nash stable partitioning of graphs and community detection in social networks · 2011-01-10 ·...

Nash Stable Partitioning of Nash Stable Partitioning of Nash Stable Partitioning of Nash Stable Partitioning of Graphs and Community Graphs and Community Graphs and Community Graphs and Community

Detection in Social NetworksDetection in Social NetworksDetection in Social NetworksDetection in Social NetworksDecember 15, 2010December 15, 2010December 15, 2010December 15, 2010

E-Commerce Lab, CSA, IISc1

December 15, 2010December 15, 2010December 15, 2010December 15, 2010

Y. NARAHARIY. NARAHARIY. NARAHARIY. NARAHARI

http://lcm.csa.iisc.ernet.in/harihttp://lcm.csa.iisc.ernet.in/harihttp://lcm.csa.iisc.ernet.in/harihttp://lcm.csa.iisc.ernet.in/hari

Computer Science and AutomationComputer Science and AutomationComputer Science and AutomationComputer Science and AutomationIndian Institute of Science, BangaloreIndian Institute of Science, BangaloreIndian Institute of Science, BangaloreIndian Institute of Science, Bangalore

OUTLINE

PART 1: Social Network Analysis

PART 2: Game Theoretic Models forSocial Network Analysis


PART 3: Community Detection and Nash Stable Partitions

PART 4: SCoDA: Stable Community Detection Algorithm

Today’s Talk is a Tribute to

John von NeumannThe 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

Game Theoretic Modelsare very natural for

modeling social networks--------------------------------------Social network nodes are

rational, intelligent--------------------------------------Social networks form in a

It would be interestingto explore

Game Theoretic Models for analyzing social networks --------------------------------------Example 1: Discovering

Communities--------------------------------------


Ramasuri Narayanam. Game Theoretic Models for Social Network Analysis,Ph.D. Dissertation, CSA, IISc, November 2010

Social networks form in adecentralized way

--------------------------------------Strategic interactions among

social network nodes---------------------------------------

--------------------------------------Example 2: Network

Formation---------------------------------------

Example 3: FindingInfluential Nodes

Why Social Network Analysis ?

§ Diffusion of Information and Innovations§ To understand spread of diseases (Epidemiology)§ E-Commerce and E-Business (selling patterns,

marketing)§ Job Finding (through referrals)§ 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…

§ Random Graphs

§ Simulation

§ Probabilistic Models

Tools and Techniques for SNA

§ Probabilistic Models

§ Multi-Level Models

§Optimization Models

§ Game Theoretic Models

Game TheoryMathematical 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

Strategic Form Games (Normal Form Games)

S1

Sn

U1 : S R

Un : S R

N = {1,…,n}

Players

S1, … , Sn

Strategy SetsPayoff functions


Players Strategy Sets

S = S1 X … X Sn

(Utility functions)

Example 1: Coordination Game

B A

IIITB MG Road

IIITB 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 ConfessNC

ConfessC

No Confess


No ConfessNC - 2, - 2 - 10, - 1

ConfessC -1, - 10 - 5, - 5

Nash Equilibrium

A profile of strategies is said to bea 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

Nash Equilibria in Coordination Game

B A

IIITB MG Road

IIITB 100,100 0,0


MG Road 0,0 10,10

Two pure strategy Nash equilibria: (IIITB, IIITB) and (MG Road, MG Road);

one mixed strategy Nash equilibrium

Nash Equilibrium in Prisoner’s Dilemma

No ConfessNC

ConfessC

No Confess


NC - 2, - 2 - 10, - 1

ConfessC -1, - 10 - 5, - 5

(C,C) is a Nash equilibrium

Nash’s Theorem

Every finite strategic form game has at least one mixed strategy Nash

equilibrium


Mixed strategy of a player ‘i’ is a probability distribution on Si .

is a mixed strategy Nash equilibrium if

is a best response against , *iσ

*i−σ

( )**2

*1 ,...,, nσσσ

ni ,...,2,1=∀

Relevance of Nash Equilibrium

Nash equilibrium has several possible implications:

(a) Players are happy the way they are; do not want to deviate unilaterally

(b) Stable, self-enforcing, self-sustaining agreement


(b) Stable, self-enforcing, self-sustaining agreement

(c) Provides a principled way of predicting a steady-state outcome of a

dynamic adjustment process

Example: Nash Equilibrium in Social Network Formation

N = { 1, 2, …, n} Nodes

Si = Subsets of N – { i }

Ui : S1 x S2 x … x Sn àààà RUi : S1 x S2 x … x Sn àààà R

Each strategy profile results in a particular network

Standard Assumptions:

• A link is formed under mutual consent• A link may be deleted unilaterally

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1) No agent can increase its payoff by deleting a link . That is , ,, Nji ∈∀

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ijgugu

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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


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 globalobjective based on centrality based measures

MEJ 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,


MEJ 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

Existing Algorithms for Community Detection: A Few Issues

Most of these algorithms work with a global objective such as

modularity, conductance, etc.


Do not take into account the strategic natureof the players and their associations

Most algorithms 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


as a graph partitioning problem and set up agraph 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 summarization


3. 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 localinformation

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) = 6 u1(S2) = 0

u2(S1) = 9.33 u2(S2) = 0

u3(S1) = 9.33 u3(S2) = 0

u4(S1) = 9 u4(S2) = 0

u5(S1) = 6 u5(S2) = 1

u6(S1) = 1 u6(S2) = 1

u7(S1) = 6 u7(S2) = 1

u8(S1) = 9 u8(S2) = 0

u9(S1) = 9.33 u9(S2) = 0

u10(S1) = 9.33 u10(S2) = 0

u11(S1) = 6 u11(S2) = 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


MEJ 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 77 508 467


Les Miserables 77 508 467Political Books 105 882 560Football 115 1226 810Jazz Musicians 198 274 17899Email 1133 5451 10687Yeast 2361 6913 5999

Karate Club Dataset with 3 Communities

Les Miserables Dataset with 7 Communities

FootBall Dataset with 12 Communities

Political Books Dataset with 5 Communities

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


highly non-trivial

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


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 ?


Sensational new algorithms for SNA problems ?Still a long way to go but the potential is good.

Calls for a much deeper study

Questions and Answers …


Thank You …

nash stable partitioning of graphs and community detection in social networks · 2011-01-10 ·...

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