Lecture5:Social Network Analysis-By Dr. Y. Narahari

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

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

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)

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.

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

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

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

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

Strategic Form Games (Normal Form Games)

U1 : S R

Un : S R

N = {1,…,n}

Players

S1, … , Sn

Strategy Sets

S = S1 X … X Sn

Payoff functions

(Utility functions)

Example 1: Coordination Game

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

Example 2: Prisoner’s Dilemma

No Confess

NCConfess

No Confess

NC - 2, - 2 - 10, - 1

Confess

C -1, - 10 - 5, - 5

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

*1 ,...,, nsss

A Nash equilibrium profile is robust to unilateral deviations and captures a stable, self-enforcing

agreement among the players

Nash Equilibria in Coordination Game

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

Nash Equilibrium in Prisoner’s Dilemma

No Confess

NCConfess

No Confess

NC - 2, - 2 - 10, - 1

Confess

C -1, - 10 - 5, - 5

(C,C) is a Nash equilibrium

SourceDestination

Example 3: Traffic Routing Game

N = {1,…,n}; S1 = S2 = … = Sn = {C,D}

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

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

Example 4: Network Formation

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

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

*1 ,...,, n

ni ,...,2,1

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

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.

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.

State-of-the-Art ReviewJ. Lescovec et al. Empirical comparison of algorithms for community detection.

WWW 2010

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

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.

Community Detection and Graph Partitioning

•Non-overlapping community detection can be viewed as a graph partitioning problem

Graph Partitioning: Applications

1. VLSI circuit design2. Resource allocation in parallel computing3.Graph visualization and summarization4.Epidemiology5.Social Network Analysis

Email Network – Visualization and Summarization

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

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

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;

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

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

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

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

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

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

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{

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.

;0;0;0:),,{(

}3,2,1{

321321

xxxxxxSSS

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({

otherwise 0

),,( if ),,( 3211321

xxxsxsssu ii

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({

otherwise 0

),,( if ),,( 32121321

xxxssxsssu ii

Either 1 and 2 should propose the same allocation or 1 and 3

should propose the same allocation

300})3,2,1({})3,1({})2,1({

0})3,2({})3({})2({})1({

otherwise 0

),,(or ),,( if ),,( 3213132121321

xxxssxxxssxsssu ii

It is enough any pair of players has the same proposal

Also called the Majority Voting Game

300})3,2,1({})3,2({})3,1({})2,1({

0})3({})2({})1({

otherwise 0

),,(or

),,( if ),,(

32121321

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

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)

CviCvN

)}(}){({|!|

)!1|||(||!|)(

))(),...,(()( 0

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

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

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

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

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

William Vickrey(1914 – 1996 )

Nobel Prize: 1996

Winner = Winner = 4 Price = 4 Price =

4400445566008800

BuyersBuyers

Mechanism Design: Example 3 Vickrey Auction

Vickrey-Clarke-Groves (VCG) Mechanisms

Only mechanisms under a quasi-linear setting satisfyingAllocative Efficiency

Dominant Strategy Incentive Compatibility

Vickrey Clarke Groves

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

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

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

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)

GROVES MYERSON

SSAOPT CBOPT

MECHANISM DESIGN SPACE

Mechanism Design for Social Networks

Social Network Monitization (QA Networks,Query Incentive Networks)

Virus Inoculation Strategies

Crowdsourcing Mechanisms,Mechanical Turk Marketplaces

CONCLUDINGREMARKS

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

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

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

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

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

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!

Questions and Answers …

Thank You …

Lecture5:Social Network Analysis-By Dr. Y. Narahari

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