1. A Social Group Utility Maximization Framework with Applications in Database Assisted Spectrum Access
Xu Chen Xiaowen Gong Lei Yang Junshan Zhang
School of Electrical, Energy, and Computer Engineering
Arizona State University
Today I will talk about my recent research results on mobile social networking. The title is exploit social trust for cooperative networking: A social group utility maximization framework.

2. Outline Introduction Social Group Utility Maximization Framework
Database-assisted Spectrum Access under SGUM
Conclusion and Future Work
I will first introduce the motivation of this work

3. When Mobile Network Meets Social Network
Mobile networks are ubiquitous
Mobile phone shipments is projected to reach 1.9 billion in 2014 (about 7 times that of desktop and laptop combined), mobile data traffic more than doubled in 2013
Advanced wireless technology (e.g., MIMO, OFDM), powerful wireless devices (e.g., smartphone, wearable smart devices)
Social networks shape people’s behavior
Social relationships have pervasive impact (e.g., social media, social recommendation)
Online social networks users in 2013 crossed 1.7 billion (about one quarter of world’s population)
As we know, nowadays we are surrounded by all kinds of mobile devices, smartphones, tablets, laptops. Mobile network has been very successful as shown by the number of mobile phone shipments and mobile data traffic. With advanced wireless technology such as MIMO, OFDM, we can enjoy fast connection speed almost anytime, anywhere. Our mobile devices are also becoming more and more powerful, from smartphones to smart glass and smart watch, and they can do a lot of jobs which could only be done by desktops in the past. So mobile network will be the dominant way to connect people in the world in the future.
On the other hand, social network is always a fundamental structure of human society. Social relationships impact people’s behavior in many ways, e.g., they affect how information is disseminated among people, and how people’s opinion are formed. With the rapid growth of internet, especially online social network services, social relationships would have more significant impact on people’s interactions.
So it is natural to ask how social networks can impact people’s behavior in mobile networks.

4. Social Dimension on Mobile Networking
Key observations on mobile network
Mobile devices are personal communication devices carried and operated by human beings
People have diverse social ties and care about their social neighbors at different levels (e.g., family, friend, acquaintance)
Our motivation is based on two observations. First, mobile devices are essentially personal communication devices operated by human beings. Therefore, their behavior should follow people’s desire. Second, people have diverse social ties such that a user may care about other people, and may care about different people at different levels. E.g., a user may care about his family and friend, and typically he cares his family more than his friend.
So social ties are hidden incentives which can affect users’ behavior. We want to explicitly make use of these incentives.

5. Virtual Social Network Underlays Physical Communication Network
Physical Domain 2
Physical Coupling 5 1 4 3
Social Domain 2 4 5
Social Coupling 1 3
Based on this motivation, we view a mobile network as a social network underlaying a physical communication network. In the physical domain, mobile devices have physical coupling depending their physical environment. E.g., two devices have physical coupling if one device can make interference to the other. In the social domain, mobile users have social coupling due to social ties. Two users have social coupling if they have a social tie.
We should note that the physical coupling structure can be diverse such that users have different abilities to affect each other. On the other hand, the social coupling structure can also be diverse such that users have different desires to affect each other. So how to cleverly use the diverse social coupling to improve users’ interactions in the communication network subject to diverse physical coupling? This is a challenge we should address.
Question: Can we exploit social ties among mobile users to improve the interactions of their mobile devices in communication networks? How can we leverage it cleverly?
Physical-social coupling among mobile devices
Physical domain: physical coupling subject to physical relationship
Social domain: social coupling due to social ties among users

