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 Networks Detection in Social Networks Detection in Social Networks Detection in Social Networks December 15, 2010 December 15, 2010 December 15, 2010 December 15, 2010 E-Commerce Lab, CSA, IISc 1 December 15, 2010 December 15, 2010 December 15, 2010 December 15, 2010 Y. NARAHARI Y. NARAHARI Y. NARAHARI Y. NARAHARI http://lcm.csa.iisc.ernet.in/hari http://lcm.csa.iisc.ernet.in/hari http://lcm.csa.iisc.ernet.in/hari http://lcm.csa.iisc.ernet.in/hari Computer Science and Automation Computer Science and Automation Computer Science and Automation Computer Science and Automation Indian Institute of Science, Bangalore Indian Institute of Science, Bangalore Indian Institute of Science, Bangalore Indian Institute of Science, Bangalore
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Nash Stable Partitioning of Graphs and Community Detection in Social Networks · 2011-01-10 · Nash Stable Partitioning of Graphs and Community Detection in Social Networks December
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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
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
E-Commerce Lab, CSA, IISc2
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)
E-Commerce Lab, CSA, IISc3
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--------------------------------------
E-Commerce Lab, CSA, IISc4
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---------------------------------------
(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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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
E-Commerce Lab, CSA, IISc21
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.
E-Commerce Lab, CSA, IISc23
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
E-Commerce Lab, CSA, IISc25
Graph Partitioning: Applications
1. VLSI circuit design2. Resource allocation in parallel computing3. Graph visualization and summarization
E-Commerce Lab, CSA, IISc26
3. Graph visualization and summarization4. Epidemiology5. Social Network Analysis
Email Network – Visualization and Summarization
E-Commerce Lab, CSA, IISc27
Graph Partitioning Game
E-Commerce Lab, CSA, IISc28
§ 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
E-Commerce Lab, CSA, IISc29
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
E-Commerce Lab, CSA, IISc30
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
E-Commerce Lab, CSA, IISc31
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
E-Commerce Lab, CSA, IISc32
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
E-Commerce Lab, CSA, IISc33
MODULARITYNormalized fraction of difference of intra-community edges