Lecture5:Social Network Analysis-By Dr. Y. Narahari
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E-Commerce Lab, CSA, IISc 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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E-Commerce Lab, CSA, IISc 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 Nodes Using Shapley Value PART 5: Social Network Monitization Using Mechanism Design PART 7: Conclusions, Promising Directions
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E-Commerce Lab, CSA, IISc 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)
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E-Commerce Lab, CSA, IISc CENTRAL IDEA Ramasuri Narayanam. Game Theoretic Models for Social Network Analysis , Ph.D. Dissertation, CSA, IISc, November 2010 Game Theoretic Models are 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 interesting to explore Game Theoretic Models for analyzing social networks -------------------------------------- Example 1: Discovering Communities -------------------------------------- Example 2: Finding Influential Nodes --------------------------------------- Example 3: Monitizing Social Networks
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<ul><li>A Social Structure made up of nodes (Individuals or Organizations or Social Entities) that are tied by one or more specific types of relationships. </li></ul><ul><li>Social Networks are ubiquitous and have been existing from times immemorial </li></ul><ul><ul><li>Family Relationship Networks </li></ul></ul><ul><ul><li>Education Networks </li></ul></ul><ul><ul><li>Professional Networks (Scientists, Doctors, Musicians) </li></ul></ul><ul><li>Social Networks have a long history – Sociologists and Psychologists have studied them extensively </li></ul>Social Networks
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Why Are Social Networks Important ? <ul><li>Diffusion of Information and Innovations </li></ul><ul><li>To understand spread of diseases (Epidemiology) </li></ul><ul><li>E-Commerce and E-Business (selling patterns, marketing) </li></ul><ul><li>Job Finding (through referrals) </li></ul><ul><li>Determine Influential Players (scientists, innovators, employees, customers, companies, genes, etc.) </li></ul><ul><li>Build effective social and political campaigns </li></ul><ul><li>Predict future events </li></ul><ul><li>Crack terrorist/criminal networks </li></ul><ul><li>Track alumni, etc… </li></ul>
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Social Network Analysis <ul><li>Find the structure of social networks </li></ul><ul><li>Understand the formation of social networks </li></ul><ul><li>Discover complex communication patterns, characteristic features </li></ul><ul><li>Graph theory (random graphs), simulation, have been extensively used </li></ul><ul><li>Social network analysis is crucial for all its applications </li></ul>
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<ul><li>Examining the structure is a fairly formidable task because of the scale and complexity </li></ul><ul><li>Extensive experimental and simulation based studies have been carried out, leading to some key observations </li></ul><ul><li>We are interested in the What, Why, and How of the structure of social networks </li></ul>Structure of Social Networks
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Small World Phenomenon (Low Diameters) Stanley Milgram
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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
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Erdos Number Paul Erdos Describes the collaborative distance between an author and Paul Erdos, celebrated and prolific mathematician who has written 1500 papers
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Power Law Degree Distribution Social networks fall into the class of scale-free networks , meaning that they have power-law (or scale-free) degree distributions.