6. From Non-cooperative Game to Network Utility Maximization
Non-cooperative game (NCG)
Each user is selfish, aiming to maximize its individual utility
Widely used to study strategic interaction among autonomous users
Network utility maximization (NUM)
Users are altruistic, aiming to maximize social welfare
Extensively studied for network resource allocation
Answer: Social group utility maximization (SGUM):
A new paradigm on mobile social networking
NCG and NUM are two extreme cases:
selfish (social-oblivious) or altruistic (fully social-aware)
Before giving our answer, let’s first look at some existing approaches. NCG is used to study the interactions of selfish users, where each user only cares about its own utility. On the other hand, NUM is suitable for network resource allocation where users are altruistic and have the same objective to maximize the social welfare. So we can see that NCG and NUM are two extreme cases where users are either social-oblivious or fully social-aware. They can not capture the case where a user may care about others, but not to the full extent, and care about different people to different extents. This diverse social coupling of mobile users lies in between these two extremes. To capture this diverse social coupling, we develop a SGUM framework, which provide a new paradigm on mobile social networking.
Question: What is between these two extremes?
Non-Cooperative Game
Network Utility Maximization
Users’ Social Awareness
Selfish
Altruistic

7. Outline Introduction Social Group Utility Maximization Framework
Database-assisted Spectrum Access under SGUM
Conclusion and Future Work
Next I will formally introduce the SGUM framework.

8. Physical Graph Model A set of mobile users 𝑁={1,2,…𝑛}
User-specific feasible strategy set 𝑋𝑖 (e.g., channel selection, power level selection)
Physical graph 𝐺 𝑝 ={𝑁, 𝐸 𝑝 }
Two users are connected by a physical edge if they have physical coupling
Capture the physical relationships among users (e.g., interference)
𝑁 𝑖 𝑝 : the set of users having physical coupling with user 𝑖
First we use a physical graph to capture the physical coupling. For a set of mobile users, each user selects a strategy from a specific set of strategies. E.g., a user may decide which channel to access from some available channels. Two users are connected in the physical graph if they have physical coupling, e.g., if one user makes interference to the other. A user’s individual utility is its own benefit, and depends on the strategies of users who have physical coupling with him. E.g., if a user’s utility is SNR or data rate, then it would depend on the total interference received from others.
Individual utility 𝑈 𝑖 (𝒙)
User’s payoff under strategy profile 𝒙 (e.g., SINR, data rate)
Depend on the physical graph (e.g., interference graph)

9. Social Graph Model Social graph 𝐺 𝑠 ={𝑁, 𝐸 𝑠 }
Two users are connected by a social edge if they have a social tie
Capture the social coupling among users (i.e., kinship, friendship)
𝑁 𝑖 𝑠 : the set of users user 𝑖 has social ties with
𝑎 𝑖𝑗 ∈[0,1]: social tie strength from user 𝑖 to user 𝑗
From individual utility to social group utility
𝑆 𝑖 𝑥 = 𝑈 𝑖 𝑥 𝑗∈ 𝑁 𝑖 𝑠 𝑎 𝑖𝑗 𝑈 𝑗 (𝑥)
user 𝑖’s individual utility
weighted sum of individual utilities of user 𝑖’s social neighbors
Next we use a social graph to capture the social coupling. Two users are connected in the social graph it they have a social tie, e.g., if they are friends or in a family. We use social tie strength to quantify how much a user cares about another. We normalize the social tie strength as between 0 and 1. As a user would cares about those who have social ties with him, we define its social group utility, as its individual utility plus the weighted sum of individual utilities of its social neighbors. We can see that each individual utility depends on the physical coupling, and the weighted sum term captures the social coupling. So social group utility captures both physical coupling and social coupling in a unified manner.

