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E-Commerce Lab, CSA, IISc1
Game Theoretic Models for Social Network AnalysisY. NARAHARI
April 29, 2011SILVER JUBILEE OF CS DEPARTMENT, MYSORE
UNIVERSITY
150th BIRTH ANNIVERSARY OF SIR M. VISVESWARAYA
E-Commerce Laboratory
Computer Science and AutomationIndian Institute of Science, Bangalore
E-Commerce Lab, CSA, IISc2
OUTLINE
PART 1: SNA: What, Why, and How?
PART 2: Introduction to Relevant Game Theory
PART 3: Community Detection using Nash Equilibrium
PART 4: Discovering Influential NodesUsing Shapley Value
PART 5: Social Network MonitizationUsing Mechanism Design
PART 7: Conclusions, Promising Directions
E-Commerce Lab, CSA, IISc3
Today’s Talk is a Tribute to
John von Neumann The Genius who created two intellectual currents in the 1930s, 1940s
Founded Game Theory with Oskar Morgenstern (1928-44)
Pioneered the Concept of a Digital Computer and Algorithms (1930s and 40s)
E-Commerce Lab, CSA, IISc4
CENTRAL IDEA
Ramasuri Narayanam. Game Theoretic Models for Social Network Analysis, Ph.D. Dissertation, CSA, IISc, November 2010
Game Theoretic Modelsare very natural for
modeling social networks--------------------------------------Social network nodes are
rational, intelligent--------------------------------------Social networks form in a
decentralized way--------------------------------------
Strategic interactions among social network nodes
---------------------------------------
It would be interestingto explore
Game Theoretic Models for analyzing social networks --------------------------------------
Example 1: DiscoveringCommunities
--------------------------------------Example 2: FindingInfluential Nodes
---------------------------------------Example 3: Monitizing
Social Networks
A Social Structure made up of nodes (Individuals or Organizations or Social Entities) that are tied by one or more specific types of relationships.
Social Networks are ubiquitous and have been existing from times immemorial
• Family Relationship Networks• Education Networks • Professional Networks (Scientists, Doctors, Musicians)
Social Networks have a long history – Sociologists and Psychologists have studied them extensively
Social Networks
A Friendship Network
Credits: google images
A Jazz Musicians Network
Credits: Dataset from MEJ Newmann Homepage
Terrorist Network of 9/11
Why Are Social Networks Important ?
Diffusion of Information and Innovations To understand spread of diseases (Epidemiology) E-Commerce and E-Business (selling patterns,
marketing) Job Finding (through referrals) Determine Influential Players (scientists, innovators,
employees, customers, companies, genes, etc.) Build effective social and political campaigns Predict future events Crack terrorist/criminal networks Track alumni, etc…
Social Network Analysis
Find the structure of social networks Understand the formation of social networks Discover complex communication patterns,
characteristic features Graph theory (random graphs), simulation, have
been extensively used
Social network analysis is crucial for all its applications
Examining the structure is a fairly formidable task because of the scale and complexity
Extensive experimental and simulation based studies have been carried out, leading to some key observations
We are interested in the What, Why, and How of the structure of social networks
Structure of Social Networks
Clustering and Communities
Small World Phenomenon(Low Diameters)
Stanley Milgram
Six Degrees of Freedom
Duncan J. Watts
Duncan J Watts, Six degrees: The Science of a
Connected age, 2004, W.W. Norton and Company
Duncan J Watts, Small worlds: The Dynamics of
Networks between Order and Randomness, 2003,
Princeton University Press
Erdos Number
Paul Erdos
Describes the collaborative
distance between an author and Paul Erdos, celebrated
and prolific mathematician who
has written 1500 papers
Power Law Degree DistributionSocial networks fall into the class of scale-free networks, meaning that they have power-law (or scale-free) degree
distributions.
kdP
kdP
cddP
same the is )(
)(
)(
Question: Do we have analytical models that explain the unique characteristics of social networks satisfactorily?