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<ul><li>Question : Do we have analytical models that explain the unique characteristics of social networks satisfactorily? </li></ul><ul><li>Such models will be useful in many ways: </li></ul><ul><li>- understand information diffusion </li></ul><ul><li>- predict future events </li></ul><ul><li>- determine influential players </li></ul><ul><li>- build effective social campaigns </li></ul>Models of Social Network Formation
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<ul><li>Provide a natural model for analysis of networks that form when links are chosen by agents </li></ul><ul><li>- game theory can model choice ; random graphs only model chance </li></ul><ul><li>Appropriate because the agents in a social network are rational and intelligent </li></ul><ul><li>- strategic actors who are discreet in choosing </li></ul><ul><li>the relationships </li></ul>Game Theoretic Models
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Game Theory <ul><li>Mathematical framework for rigorous study of conflict and cooperation among rational, intelligent agents </li></ul>E-Commerce Lab, CSA, IISc 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
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E-Commerce Lab, CSA, IISc 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
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E-Commerce Lab, CSA, IISc GAME THEORY IN SNA: TWO VIEWPOINTS Game Theoretic Models are very natural for many SNA problems (Rationality of Internet Users) -------------------------------------- Example 1: Social Network Formation -------------------------------------- Example 2: Modeling Incentives --------------------------------------- Example 3: Extracting Knowledge Accurately Game Theoretic Solution Concepts Lead to More Efficient Algorithms -------------------------------------- Example 1: Mining Influential Nodes -------------------------------------- Example 2: Clustering Large Data Sets --------------------------------------- Example 3: Discovering Communities
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Strategic Form Games (Normal Form Games) E-Commerce Lab, CSA, IISc S 1 S n U 1 : S R U n : S R N = {1,…,n} Players S 1 , … , S n Strategy Sets S = S 1 X … X S n Payoff functions (Utility functions)
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Example 1: Coordination Game E-Commerce Lab, CSA, IISc Models the strategic conflict when two players have to choose their priorities B A IISc MG Road IISc 100,100 0,0 MG Road 0,0 10,10
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Example 2: Prisoner’s Dilemma E-Commerce Lab, CSA, IISc No Confess NC Confess C No Confess NC - 2, - 2 - 10, - 1 Confess C -1, - 10 - 5, - 5
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Pure Strategy Nash Equilibrium <ul><li>A profile of strategies is said to be </li></ul><ul><li>a pure strategy Nash Equilibrium if is a best </li></ul><ul><li>response strategy against </li></ul>E-Commerce Lab, CSA, IISc A Nash equilibrium profile is robust to unilateral deviations and captures a stable, self-enforcing agreement among the players
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Nash Equilibria in Coordination Game E-Commerce Lab, CSA, IISc Two pure strategy Nash equilibria: (IISc, IISc) and (MG Road, MG Road); one mixed strategy Nash equilibrium B A IISc MG Road IISc 100,100 0,0 MG Road 0,0 10,10
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Nash Equilibrium in Prisoner’s Dilemma E-Commerce Lab, CSA, IISc (C,C) is a Nash equilibrium No Confess NC Confess C No Confess NC - 2, - 2 - 10, - 1 Confess C -1, - 10 - 5, - 5
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45 C 2 45 x/ 100 x/ 100 B D A Source Destination Example 3: Traffic Routing Game N = {1,…,n}; S 1 = S 2 = … = S n = { C,D }
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45 C 2 45 x/ 100 x/ 100 B D A Source Destination Traffic Routing Game: Nash Equilibrium Assume n = 4000 U 1 (C,C, …, C) = - (40 + 45) = - 85 U 1 (D,D, …, D) = - (45 + 40) = - 85 U 1 (D,C, …, C) = - (45 + 0.01) = - 45.01 U1 (C, …,C;D, …,D) = - (20 + 45) = - 65 Any Strategy Profile with 2000 C’s and 2000 D’s is a Nash Equilibrium
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45 C 2 45 x/ 100 x/ 100 B D A Source Destination Traffic Routing Game: Braess’ Paradox Assume n = 4000 S 1 = S 2 = … = S n = {C,CD, D} U 1 (CD,CD, …, CD) = - (40+0+40) = - 80 U 1 (C,CD, …, CD) = - (40+45) = - 85 U1 (D,CD, …, CD) = - (45+40) = - 85 Strategy Profile with 4000 CD’s is the unique Nash Equilibrium 0