10. Social Group Utility Maximization Game
Distributed decision making among users
Each user aims to maximize its social group utility
Social group utility maximization (SGUM) game
𝑁  player set
𝑋𝑖  strategy space of player 𝑖
𝑆 𝑖 𝑥  payoff function of player 𝑖
Social-aware Nash equilibrium (SNE)
𝑥 𝑖 𝑆𝑁𝐸 = argmax 𝑥 𝑖 ∈ 𝑋 𝑖 𝑆 𝑖 ( 𝑥 𝑖 , 𝑥 −𝑖 𝑆𝑁𝐸 ) ,∀𝑖∈𝑁
( 𝑥 1 𝑆𝑁𝐸 ,…, 𝑥 𝑛 𝑆𝑁𝐸 ) is a SNE if no user can improve its social group utility by unilaterally changing its strategy
As users in general have different social groups, we formulate their decision making as a game. In the SGUM game, each user aims to maximize its social group utility. Similar to a standard non-cooperative game, we generalize the concept of NE to SNE, where no user can improve its social group utility by changing its strategy.

11. Social Group Utility Maximization
SGUM provides rich modeling flexibility
If no social tie exists (i.e., 𝑎 𝑖𝑗 =0,∀𝑖,𝑗), SGUM degenerates to NCG as 𝑆 𝑖 𝑥 𝑖 , 𝒙 −𝑖 =𝑈 𝑖 𝑥 𝑖 , 𝒙 −𝑖
If all social ties have the maximum strength (i.e., 𝑎 𝑖𝑗 =1,∀𝑖,𝑗), SGUM
becomes NUM as 𝑆 𝑖 ( 𝑥 𝑖 , 𝒙 −𝑖 )= 𝑗=1 𝑛 𝑈 𝑗 ( 𝑥 𝑗 , 𝒙 −𝑗 )
Span the continuum space between NCG and NUM
Social group utility maximization (SGUM) framework captures diverse social ties of mobile users and diverse physical relationships of mobile devices; it spans the continuum space between non-cooperative game and network utility maximization.
𝑎 21 1 2
𝑎 23
NCG
NUM
Social-aware
Altruistic
SGUM
Selfish 3 4
A primary advantage of this general framework is that it provides rich modeling flexibility. If no social ties exists, then the game is equivalent to a standard NCG. If all social ties have the same strength equal to 1, then the game is equivalent to NUM. So this general framework capture NCG and NUM as two special cases, and furthermore, it spans the continuum space between these two extremes, which are two traditionally disjoint paradigms for network optimization.
So here is the take-away message: the SGUM framework captures the diverse social ties of mobile users and the diverse physical coupling of their devices, and it turns out to span the continuum space between NCG and NUM.
𝑎 34
SGUM 1 1 2 1 2
selfish 1 1 1 1
altruistic 3 4 3 4 1
NCG
NUM

12. Related Work Explore social aspect for wireless networks
Exploit social contact pattern for efficient data forwarding [Gao et al, 2009]; leverage social trust and reciprocity to improve D2D communication [Chen et al, 2013]
Routing game among altruistic users [Chen et al, 2008] [Hoefer et al, 2009], random access game between two symmetrically social-aware users [Kesidis 2010]
SGUM game is not coalition game (CG)
Each user in a CG only cares about its own utility (though it is achieved through cooperation with others)
A user in a CG can only join one coalition, while it can be in multiple social groups in a SGUM game
There has been much effort on using social information for wireless networks, e.g., social contact pattern, social trust and reciprocity have been used. However, the idea of “social tie” has been considered in a few works, which assume altruistic or symmetrically social-aware users. These models can be captured as special cases under our framework.
We should note that SGUM game is different from coalitional game. This is because each user in a CG still only cares about its own utility. In addition, a user in a CG can only join one coalition, while a user in a SGUM game, such as user 3 in this example, can be in multiple social groups. 1 4 1 4 3 3
coalitions in a CG:
{1,2,3}, {4,5}
social groups under SGUM:
{1,2,3},{3,4,5} 2 5 2 5

13. Outline Introduction Social Group Utility Maximization Framework
Database Assisted Spectrum Access under SGUM
Conclusion and Future Work
So far we have introduced a general framework. Next we talk about how to apply this framework to study a concrete example, which is database assisted spectrum access.

