nash stable partitioning of graphs and community detection in social networks · 2011-01-10 ·...
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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
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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)
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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--------------------------------------
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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
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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
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Players Strategy Sets
S = S1 X … X Sn
(Utility functions)
Example 1: Coordination Game
B A
IIITB MG Road
IIITB 100,100 0,0
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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
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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
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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
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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
E-Commerce Lab, CSA, IISc13
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
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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
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(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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Pairwise Nash Stability
1) No agent can increase its payoff by deleting a link . That is , ,, Nji ∈∀
)( )(
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ijgugu
ijgugu
jj
ii
−≥
−≥
2) No two agents can both benefit, at least one of them strictly, by adding a link between themselves. That is, the following is not possible.
with at least one inequality strict.
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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
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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,
E-Commerce Lab, CSA, IISc22
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.
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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
E-Commerce Lab, CSA, IISc24
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
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Graph Partitioning: Applications
1. VLSI circuit design2. Resource allocation in parallel computing3. Graph visualization and summarization
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3. Graph visualization and summarization4. Epidemiology5. Social Network Analysis
Email Network – Visualization and Summarization
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Graph Partitioning Game
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§ 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
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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
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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
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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
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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
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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
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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
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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
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highly non-trivial
SOME MYTHS
Game theory is a panacea for solving SNA problems
Game theory makes all SNA algorithms much more efficient
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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
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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
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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 ?
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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 …
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Thank You …