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2 1 Example 4: Network Formation 2 1 2 1 2 1 N = {1,2} ; S 1 = {null, 2}; S 2 = {null, 1} s 1 = s 2 = null U 1 = 0; U 2 = 0 NE if b <= c s 1 = 2; s 2 = null U 1 = b - c; U 2 = 0 NE if b = c s 1 = null; s 2 = 1 U 1 = 0; U 2 = b - c NE if b = c s 1 = 2; s 2 = 1 U 1 = b - c; U 2 = b – c NE if b >= c
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Nash’s Theorem <ul><li>Every finite strategic form game has at least one mixed strategy Nash equilibrium </li></ul>E-Commerce Lab, CSA, IISc Mixed strategy of a player ‘i’ is a probability distribution on S i . is a mixed strategy Nash equilibrium if is a best response against ,
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E-Commerce Lab, CSA, IISc 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 correspond to a socially optimal or Pareto optimal solution
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Community Detection using Nash Stable Partitions
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E-Commerce Lab, CSA, IISc Community Detection Problem <ul><li>Discover natural components such that connections </li></ul><ul><li>within a component are dense and across components </li></ul><ul><li>are sparse </li></ul><ul><li>Important for social campaigns, viral marketing, search, </li></ul><ul><li>and a variety of applications </li></ul><ul><li>Extensively investigated problem </li></ul><ul><li>Communities could be overlapping or non-overlapping . </li></ul><ul><li>We are interested in non-overlapping communities . </li></ul>
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E-Commerce Lab, CSA, IISc Community Detection: Relevant Work Optimization based approaches using global objective based on centrality based measures MEJ Newman. Detecting Community Structure in Networks. European Physics Journal. 2004. Spectral methods, Eigen vector based methods MEJ Newman. Finding community structure in networks using eigen vectors, Physical Review-E, 2006 Multi-level Approaches B. Hendrickson and R. Leland. A multi-level algorithm for partitioning graphs. 1993 . State-of-the-Art Review J. Lescovec et al. Empirical comparison of algorithms for community detection. WWW 2010
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E-Commerce Lab, CSA, IISc Existing Algorithms for Community Detection: A Few Issues Most of these work with a global objective such as modularity, conductance, etc. Do not take into account the strategic nature of the players and their associations Invariably require the number of communities To be provided as an input to the algorithm
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E-Commerce Lab, CSA, IISc Our Approach We use a strategic form game to model the formation of communities We view detection of non-overlapping communities as a graph partitioning problem and set up a graph partitioning game Only relevant existing work W. Chen et al. A game theoretic framework to identify overlapping Communities in social networks. DMKD, 2010.
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E-Commerce Lab, CSA, IISc Community Detection and Graph Partitioning <ul><li>Non-overlapping community detection can be viewed as a </li></ul><ul><li>graph partitioning problem </li></ul>
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E-Commerce Lab, CSA, IISc Graph Partitioning Game <ul><li>Nodes in the network are the players </li></ul><ul><li>Strategy of a node is to choose its community </li></ul><ul><li>Utilities to be defined to reflect the network structure </li></ul><ul><li>and the problem setting; preferably should use only local </li></ul><ul><li>information </li></ul>
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E-Commerce Lab, CSA, IISc Proposed Utility Function U i (S) is the sum of number of neighbours of node i in the community plus a normalized value of the neighbours who are themselves connected The proposed utility function captures the Degree 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
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E-Commerce Lab, CSA, IISc SCoDA: Stable Community Detection Algorithm Start with an initial partition where each community has a small number of nodes Choose nodes in a non-decreasing order of degrees and investigate if it is better to defect to a neighbouring community The algorithm terminates in a Nash stable partition
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E-Commerce Lab, CSA, IISc Comparison of SCoDA with other Algorithms Girvan and Newman M Girvan and MEJ Newman. PNAS 2002 Greedy Algorithm MEJ Newman. Physical Review E, 2004 Spectral Algorithm MEJ Newman. PNAS 2006 RGT Algorithm W. Chen et al. DMKD, 2010
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E-Commerce Lab, CSA, IISc Performace Metrics COVERAGE Fraction of edges which are of intra-community type MODULARITY Normalized fraction of difference of intra-community edges In the given graph and a random graph