14. Database Assisted Spectrum Access
FCC recent ruling on TV spectrum utilization
White-space users determine vacant channels via geo-location database
Obviate the need of spectrum sensing for individual users
As we know, DASA is a promising solution for improving spectrum utilization. In such a system, a white-space user obtains a set of vacant channels from a geo-location database. In this way, spectrum sensing is not needed for individual users, which is typically a challenging task. However, to achieve efficient spectrum sharing, one challenge is that if users access the same channel, they can make severe interference to each other. So we need cooperation incentives for users to reduce their mutual interference. Therefore, we can use social ties as the incentives, and apply the SGUM framework here.
Challenges for achieving efficient shared spectrum access
Access the same vacant channel  cause severe interference
Effective cooperation incentives for spectrum access is needed

15. SGUM-based Spectrum Access Game
A set of white-space users 𝑁={1,2,…𝑛}
Each user selects a vacant channel from a specific set 𝑋 𝑖
Physical graph 𝐺 𝑝 ={𝑁, 𝐸 𝑝 }
Two users are connected by a physical edge  they can cause interference to each other
Individual utility
𝑈 𝑖 𝑥 =− 𝑗∈ 𝑁 𝑖 𝑝 𝑃 𝑗 𝑑 𝑗𝑖 −𝛼 𝐼 𝑥 𝑖 = 𝑥 𝑗 − 𝑤 𝑥 𝑖 𝑖
Each user aims to minimize its total interference plus noise
In the SGUM-based spectrum access game, each user selects a vacant channel to access. Two users have physical coupling if they can make interference to each other. A user’s utility is defined as the total interference on its selected channel plus the noise on that channel. We assume that the interference level depends on the distances among users. Then social group utility is then defined based on individual utilities and social ties.
Social group utility
𝑆 𝑖 𝑥 = 𝑈 𝑖 (𝑥) + 𝑗∈ 𝑁 𝑖 𝑠 𝑎 𝑖𝑗 𝑈 𝑗 (𝑥)

16. SGUM-based Spectrum Access Game
THEOREM: Social group utility maximization game for database assisted spectrum access is a potential game and always admits a SNE.
Potential game: if the game has a potential function ɸ 𝑥 such that
𝑆 𝑖 𝑥 𝑖 , 𝑥 −𝑖 − 𝑆 𝑖 𝑦 𝑖 , 𝑥 −𝑖 =ɸ 𝑥 𝑖 , 𝑥 −𝑖 −ɸ 𝑦 𝑖 , 𝑥 −𝑖
Property: any strategy that locally maximizes the potential function is a Nash equilibrium
For this game, we can show that it is a potential game. A game is a potential game if any user’s payoff change by changing its strategy is equal to the change of some potential function. A good property of a potential game is it always has a NE. In particular, for this game, we can see that the potential function has an intuitive structure with two parts. The first part is due the physical coupling. The second part is due to the social coupling.
Potential function of the SGUM game
ɸ 𝑥 =− 1 2 𝑖=1 𝑛 𝑗∈ 𝑁 𝑖 𝑝 𝑃 𝑗 𝑑 𝑗𝑖 −𝛼 𝐼 𝑥 𝑖 = 𝑥 𝑗 − 𝑖=1 𝑛 𝑤 𝑥 𝑖 𝑖 − 1 2 𝑖=1 𝑛 𝑗∈ 𝑁 𝑖 𝑝 ∩ 𝑁 𝑖 𝑠 𝑎 𝑖𝑗 𝑃 𝑗 𝑑 𝑗𝑖 −𝛼 𝐼 𝑥 𝑖 = 𝑥 𝑗
Due to physical coupling
Due to social coupling