Such models will be useful in many ways: - understand information diffusion - predict future events - determine influential players - build effective social campaigns
Models of Social Network Formation
Provide a natural model for analysis of networks that form when links are chosen by agents- game theory can model choice; random graphs only model chance
Appropriate because the agents in a social network are rational and intelligent - strategic actors who are discreet in choosing the relationships
Game Theoretic Models
E-Commerce Lab, CSA, IISc19
Game Theory
Mathematical framework for rigorous study of conflict and cooperation among rational, intelligent agents
Market
Buying Agents (rational and intelligent)
Selling Agents (rational and intelligent)
Social Planner
In the Internet Era, Game Theory has become a valuable tool for analysis and design
E-Commerce Lab, CSA, IISc20
Microeconomics, Sociology, Evolutionary Biology
Auctions and Market Design: Spectrum Auctions, Procurement Markets, Double Auctions
Industrial Engineering, Supply Chain Management, E-Commerce, Resource Allocation
CS: Algorithmic Game Theory, Internet and Network Economics, Protocol Design, etc.
---------------------------------------------------------------------There has been a surge of interest in applying
Game Theory to SNA and KDD Problems
Applications of Game Theory
E-Commerce Lab, CSA, IISc21
GAME THEORY IN SNA: TWO VIEWPOINTS
Game Theoretic Modelsare very natural for
many SNA problems (Rationality of Internet Users)
--------------------------------------Example 1: Social Network
Formation--------------------------------------
Example 2: Modeling Incentives
---------------------------------------Example 3: ExtractingKnowledge Accurately
Game Theoretic SolutionConcepts Lead to
More Efficient Algorithms --------------------------------------
Example 1: MiningInfluential Nodes
--------------------------------------Example 2: Clustering
Large Data Sets---------------------------------------
Example 3: DiscoveringCommunities
E-Commerce Lab, CSA, IISc22
Strategic Form Games (Normal Form Games)
S1
Sn
U1 : S R
Un : S R
N = {1,…,n}
Players
S1, … , Sn
Strategy Sets
S = S1 X … X Sn
Payoff functions
(Utility functions)
E-Commerce Lab, CSA, IISc23
Example 1: Coordination Game
B
A
IISc MG Road
IISc 100,100 0,0
MG Road 0,0 10,10
Models the strategic conflict when two players have to choose their priorities
E-Commerce Lab, CSA, IISc24
Example 2: Prisoner’s Dilemma
No Confess
NCConfess
C
No Confess
NC - 2, - 2 - 10, - 1
Confess
C -1, - 10 - 5, - 5
E-Commerce Lab, CSA, IISc25
Pure Strategy Nash Equilibrium
A profile of strategies is said to be
a pure strategy Nash Equilibrium if is a best
response strategy against *is ni ,...,2,1
**2
*1 ,...,, nsss
*is
A Nash equilibrium profile is robust to unilateral deviations and captures a stable, self-enforcing
agreement among the players
E-Commerce Lab, CSA, IISc26
Nash Equilibria in Coordination Game
B
A
IISc MG Road
IISc 100,100 0,0
MG Road 0,0 10,10
Two pure strategy Nash equilibria: (IISc, IISc) and (MG Road, MG Road);
one mixed strategy Nash equilibrium
E-Commerce Lab, CSA, IISc27
Nash Equilibrium in Prisoner’s Dilemma
No Confess
NCConfess
C
No Confess
NC - 2, - 2 - 10, - 1
Confess
C -1, - 10 - 5, - 5
(C,C) is a Nash equilibrium
45C
2
45
x/100
x/100
B
D
A
SourceDestination
Example 3: Traffic Routing Game
N = {1,…,n}; S1 = S2 = … = Sn = {C,D}
45C
2
45
x/100
x/100
B
D
A
SourceDestination
Traffic Routing Game: Nash Equilibrium
Assume n = 4000
U1 (C,C, …, C) = - (40 + 45) = - 85
U1 (D,D, …, D) = - (45 + 40) = - 85
U1 (D,C, …, C) = - (45 + 0.01) = - 45.01
U1 (C, …,C;D, …,D) = - (20 + 45) = - 65
Any Strategy Profilewith 2000 C’s and 2000 D’s is a Nash Equilibrium
45C
2
45
x/100
x/100
B
D
A
SourceDestination
Traffic Routing Game: Braess’ Paradox
Assume n = 4000
S1 = S2 = … = Sn = {C,CD, D}
U1 (CD,CD, …, CD) = - (40+0+40) = - 80
U1 (C,CD, …, CD) = - (40+45) = - 85
U1 (D,CD, …, CD) = - (45+40) = - 85
Strategy Profile with 4000 CD’s is the uniqueNash Equilibrium
0
21
Example 4: Network Formation
21
21
21
N = {1,2} ; S1 = {null, 2}; S2 = {null, 1}
s1 = s2 = nullU1 = 0; U2 = 0NE if b <= c
s1 = 2; s2 = nullU1 = b - c; U2 = 0NE if b = c
s1 = null; s2 = 1U1 = 0; U2 = b - c NE if b = c
s1 = 2; s2 = 1U1 = b - c; U2 = b – cNE if b >= c
E-Commerce Lab, CSA, IISc32
Mixed strategy of a player ‘i’ is a probability distribution on Si .