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E-Commerce Lab, CSA, IISc DATASETS Data Set Nodes Edges Triangles Karate 34 78 45 Dolphins 62 318 95 Les Miserables 77 508 467 Political Books 105 882 560 Football 115 1226 810 Jazz Musicians 198 274 17899 Email 1133 5451 10687 Yeast 2361 6913 5999
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E-Commerce Lab, CSA, IISc SOME INSIGHTS SCoDA has comparable computational complexity and running time SCoDA maintains a good balance between Coverage and modularity SCoDA uses only local information Game theory helps solve certain KDD problems with Incomplete information
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E-Commerce Lab, CSA, IISc POSSIBLE EXTENSIONS Extend to weighted graphs, directed graphs, overlapping communities There could be multiple Nash stable Partitions – choosing the best one is highly non-trivial
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Discovering Influential Nodes using Shapley Value
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E-Commerce Lab, CSA, IISc Solution Concepts in Cooperative Game Theory Solution Concepts in Non-cooperative Game Theory Nash Equilibrium, Strong Nash Equilibrium Dominant Strategy Equilibrium Subgame Perfect Equilibrium, etc. Solution Concepts in Cooperative Game Theory The Core Shapley Value, Myerson Value, The Kernel, Nucleolus, etc.
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Divide the Dollar Game There are three players who have to share 300 dollars. Each one proposes a particular allocation of dollars to players.
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Divide the Dollar : Version 1 <ul><li>The allocation is decided by what is proposed by player 0 </li></ul><ul><li>Characteristic Function </li></ul>
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Divide the Dollar : Version 2 <ul><li>It is enough 1 and 2 propose the same allocation </li></ul><ul><li>Players 1 and 2 are equally powerful; Characteristic Function is: </li></ul>
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Divide the Dollar : Version 3 <ul><li>Either 1 and 2 should propose the same allocation or 1 and 3 should propose the same allocation </li></ul><ul><li>Characteristic Function </li></ul>
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Divide the Dollar : Version 4 <ul><li>It is enough any pair of players has the same proposal </li></ul><ul><li>Also called the Majority Voting Game </li></ul><ul><li>Characteristic Function </li></ul>
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Shapley Value of a Cooperative Game <ul><li>Captures how competitive forces influence the outcomes of a game </li></ul><ul><li>Describes a reasonable and fair way of dividing the gains from cooperation given the strategic realities </li></ul><ul><li>Shapley value of a player finds its average marginal contribution across all permutation orderings </li></ul><ul><li>Unique solution concept that satisfies symmetry, preservation of carrier , additivity , and Pareto optimality </li></ul>E-Commerce Lab, CSA, IISc Lloyd Shapley
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Shapley Value : A Fair Allocation Scheme <ul><li>Provides a unique payoff allocation that describes a fair way of dividing the gains of cooperation in a game ( N , v ) </li></ul>
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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)
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Example: Information Diffusion Game <ul><li>Consider each individual node in the social network as a player n nodes n players </li></ul><ul><li>Assume that the influence probabilities are known. </li></ul><ul><li>For each subset C of N that is chosen as a target set, define </li></ul><ul><li>v(C) = Expected number of active nodes at the end of the </li></ul><ul><li>information diffusion process </li></ul><ul><li>The Shapley value of a player now gives the marginal contribution of the player to the information diffusion activity </li></ul>E-Commerce Lab, CSA, IISc
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E-Commerce Lab, CSA, IISc Top-10 Influential Nodes in Jazz Musicians Data Set Top-10 Influential Nodes in NIPS Co-authorship Network
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DARPA Red Balloon Contest E-Commerce Lab, CSA, IISc Mechanism Design Meets Computer Science, Communications of the ACM , August 2010
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Amazon Mechanical Turk A Plea to Amazon: Fix Mechanical Turk! Noam Nisan’s Blog – October 21, 2010
