Networks Xiaowen Gong Xu Chen Junshan Zhang Arizona State University Allerton Conference 2013 Oct 4th 2013 Outline Introduction Social Group Utility Maximization Framework ID: 658801
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Slide1
Social Group Utility Maximization Game with Applications in Mobile Social Networks
Xiaowen Gong, Xu Chen, Junshan ZhangArizona State University
Allerton Conference 2013 Oct. 4th, 2013Slide2
OutlineIntroduction
Social Group Utility Maximization FrameworkRandom Access Control Game under SGUMPower Control Game under SGUMConclusion2Slide3
Non-cooperative Game v.s
. Network Utility Maximization3
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
Non-cooperative game (NCG)
Each user is selfish, aiming to maximize its individual utilityWidely 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?Slide4
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 networkPhysical domain: physical coupling based on physical relationships
Social domain: social coupling due to social ties among usersSlide5
Social Group Utility Maximization Framework
5
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 are players
: 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
Slide6
Social Group Utility Maximization Framework
6
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 casesIf 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
Slide7
Related Work
7
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
Little attention paid to the continuum space between NCG and NUMRouting 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 networkingExploit 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]
c
oalitions in a CG:
{1,2,3}, {4,5}
s
ocial groups under SGUM:
{1,2,3},{3,4,5}Slide8
Random Access Control Model
8
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 modelEach user decides access
probability
to
contend
for data transmission
If
multiple
users contend, a collision occurs and no user can grab the transmission opportunity Slide9
Random Access Control Game under SGUM
9
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 strategySlide10
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 .
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
Remark
Users’ individual utilities are
equally weighted
in the
n
etwork 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
Slide11
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 NUM SGUM spans the continuum space between NCG and NUM
An example of two-user game with Slide12
Power Control Model
12
Physical interference model
Each user
is a link consisting of a transmitter
and a receiver
:
t
ransmission channel gain of link
:
interference channel gain from to : noise at
Power control modelEach user decides transmit power
of
:
signal-to-interference-plus-noise
ratio (SINR)
Slide13
Power Control Game under SGUM
13
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 response from , such that it will monotonically converge to the SNE
Slide14
Power Control Game under SGUM
14
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
Slide15
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
Slide16
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 utilityShowed 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 workSGUM provides rich modeling flexibility by spanning the continuum space between NCG and NUMStudy SGUM game for more applications (e.g., spectrum access) and investigate the impact of social ties on different performance metrics (e.g., fairness)Slide17