is a mixed strategy Nash equilibrium if
is a best response against ,
Nash’s Theorem
Every finite strategic form game has at least one mixed strategy Nash
equilibrium
*i *
i
**2
*1 ,...,, n
ni ,...,2,1
E-Commerce Lab, CSA, IISc33
Relevance/Implications of Nash Equilibrium
Players are happy the way they are;Do not want to
deviate unilaterally
Stable, self-enforcing,self-sustaining
agreement
Provides a principled way of predicting a
steady-state outcome of a dynamic
Adjustment process
Need not correspondto a socially optimal or
Pareto optimalsolution
Community Detection using Nash Stable Partitions
E-Commerce Lab, CSA, IISc35
Community Detection Problem
• Discover natural components such that connections within a component are dense and across components are sparse
• Important for social campaigns, viral marketing, search, and a variety of applications
• Extensively investigated problem
• Communities could be overlapping or non-overlapping. We are interested in non-overlapping communities.
E-Commerce Lab, CSA, IISc36
Community Detection: Relevant WorkOptimization based approaches using global
objective based on centrality based measuresMEJ Newman. Detecting Community Structure in Networks.
European Physics Journal. 2004.
Spectral methods, Eigen vector based methodsMEJ Newman. Finding community structure in networks using eigen vectors,
Physical Review-E, 2006
Multi-level ApproachesB. Hendrickson and R. Leland. A multi-level algorithm for partitioning graphs.
1993.
State-of-the-Art ReviewJ. Lescovec et al. Empirical comparison of algorithms for community detection.
WWW 2010
E-Commerce Lab, CSA, IISc37
Existing Algorithms for Community Detection: A Few Issues
Most of these work with a global objective such asmodularity, conductance, etc.
Do not take into account the strategic natureof the players and their associations
Invariably require the number of communitiesTo be provided as an input to the algorithm
E-Commerce Lab, CSA, IISc38
Our Approach
We use a strategic form game to model theformation of communities
We view detection of non-overlapping communitiesas a graph partitioning problem and set up a
graph partitioning game
Only relevant existing workW. Chen et al. A game theoretic framework to identify overlapping
Communities in social networks. DMKD, 2010.