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E-Commerce Lab, CSA, IISc Mechanism Design Game Theory involves analysis of games – computing equilibria and analyzing equilibrium behaviour Mechanism Design is the design of games or reverse 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
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Mechanism Design: Example 1 Fair Division of a Cake Mother Social Planner Mechanism Designer Kid 1 Rational and Intelligent Kid 2 Rational and Intelligent
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Mechanism Design: Example 2 Truth Elicitation through an Indirect Mechanism Tenali Rama (Birbal) Mechanism Designer Mother 1 Rational and Intelligent Player Mother 2 Rational and Intelligent Player Baby
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E-Commerce Lab, CSA, IISc Vickrey-Clarke-Groves (VCG) Mechanisms Only mechanisms under a quasi-linear setting satisfying Allocative Efficiency Dominant Strategy Incentive Compatibility Vickrey Clarke Groves
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E-Commerce Lab, CSA, IISc Robert Aumann Nobel 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 Hurwicz Nobel 2007 Eric Maskin Nobel 2007 Roger Myerson Nobel 2007
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E-Commerce Lab, CSA, IISc PROPERTIES OF SOCIAL CHOICE FUNCTIONS DSIC (Dominant Strategy Incentive Compatibility ) Reporting Truth is always good BIC (Bayesian Nash Incentive Compatibility) Reporting truth is good whenever others 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-Dictatorship No single agent is favoured all the time Individual Rationality Players participate voluntarily since they do not incur losses
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E-Commerce Lab, CSA, IISc POSSIBILITIES AND IMPOSSIBILITIES - 1 Gibbard-Satterthwaite Theorem When the preference structure is rich, a social choice function is DSIC iff it is dictatorial Groves Theorem In the quasi-linear environment, there exist social choice functions which are both AE and DSIC The dAGVA Theorem In the quasi-linear environment, there exist social choice functions which are AE, BB, and BIC
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E-Commerce Lab, CSA, IISc POSSIBILITIES AND IMPOSSIBILITIES -2 Green- Laffont Theorem When the preference structure is rich, a social choice function cannot be DSIC and BB and AE Myerson-Satterthwaite Theorem In the quasi-linear environment, there cannot exist a social choice function that is BIC and BB and AE and IR Myerson’s Optimal Mechanisms Optimal mechanisms are possible subject to IIR and BIC (sometimes even DSIC)
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E-Commerce Lab, CSA, IISc 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 to efficient algorithms in some contexts Game theory helps solve certain SNA problems with incomplete information
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E-Commerce Lab, CSA, IISc 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 to SNA problem solving
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E-Commerce Lab, CSA, IISc 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
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E-Commerce Lab, CSA, IISc SOME PROMISING DIRECTIONS Designing scalable approximation algorithms with worst case guarantees Explore numerous solution concepts available in the ocean of game theory literature Exploit games with special structure such as convex games, potential games, matrix games, etc. Problems such as incentive compatible learning and social network monitization are at the cutting edge
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E-Commerce Lab, CSA, IISc SUMMARY Game Theory imparts more power, more efficiency, 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
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<ul><li>Game Theory in SNA is now an Active Area with plenty of promising problems </li></ul><ul><li>Matthew O. Jackson. Social and Economic Networks. Princeton University Press, 2008. </li></ul><ul><li>Sanjeev Goyal. Connections: An Introduction to the Economics of Networks. Princeton University Press. 2007. </li></ul><ul><li>David Easley and Jon Kleinberg. Networks, Crowds, and Markets, Cambridge University Press, 2010 </li></ul><ul><li>Ramasuri Narayanam, Game Theoretic Models for Social Network Analysis </li></ul>
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E-Commerce Lab, CSA, IISc BEST WISHES TO [email_address] 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 target is 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!
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E-Commerce Lab, CSA, IISc Questions and Answers … Thank You …
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