17. Distributed Spectrum Access Algorithm
How to achieve a SNE with a good social welfare?
The strategy that (globally) maximizes the potential function is appealing, but it is a combinatorial problem that is hard to solve in general
Distributed algorithm is desirable
Distributed spectrum access algorithm
Inspired by adaptive CSMA for NUM [Jiang et al, 2010]
Key idea: coordinate users’ asynchronous channel selection updates to form a Markov chain, and drive it to the stationary distribution, which asymptotically maximizes the potential function
We have shown that the games has NEs, but the social welfare of a NE can be very different from one to another. So how to achieve a SNE with a good social welfare? Intuitively, the strategy that maximize the potential function would be a good solution, but finding it is a combinatorial problem that is hard to solve in general. Another challenge is that the NE should be achieved in a distributed manner.
To this end, we develop a distributed spectrum access algorithm. It is inspired the adaptive CSMA mechanism proposed by Libin Jiang in 2010 for network utility maximization. The key idea is that we can coordinate users’ asynchronous channel selection to form a Markov chain and drive it to the stationary distribution, which can asymptotically maximize the potential function.

18. Distributed Spectrum Access Algorithm
Each user 𝑖 repeats following steps in parallel:
Compute the social group utility 𝑆 𝑖 𝑥 𝑖 , 𝑥 −𝑖 on the current channel 𝑥 𝑖 based on the individual utilities reported by social neighbors
Set a random timer following the exponential distribution
Count down until the timer expires
The algorithm works in a distributed manner. Each user repeats the following steps in parallel. The user first computes its social group utility on the current channel based on the individual utilities of its social neighbors. Then the user sets a random timer following the exponential distribution and count down the timer.

19. Distributed Spectrum Access Algorithm
Each user 𝑖 repeats following operations in parallel:
When the timer expires, choose a new channel 𝑦 𝑖 randomly
Compute the social group utility 𝑆 𝑖 𝑦 𝑖 , 𝑥 −𝑖 on the new channel 𝑦 𝑖
Decision update: stay in the new channel 𝑦 𝑖 with probability
𝑄≜ exp(𝜃 𝑆 𝑖 𝑦 𝑖 , 𝑥 −𝑖 ) max{exp(𝜃 𝑆 𝑖 𝑦 𝑖 , 𝑥 −𝑖 ),exp(𝜃 𝑆 𝑖 x 𝑖 , 𝑥 −𝑖 )} ;
or move back to the original channel 𝑥 𝑖 with probability 1−𝑄
When the timer expires, the user selects a new channel randomly, and move to the new channel with a certain probability. This probability is determined by the social group utility for the new channel and for the if the new channel is better than the current channel, the user moves to the new channel with probability 1; otherwise, the user still moves to the new channel with a non-zero probability. The idea of this design is that, to globally maximize the potential function, we need some randomness to avoid being trapped in a local maximum point.

20. Distributed Spectrum Access Algorithm
The distributed algorithm induces a Markov chain
System state: the channel selection profile of all users
Each state transition involves one user: due to the property of exponential distribution for channel update countdown
Two-user Markov chain example:
Channel of User A:
Channel of User B:
Vacant Channel Set
{1, 2}
{2, 3}
(1,2)
(1,3)
(2,2)
(2,3)
Markov Chain For Dynamic Channel Selection
Channel 1
Channel 2
The algorithm induces a Markov chain. The system state the channel selections of all users. Due to the exponential distribution of the random timer, each state transition involves only one user.
In this example of two users, each user A and B has two vacant channels for access. Initially User A is on channel 1 and user B is on channel 2. So the Markov chain is in state (1,2). Next user B moves to channel 3, so the state changes to (1,3). Then user A moves to channel 2…so on so forth. Since each user can access two channels, the Markov chain has 4 states.

21. Distributed Spectrum Access Algorithm
The distributed algorithm induces a Markov chain
System state: the channel selection profile of all users
Each state transition involves one user: due to the property of exponential distribution for channel update count-down
Two-user Markov chain example:
Channel of User A:
Channel of User B:
Vacant Channel Set
{1, 2}
{2, 3}
(1,2)
(1,3)
(2,2)
(2,3)
Markov Chain For Dynamic Channel Selection
User B Updates
Channel Selection
Channel 1
Channel 3
Then if user B updates the channel selection from channel 2 to 3, the markov chain will move to state (1,3).