E-Commerce Lab, CSA, IISc39
Community Detection and Graph Partitioning
•Non-overlapping community detection can be viewed as a graph partitioning problem
E-Commerce Lab, CSA, IISc40
Graph Partitioning: Applications
1. VLSI circuit design2. Resource allocation in parallel computing3.Graph visualization and summarization4.Epidemiology5.Social Network Analysis
E-Commerce Lab, CSA, IISc41
Email Network – Visualization and Summarization
E-Commerce Lab, CSA, IISc42
Graph Partitioning Game
Nodes in the network are the players Strategy of a node is to choose its community Utilities to be defined to reflect the network structure and the problem setting; preferably should use only local information
E-Commerce Lab, CSA, IISc43
Proposed Utility Function
Ui (S) is the sum of number of neighbours of node iin the community plus a normalized value of the
neighbours who are themselves connected
The proposed utility function captures theDegree of connectivity of the node and also the
density of its neighbourhood
A Nash Stable Partition is one in which no node has incentive to defect to any other community
E-Commerce Lab, CSA, IISc44
Nash Stable Partition: An Example
u1(S1) = 3; u1(S2) = 0;
u2(S1) = 8; u2(S2) = 0;
u3(S1) = 8; u3(S2) = 0;
u4(S1) = 6; u4(S2) = 0;
u5(S1) = 7; u5(S2) = 1;
u6(S1) = 1; u6(S2) = 1;
u7(S2) = 7; u7(S1) = 3;
u8(S2) = 6; u8(S1) = 0;
u9(S2) = 8; u9(S1) = 0;
u10(S2) = 8; u10(S1) = 0;
u11(S2) = 3; u11(S1) = 0;
E-Commerce Lab, CSA, IISc45
SCoDA: Stable Community Detection Algorithm
Start with an initial partition where each community hasa small number of nodes
Choose nodes in a non-decreasing order of degreesand investigate if it is better to defect to a neighbouring
community
The algorithm terminates in a Nash stable partition
E-Commerce Lab, CSA, IISc46
Comparison of SCoDA with other Algorithms
Girvan and Newman M Girvan and MEJ Newman. PNAS 2002
Greedy AlgorithmMEJ Newman. Physical Review E, 2004
Spectral AlgorithmMEJ Newman. PNAS 2006
RGT AlgorithmW. Chen et al. DMKD, 2010
E-Commerce Lab, CSA, IISc47
Performace Metrics
COVERAGEFraction of edges which are of intra-community type
MODULARITYNormalized fraction of difference of intra-community edges
In the given graph and a random graph
E-Commerce Lab, CSA, IISc48
DATASETS
Data Set Nodes Edges Triangles
Karate 34 78 45Dolphins 62 318 95Les Miserables 77 508 467Political Books 105 882 560Football 115 1226 810Jazz Musicians 198 274 17899Email 1133 5451 10687Yeast 2361 6913 5999
E-Commerce Lab, CSA, IISc49
SOME INSIGHTS
SCoDA has comparable computational complexityand running time
SCoDA maintains a good balance betweenCoverage and modularity
SCoDA uses only local information
Game theory helps solve certain KDD problems withIncomplete information
E-Commerce Lab, CSA, IISc50
POSSIBLE EXTENSIONS
Extend to weighted graphs, directed graphs,overlapping communities
There could be multiple Nash stablePartitions – choosing the best one is
highly non-trivial
Discovering Influential Nodes using Shapley Value
E-Commerce Lab, CSA, IISc52
Solution Concepts in Cooperative Game Theory
Solution Concepts in Non-cooperative Game Theory Nash Equilibrium, Strong Nash Equilibrium
Dominant Strategy EquilibriumSubgame Perfect Equilibrium, etc.
Solution Concepts in Cooperative Game Theory The Core
Shapley Value, Myerson Value,The Kernel, Nucleolus, etc.
Cooperative Game with Transferable Utilities
coalitions possible 12 are There
. a called is
0)( 2:
},...,2,1{
),(
||
N
N
NC
vv
nN
vNT
coalition
; functionsticcharacteri
players of set
Divide the Dollar GameThere are three players who have to share 300 dollars. Each one proposes a particular allocation of dollars to
players.
