E-Commerce Lab, CSA, IISc 1 Game Theoretic Models for Social Network Analysis Y. NARAHARI April 29, 2011 SILVER JUBILEE OF CS DEPARTMENT, MYSORE UNIVERSITY 150 th BIRTH ANNIVERSARY OF SIR M. VISVESWARAYA E-Commerce Laboratory Computer Science and Automation Indian Institute of Science, Bangalore
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Lecture5:Social Network Analysis-By Dr. Y. Narahari
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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
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
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