22. Distributed Spectrum Access Algorithm
The distributed algorithm induces a Markov chain
System state: the channel selection profile of all users
Each state transition involves one user: due to the property of exponential distribution for channel update count-down
Two-user Markov chain example:
Channel of User A:
Channel of User B:
Vacant Channel Set
{1, 2}
{2, 3}
(1,2)
(1,3)
(2,2)
(2,3)
Markov Chain For Dynamic Channel Selection
Channel 2
Channel 3
User A Updates
Channel Selection
Then if user A updates the channel selection from channel 1 to 2, the markov chain will move to state (2,3).

23. Distributed Spectrum Access Algorithm
The distributed algorithm induces a Markov chain
System state: the channel selection profile of all users
Each state transition involves one user: due to the property of exponential distribution for channel update count-down
Two-user Markov chain example:
Channel of User A:
Channel of User B:
Vacant Channel Set
{1, 2}
{2, 3}
(1,2)
(1,3)
(2,2)
(2,3)
Markov Chain For Dynamic Channel Selection
Channel 2
User B Updates
Channel Selection
And so on so forth.

24. Distributed Spectrum Access Algorithm
The distributed algorithm induces a Markov chain
System state: the channel selection profile of all users
Each state transition involves one user: due to the property of exponential distribution for channel update count-down
Two-user Markov chain example: diagram of all feasible state transitions
User A
User B
Vacant Channel Set
{1, 2}
{2, 3}
(1,2)
(1,3)
(2,2)
(2,3)
Markov Chain For Dynamic Channel Selection
We show the diagram of all feasible state transitions as this.

25. Distributed Spectrum Access Algorithm
The distributed algorithm induces a Markov chain
System state: the channel selection profile of all users
Each state transition involves one user: due to the property of exponential distribution for channel update count-down
Two-user Markov chain example
THEOREM: The distributed spectrum access algorithm induces a time-reversible Markov chain with the unique stationary distribution given as
𝑞 𝑥 ∗ = exp⁡(𝜃ɸ(𝑥)) 𝑦 exp⁡(𝜃ɸ(𝑦))
For the Markov chain induced by the algorithm, we can derive its stationary distribution in this form. Based on this expression, we can see that when parameter \theta goes to infinity, the strategy that maximize the potential function will occur with probability 1. If the \theta is finite, we can provide a bound on the gap from the maximum potential function. We can see that as long as we choose a large enough \theta, the gap can be arbitrarily close to 0.
As 𝜃→∞, the SNE 𝑥 𝑆𝑁𝐸 = argmax 𝑥 ɸ 𝑥 can be achieved
For finite 𝜃, the gap from max 𝑥 ɸ 𝑥 is bounded by 1 𝜃 lnK, where K is the number of states in Markov chain

26. Performance Gap
Performance gap from the social optimal strategy by NUM
maximum social welfare: 𝑣 ∗ = max 𝒙 𝑣 𝒙 where 𝑣 𝒙 ≜ 𝑖=1 𝑛 𝑈 𝑖 (𝒙)
THEOREM: The performance gap 𝜌≜ 𝑣 ∗ −𝑣( 𝑥 𝑆𝑁𝐸 ) of the SNE found by the distributed spectrum access algorithm is at most
1 2 i=1 n j∈ N i p ∩ N i s (1− a ij )P j d ji −α i=1 n j∈ N i p \ N i p ∩ N i s P j d ji −α
Based on this result, we can provide a performance bound of the SNE achieved by the algorithm. The difference between the social welfare of the SNE and the maximum social welfare is bounded by this expression. We can see that this bound decreases as social ties increase. In particular, the first term decreases to 0 as a_ij increases to 1. For the second term, as social ties increase, this set difference will get smaller, and becomes empty set when all social ties are equal to 1. So the second term decreases to 0 as social ties increase to 1.
𝜌 decreases as the social tie strength 𝑎 𝑖𝑗 increases
𝜌=0 when all users are altruistic, i.e., 𝑎 𝑖𝑗 =1,∀𝑖,𝑗

27. Numerical Results N=100 users randomly scatter over a square area
Physical graph is generated based on users’ distances
Finally we show the performance with numerical results. We assume users are randomly located and we construct the physical graph is based on their distances.