}300
;0;0;0:),,{(
}3,2,1{
321
3213
321321
xxx
xxxxxxSSS
N
Divide the Dollar : Version 1
The allocation is decided by what is proposed by player 0
Characteristic Function
300})3,2,1({})3,1({})2,1({
0})3,2({})3({})2({
300})1({
vvv
vvv
v
otherwise 0
),,( if ),,( 3211321
xxxsxsssu ii
Divide the Dollar : Version 2
It is enough 1 and 2 propose the same allocation
Players 1 and 2 are equally powerful; Characteristic Function is:
300})3,2,1({
0})3,2({})3,1({
300})2,1({
0})3({})2({})1({
v
vv
v
vvv
otherwise 0
),,( if ),,( 32121321
xxxssxsssu ii
Divide the Dollar : Version 3
Either 1 and 2 should propose the same allocation or 1 and 3
should propose the same allocation
Characteristic Function
300})3,2,1({})3,1({})2,1({
0})3,2({})3({})2({})1({
vvv
vvvv
otherwise 0
),,(or ),,( if ),,( 3213132121321
xxxssxxxssxsssu ii
Divide the Dollar : Version 4
It is enough any pair of players has the same proposal
Also called the Majority Voting Game
Characteristic Function
300})3,2,1({})3,2({})3,1({})2,1({
0})3({})2({})1({
vvvv
vvv
otherwise 0
),,(or
),,(or
),,( if ),,(
32132
32131
32121321
xxxss
xxxss
xxxssxsssu ii
Shapley Value of a Cooperative Game
Captures how competitive forces influence the outcomes of a game
Describes a reasonable and fair way of dividing the gains from cooperation given the strategic realities
Shapley value of a player finds its average marginal contribution across all permutation orderings
Unique solution concept that satisfies symmetry, preservation of carrier, additivity, and Pareto optimality
E-Commerce Lab, CSA, IISc59
Lloyd Shapley
Shapley Value : A Fair Allocation Scheme
Provides a unique payoff allocation that describes a fair way
of dividing the gains of cooperation in a game (N, v)
iNCi
n
CviCvN
CNCv
vvv
)}(}){({|!|
)!1|||(||!|)(
))(),...,(()( 0
where
Shapley Value: Examples
Version of Divide-the-Dollar Shapley Value
Version 1
Version 2
Version 3
Version 4
(150, 150, 0)
(300, 0, 0)
(200, 50, 50)
(100, 100, 100)
Example: Information Diffusion Game
• Consider each individual node in the social network as a player n nodes n players
• Assume that the influence probabilities are known.
• For each subset C of N that is chosen as a target set, define
v(C) = Expected number of active nodes at the end of the
information diffusion process
• The Shapley value of a player now gives the marginal contribution of the player to the information diffusion activity
E-Commerce Lab, CSA, IISc62
E-Commerce Lab, CSA, IISc63
Top-10 Influential Nodes in Jazz Musicians Data Set
Top-10 Influential Nodes in NIPS Co-authorship Network
Monitization of Social Networks
DARPA Red Balloon Contest
E-Commerce Lab, CSA, IISc65
Mechanism Design Meets Computer Science, Communications of the ACM, August 2010
Amazon Mechanical Turk
A Plea to Amazon: Fix Mechanical Turk! Noam Nisan’s Blog – October 21, 2010
E-Commerce Lab, CSA, IISc67
Mechanism Design
Game Theory involves analysis of games – computing equilibria and
analyzing equilibrium behaviour
Mechanism Design is the design of games orreverse engineering of games; could be called
Game Engineering
Involves inducing a game among the players such that in some equilibrium of the game,
a desired social choice function is implemented
E-Commerce Lab, CSA, IISc68
Mechanism Design: Example 1 Fair Division of a Cake
MotherSocial PlannerMechanism Designer
Kid 1Rational and Intelligent
Kid 2Rational and Intelligent
Mechanism Design: Example 2 Truth Elicitation through an Indirect Mechanism
Tenali Rama(Birbal)Mechanism Designer
Mother 1Rational and Intelligent Player
Mother 2Rational and Intelligent Player
Baby
E-Commerce Lab, CSA, IISc71
William Vickrey(1914 – 1996 )
Nobel Prize: 1996
Winner = Winner = 4 Price = 4 Price =
6060
11
22
33
44
4400445566008800
BuyersBuyers
11
Mechanism Design: Example 3 Vickrey Auction
E-Commerce Lab, CSA, IISc72
Vickrey-Clarke-Groves (VCG) Mechanisms
Only mechanisms under a quasi-linear setting satisfyingAllocative Efficiency
Dominant Strategy Incentive Compatibility
Vickrey Clarke Groves
E-Commerce Lab, CSA, IISc73
Robert AumannNobel 2005
Recent Excitement : Nobel Prizes for Game Theory and Mechanism Design
The Nobel Prize was awarded to two Game Theorists in 2005
The prize was awarded to three mechanism designers in 2007 Thomas Schelling
