utility maximization (NUM)o Users are altruistic, with the same objective of maximizing the total

utility of all userso Extensively studied for network resource allocation

• Non-cooperative game (NCG)o Each user is selfish, aiming to maximize its individual utilityo Widely applied in networking field to model strategic interaction

among autonomous network entities

• NCG v.s. NUM are two extreme cases: socially oblivious v.s. fully social-ware

Question: What is between these two extremes?

Mobile Social Network• Mobile social network

o Hand-held mobile devices are operated by human beingso People have diverse social relationships and care about their social

neighbors at different levels (e.g., family, friends, acquaintances)o New framework between NCG and NUM is needed

• Social network overlaying mobile networko Physical domain: physical coupling based on physical relationshipso Social domain: social coupling due to social ties among users

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Social Group Utility Maximization Framework • Social graph model

o Two users are connected by a directed edge if one has social tie towards the other

o : strength of the social tie from user to user with o The social tie strength of user to itself is ,

• Social group utility maximization game (SGUM)o User are playerso : user ’s strategy, : all users’ strategies except user ’so : individual utility of user o : social group utility of user

o Each user aims to maximize its social group utility

𝑓 𝑖(𝑥 𝑖 , 𝒙− 𝑖)≜∑𝑗=1

𝑁

𝑎𝑖𝑗𝑢 𝑗 (𝑥 𝑗 , 𝒙− 𝑗)  

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Social Group Utility Maximization Framework • Social-aware Nash equilibrium (SNE)

o 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 caseso If no social tie exists (i.e., ), SGUM degenerates to NCG as o is a Nash equilibrium (NE) if no user can improve its individual utility by unilaterally changing its strategyo If all social ties have the maximum strength (i.e., ), SGUM degenerates to NUM

as o is network optimal (NO) if it maximizes the network utility

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Related Work

• SGUM is different from cooperative game (CG)o Each user in a CG only cares individual utility, although it is achieved

through cooperation with other userso A user in a CG can only participate in one coalition, while it can be in

multiple social groups under SGUM

• Little attention paid to the continuum space between NCG and NUMo Routing game among altruistic users [Chen et al, 2008] [Hoefer et al,

2009], random access game between two symmetrically altruistic users [Kesidis et al, 2010]

• Explore social aspects in networkingo 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]

1 134

522

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5coalitions in a CG: {1,2,3}, {4,5}

social groups under SGUM: {1,2,3},{3,4,5}

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Random Access Control Model• Protocol interference model

o Each user is a link consisting of a transmitter and a receiver o causes interference to if is in the interference range of o : set of the receivers that causes interference too : set of the transmitters that causes interference to

• Random access control modelo Each user decides access probability to contend for data transmission o If multiple users contend, a collision occurs and no user can grab the

transmission opportunity

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Random Access Control Game under SGUM• Random access control game under SGUM:

o : the successful contention probability of user

o User ’s individual utility o : user ’s efficiency of utilizing the transmission opportunity (e.g.,

transmission rate)o The log function is widely used to model utility of wireless users

𝑏𝑖 (𝑞𝑖 ,𝒒−𝑖)=𝑞𝑖 ∏𝑗∈ 𝑰 𝑖

−

(1−𝑞 𝑗)

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

Random Access Control Game under SGUM

LEMMA 1: The SNE strategy is decreasing in .

LEMMA 2: The network utility at the SNE is increasing in , and is optimal when .

• Remarko 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

• Remarko 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

Random Access Control Game under SGUM

• Remarko 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 NUM SGUM spans the continuum space between NCG and NUM

• An example of two-user game with

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Power Control Model• Physical interference model

o Each user is a link consisting of a transmitter and a receiver o : transmission channel gain of link o : interference channel gain from to o : noise at

• Power control modelo Each user decides transmit power of o : signal-to-interference-plus-noise ratio (SINR)

𝛾𝑖 (𝑝𝑖 ,𝒑−𝑖)=h𝑖𝑝𝑖

𝑛𝑖+∑𝑗≠ 𝑖

𝑔𝑖𝑗𝑝𝑖

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Power Control Game under SGUM• Power control game under SGUM:

o User ’s individual utility o : user ’s cost per unit power consumptiono 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.

• Remarko The game is supermodular if o Since the game is supermodular, each user can update its strategy with

best response from , such that it will monotonically converge to the SNE

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Power Control Game under SGUM• We focus on two-user power control game under SGUM

o Provide useful insight into the impact of social tieso 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 .

• Remarko Each user decreases the SINR of another user if it increases its transmit

power user decreases when the social tie increases

Power Control Game under SGUM

LEMMA 4: The network utility at the SNE is increasing in and , and is optimal when .

• Remarko 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

o SGUM spans the continuum space between NCG and NUM

• An example of two-user game with

Conclusion• Contribution

o Developed social group utility maximization (SGUM) framework that bridges the gap between non-cooperative game and network utility maximization, two traditionally disjoint paradigms

o 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

o 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 worko SGUM provides rich modeling flexibility by spanning the continuum

space between NCG and NUMo Study SGUM game for more applications (e.g., spectrum access) and

investigate the impact of social ties on different performance metrics (e.g., fairness)