28. Numerical Results
Social graph is generated by Erdos-Renyi random graph model
There exists a social link between two users with probability 𝑃 𝐿
We construct the social graph using the ER random graph mode, where any pair of users have a social link with a given probability P_L. This figure shows the social welfare of the SNE achieved by the distributed algorithm. We can see that as the social link probability increases from 0 to 1, the performance of the SNE increases from that of NCG to NUM. This shows that the SNE migrates monotonically from the NE for NCG to the social optimal strategy for NUM. It also shows that the SGUM framework spans the continuum space between these two extreme models.
Performance improves as the social link probability increases
The SNE for the SGUM game migrates monotonically from the NE for NCG to the social optimal strategy for NUM

29. Outline Introduction Social Group Utility Maximization Framework
Database-assisted Spectrum Access under SGUM
Conclusion and Future Work
Now we make a conclusion.

30. Conclusion Contribution
Developed the social group utility maximization (SGUM) framework, which captures diverse social ties of mobile users and diverse physical relationships of mobile devices, and spans the continuum between non-cooperative game and network utility maximization
Studied SGUM game for database assisted spectrum access, showed that it is a potential game, developed a CSMA-like distributed algorithm to achieve a social-aware Nash equilibrium, and quantified the impact of social ties
In this paper, we developed …

31. Future Work Study a variety of applications under the SGUM framework
E.g, power control, random access control
Extend the SGUM framework to “negative” social ties
Social tie can be “negative” (i.e., 𝑎 𝑖𝑗 ∈[−∞,0]) such that a user intends to damage another’s welfare (e.g., against opponent or enemy)
Zero-sum game (ZSG)
Users aim to minimize others’ welfare
Employed for security applications
If total strength of social ties to each user is zero (i.e., 𝑗=1 𝑛 𝑎 𝑗𝑖 =0, ∀𝑖∈𝑁), SGUM degenerates to ZSG as 𝑖=1 𝑛 𝑓 𝑖 ( 𝑥 𝑖 , 𝒙 −𝑖 ) =0
(e.g, 𝑓 1 = 𝑢 1 − 𝑢 2 , 𝑓 2 = 𝑢 2 − 𝑢 1 )
For future work, we can apply this framework to study different applications in mobile networks, such as power control, random access control. Another interesting direction is to extend this framework to negative social ties. In this paper, we have assumed positive social ties such that a user cares about another’s welfare. Social ties can also be negative such that a users wants to damage another’s welfare. In this case, a relevant approach is ZSG. In a ZSG, users aim to minimize others’ welfare, so it is suitable for many security applications. For the SGUM framework with negative social ties, we can show that when the social ties satisfies some conditions, the game is equivalent to a ZSG. E.g., for the case with two users, the game is a ZSG if their social group utilities are inverse.

32. Future Work
NCG
NUM
Positive Social Tie
Malicious
Altruistic
Selfish
Negative Social Tie
ZSG
SGUM
SGUM
𝑎 21 1 2
𝑎 23 3 4
𝑎 34
Therefore, by integrating negative social ties, we can extend this framework such that it captures ZSG as a special case, and also spans the continuum space between ZSG and NCG. This will provide a new angle to study network security issues.
SGUM −1 1 1 2 1 2 1 2
selfish
selfish −1 −1 1 1 1 1
malicious
altruistic 3 4 3 4 3 4 −1 1
ZSG
NCG
NUM

33. Thank You!