Nobel 2005
Leonid HurwiczNobel 2007
Eric MaskinNobel 2007
Roger MyersonNobel 2007
E-Commerce Lab, CSA, IISc74
PROPERTIES OF SOCIAL CHOICE FUNCTIONS
DSIC (Dominant Strategy
Incentive Compatibility)Reporting Truth is always good
BIC (Bayesian NashIncentive Compatibility)
Reporting truth is good wheneverothers also report truth
AE (Allocative Efficiency)Allocate items to those who
value them most
BB (Budget Balance)Payments balance receipts and
No losses are incurred
Non-DictatorshipNo single agent is favoured all
the time
Individual RationalityPlayers participate voluntarilysince they do not incur losses
E-Commerce Lab, CSA, IISc75
POSSIBILITIES AND IMPOSSIBILITIES - 1
Gibbard-Satterthwaite Theorem When the preference structure is rich,
a social choice function is DSIC iff it is dictatorial
Groves TheoremIn the quasi-linear environment, there exist social
choice functions which are both AE and DSIC
The dAGVA TheoremIn the quasi-linear environment, there exist social
choice functions which are AE, BB, and BIC
E-Commerce Lab, CSA, IISc76
POSSIBILITIES AND IMPOSSIBILITIES -2
Green- Laffont TheoremWhen the preference structure is rich, a social
choice function cannot be DSIC and BB and AE
Myerson-Satterthwaite TheoremIn the quasi-linear environment, there cannot exist
a social choice function that isBIC and BB and AE and IR
Myerson’s Optimal MechanismsOptimal mechanisms are possible subject to
IIR and BIC (sometimes even DSIC)
E-Commerce Lab, CSA, IISc77
BIC
AE
WBB
IR
SBB
dAGVA
DSIC
EPE
GROVES MYERSON
VDOPT
SSAOPT CBOPT
MECHANISM DESIGN SPACE
E-Commerce Lab, CSA, IISc78
Mechanism Design for Social Networks
Social Network Monitization (QA Networks,Query Incentive Networks)
Virus Inoculation Strategies
Crowdsourcing Mechanisms,Mechanical Turk Marketplaces
CONCLUDINGREMARKS
E-Commerce Lab, CSA, IISc80
SOME FACTS
Game Theory captures many phenomena in SNA in a natural way and leads to better insights
Many game theory solution concepts (Nash equilibrium, Shapley value, Core, etc.)
have good relevance
The game theoretic approach leads toefficient algorithms in some contexts
Game theory helps solve certain SNA problems withincomplete information
E-Commerce Lab, CSA, IISc81
SOME MYTHS
Game theory is a panacea for solving SNA problems
Game theory makes all SNA algorithms much more efficient
Game Theory provides a complete alternative toSNA problem solving
E-Commerce Lab, CSA, IISc82
SOME CHALLENGES
Game theory computations are among the hardest;For example, computing NE of even 2 player games
is not even NP-hard!
Deciding when to use a game theoretic approach and mapping the given SNA problem into a
suitable game could be non-trivial
E-Commerce Lab, CSA, IISc83
SOME PROMISING DIRECTIONS
Designing scalable approximation algorithmswith worst case guarantees
Explore numerous solution concepts availablein the ocean of game theory literature
Exploit games with special structure such asconvex games, potential games, matrix games, etc.
Problems such as incentive compatiblelearning and social network monitization are at
the cutting edge
E-Commerce Lab, CSA, IISc84
SUMMARY
Game Theory imparts more power, moreefficiency, more naturalness, and more glamour
To social network analysis
Sensational new algorithms for SNA problems ?Still a long way to go but the potential is good.
Calls for a much deeper study
Game Theory in SNA is now an Active Area with plenty of promising problems
Matthew O. Jackson. Social and Economic Networks. Princeton University Press, 2008.
Sanjeev Goyal. Connections: An Introduction to the Economics of Networks. Princeton University Press. 2007.
David Easley and Jon Kleinberg. Networks, Crowds, and Markets, Cambridge University Press, 2010
Ramasuri Narayanam, Game Theoretic Models for Social Network Analysis
E-Commerce Lab, CSA, IISc86
BEST WISHES TO CS@MysoreUniv
Silver Jubilee is a significant milestone;Question to Ask: Are we in the Top 25 in the World?
IISc CS : Currently we are in 76-100. Our targetis to break into Top 25 in the next 5 years
A good target for CS@MysoreUniv will be to break into top 25 in the world in the next decade
and give IISc a run for its money!
E-Commerce Lab, CSA, IISc87
Questions and Answers …
Thank You …