Game-Theoretic Network Centrality Tomasz P. Michalak Department of Computer Science, University of Oxford Institute of Informatics, University of Warsaw
Plan of the Talk 1. Introduction to the Shapley value & its computation 2. The Shapley value as a game-theoretic network centrality measure 3. Applications and computations
Shapley value & its computational aspects
Characteristic Function Games Given 3 agents, the set of agents is: N = { a 1 , a 2 , a 3 } The possible coalitions are: a 1 a 1 a 2 a 2 a 3 a 3 a 1 a 2 a 1 a 2 a 1 a 3 a 1 a 3 a 1 a 2 a 3 a 1 a 2 a 3 a 2 a 3 a 2 a 3 5 5 5 12 12 12 24 <? ? ?> A solution of a coalitional game: STABILITY THE CORE STABILITY
Characteristic Function Games Given 3 agents, the set of agents is: N = { a 1 , a 2 , a 3 } The possible coalitions are: a 1 a 1 a 2 a 2 a 3 a 3 a 1 a 2 a 1 a 2 a 1 a 3 a 1 a 3 a 2 a 3 a 2 a 3 5 5 5 12 12 12 A solution of a coalitional game: a 2 : Wait! But it is not fair! STABILITY THE CORE Such a division of payoff a 1 a 2 a 3 a 1 a 2 a 3 which no sub-coalition wants to deviate from 24 a 3 : My contribution to every coalition in the <10 7 7> <13 7 4> a 1 : Great ! game is the same as a 1 I like this core division!`
Characteristic Function Games Given 3 agents, the set of agents is: N = { a 1 , a 2 , a 3 } The possible coalitions are: a 1 a 1 a 2 a 2 a 3 a 3 a 1 a 2 a 1 a 2 a 1 a 3 a 1 a 3 a 2 a 3 a 2 a 3 5 5 5 12 12 12 A solution of a coalitional game: Fairness criteria: FAIRNESS SHAPLEY VALUE Symmetry A unique division of payoff a 1 a 2 a 3 a 1 a 2 a 3 That meets fairness criteria Null-player (axioms) 24 Additivity <8 8 8> <? ? ?> Efficiency
Shapley Value – Definition a 1 a 1 a 2 a 2 a 3 a 3 a 1 a 1 a 2 a 2 a 3 a 3 a 2 a 2 a 1 a 1 a 3 a 3 a 2 a 2 a 3 a 3 a 1 a 1 a 3 a 3 a 1 a 1 a 2 a 2 a 3 a 3 a 2 a 2 a 1 a 1 a 2 a 2 a 1 a 1 a 3 a 3 a 1 a 2 a 1 a 2 a 1 a 3 a 1 a 3 a 2 a 3 a 2 a 3 5 5 5 12 12 12
Shapley Value – Definition a 1 a 1 a 2 a 2 a 3 a 3 a 1 a 1 a 2 a 2 a 3 a 3 a 2 a 2 a 1 a 1 a 3 a 3 a 2 a 2 a 3 a 3 a 1 a 1 a 3 a 3 a 1 a 1 a 2 a 2 a 3 a 3 a 2 a 2 a 1 a 1 a 2 a 2 a 1 a 1 a 3 a 3 a 1 a 2 a 1 a 2 a 1 a 3 a 1 a 3 a 2 a 3 a 2 a 3 5 5 5 12 12 12
Shapley Value – Definition Shapley Value – Formulas MC( a 1 ) +5 0 a 1 a 1 a 2 a 2 a 3 a 3 0 a 1 a 1 a 2 a 2 a 3 a 3 +5 +7 a 2 a 2 a 2 a 1 a 1 a 3 a 3 +12 48/6 = 8 a 2 a 2 a 2 a 3 a 3 a 3 a 1 a 1 = SV 1 ( v ) +7 a 3 a 3 a 3 a 1 a 1 a 2 a 2 +12 a 3 a 3 a 3 a 2 a 2 a 2 a 1 a 1 a 2 a 2 a 1 a 1 a 3 a 3 a 1 a 2 a 1 a 2 a 1 a 3 a 1 a 3 a 2 a 3 a 2 a 3 5 5 5 12 12 12
Shapley Value – Formulas Marginal contribution of � to coalition made of agents in the left part of the permutation n! The part of the permutation before The part of the permutation before agent agent � (left part of permutation) � (left part of permutation)
Shapley Value – Formulas n! 2 n � Computational Challenge
Circumventing intractability of the Characteristic Function New, more concise representations of coalitional games: General idea: Find a new model of a … is solved on the model level coalitional game. That is: concise expressive effective the computational problem … simple
Circumventing intractability of the Characteristic Function New, more concise representations of coalitional games: Induced Subgraph Representation Always concise but not fully expressive Marginal Contribution Nets Fully expressive but not Algebraic Decision Diagrams always concise Note: There are, of course, other representations – for specific types of games See more G. Chalkiadakis, E. Elkind, and M. Wooldrdidge. Computational Aspects of Cooperative Game Theory . Morgan & Claypool Publishers, 2011
Induced Subgraph Representation Deng and Papadimitriu (1994) agents are nodes and edges represent the value of cooperation between nodes in a coalition the value of a coalition is basically the value of the induced subgraph by this coalition v ({ a 1 }) = v ({ a 2 }) = v ({ a 1 }) = 0 4 v ({ a 4 }) = 1 a 1 a 1 a 2 v ({ a 1 ,a 2 }) = 4 v ({ a 1 ,a 3 }) = 8 8 : 3 10 5 v ({ a 3 ,a 4 }) = 1+6 v ({ a 1 ,a 2 ,a 3 }) = 4+5+8 : a 4 a 3 6 v ({ a 2 ,a 3 ,a 4 }) = 1+5+3+6 1 v ({ a 1 ,a 2 ,a 3 ,a 4 }) = 1+4+5+6+10+3+8 ? (1) Expressivity (2) Conciseness (3) Simplicity (4) Effectiveness 4 8 10 representation is not fully expressive SV 1 2
Induced Subgraph Representation Deng and Papadimitriu (1994) Let us consider the following intuition for the Shapley value formula under this representation 4 a 1 a 1 a 1 a 2 a permutation: 8 3 10 5 � will contribute his edge with � to the coalition to the left in the permutation iff � is already a a 4 a 3 member of this coalition 6 1 � The probability of this event is �
Marginal Contribution Nets Ieong and Shoham (2005) 5 a Simple Fully Expressive Efficient? Often Concise 1 5 a 1 a 1 4 a 2 Sv 1 = 5 + 3/2 1 a 4 a 2 a 2 3 Sv 2 = 4 + 3/2 a a 3 1 2 1 1 a 3 a 3 Sv 3 = 5 + 4 + 3 a 1 a 2 a 1 a 2 5 + 1 a 1 a 3 a 1 a 3 + 1 4 a 2 a 3 a 2 a 3 Additivity axiom Symmetry axiom 5 + 4 + 1 + 3 a 1 a 2 a 3 a 1 a 2 a 3
Marginal Contribution Nets Ieong and Shoham (2005) Such spectacular computational properties were initially shown for very simple rules, where only ˄ and ¬ are allowed. Such representation is called simple MC-Nets. But what about more complex rules? Elkind, Wooldridge, Goldberg and Goldberg (2009) proposed MC-Nets with arbitrary logical connectives but which are read-once. Still, polynomial computation of the Shapley value.
Algebraic Decision Diagrams Aadithya Michalak Jennings (2011) In general, a decision tree is of size exponential in the number of decision ADDs are, in essence, highly optimized representations for ordered decision variables. trees on boolean decision variables. However...
Algebraic Decision Diagrams Aadithya Michalak Jennings (2011) Rule 1: Merge isomorphic terminal nodes In most practically encountered decision trees contain a significant amount of duplication There exist many sub-trees within the decision tree that are isomorphic to Rule 2: Merge identical one another. decision nodes
Algebraic Decision Diagrams Aadithya Michalak Jennings (2011) Unlike MC-Nets, ADDs can be used for a whole range of problems In particular ADDs are the only representation formalism under which polynomial time computation of the core related problems is possible. The algorithm to compute the Shapley Value is based on dynamic programming principle ZDD zero-supressed decision diagrams Complexity: O(n 3 ) Sakurai, Ueda, Iwasaki, Minato, and Yokoo (2011)
Not only the Shapley value… the Shapley value: 2 n � the Banzhaf index 2 n � Semivalues = {Shapley, Banzhaf, …} 2 n � Generalized characteristic function the Nowak & Radzik value: n!
Myerson’s game 3 6 8 What if the cooperation is restricted by a graph? If a coalition is connected then players in can 1 5 7 communicate and create an arbitrary value added If a coalition is disconnected then players in cannot 2 4 communicate; hence, creating value added is restricted to connected components Communication Graph Myerson’s graph-restricted game
The Myerson value 3 6 8 There exist the unique value that satisfies: Axiom 1: fairness - any two agents connexted 1 5 7 with an edge profit from this connection equally Axiom 2: efficiency - the value of any connected 2 4 component is distributed among the agents within this components the Myerson value
Game-theoretic Network Centrality
Centrality Measures 6 4 9 Which node is the 1 7 2 3 most important in this network 5 10 8 12 11 13 Informal definition: methods to determine the role played by a node in the network. They differ depending on the application. Three, mostly used are:
Centrality Measures 6 4 9 Which node is the 1 7 2 3 most important in this network 5 10 8 12 11 13 Informal definition: methods to determine the role played by a node in the network. They differ depending on the application. Three, mostly used are: 1. Degree centrality – how many adjacent edges node has
Centrality Measures 6 4 9 1 Which node is the 3 2 1 7 2 3 most important in this network 1 5 10 8 12 11 13 Informal definition: methods to determine the role played by a node in the network. They differ depending on the application. Three, mostly used are: 1. Degree centrality – how many adjacent edges node has 2. Closeness centrality – how many edges, on overage, one needs to traverse to reach � � � + � + … from other nodes in the network
Centrality Measures 6 4 9 Which node is the 1 7 2 3 most important in this network 5 10 8 12 11 13 Informal definition: methods to determine the role played by a node in the network. They differ depending on the application. Three, mostly used are: 1. Degree centrality – how many adjacent edges node has 2. Closeness centrality – how many edges, on overage, one needs to traverse to reach from other nodes in the network 3. Betweenness centrality – what proportion of the shortest paths between any two nodes traverse through node
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