Choosing Products in Social Networks Krzysztof R. Apt CWI and University of Amsterdam based on joint works with Evangelos Markakis Athens University of Economics and Business Sunil Simon CWI Choosing Products in Social Networks – p. 1/33
A story should have a beginning, a middle and an end, but not necessarily in that order. Jean-Luc Godard Choosing Products in Social Networks – p. 2/33
Social Networks Facebook, Hyves, LinkedIn, Nasza Klasa (Our Class), ... Choosing Products in Social Networks – p. 3/33
But also ... An area with links to sociology (spread of patterns of social behaviour) economics (effects of advertising, emergence of ‘bubbles’ in financial markets, ... ), epidemiology (epidemics), computer science (complexity analysis), mathematics (graph theory). Choosing Products in Social Networks – p. 4/33
Example 1 (From D. Easley and J. Kleinberg, 2010). Spread of the tuberculosis outbreak. Choosing Products in Social Networks – p. 5/33
Example 2 (From D. Easley and J. Kleinberg, 2010). Pattern of e-mail communication among 436 employees of HP Research Lab. Choosing Products in Social Networks – p. 6/33
Example 3 (From D. Easley and J. Kleinberg, 2010). Collaboration of mathematicians centered on Paul Erd˝ os. Drawing by Ron Graham. Choosing Products in Social Networks – p. 7/33
Books C. P . Chamley. Rational herds: Economic models of social learning. Cambridge University Press, 2004. S. Goyal. Connections: An introduction to the economics of networks. Princeton University Press, 2007. F . Vega-Redondo. Complex Social Networks. Cambridge University Press, 2007. M. Jackson. Social and Economic Networks. Princeton University Press, Princeton, 2008. D. Easley and J. Kleinberg. Networks, Crowds, and Markets. Cambridge University Press, 2010. M. Newman. Networks: An Introduction. Oxford University Press, 2010. Choosing Products in Social Networks – p. 8/33
Some Research Topics Spread of a disease. Viral marketing. Possible impact of a product. Choosing Products in Social Networks – p. 9/33
Our Model Assumptions Weighted directed graph G = ( V , E ) , w i j ∈ [ 0 , 1 ] : weight of edge ( i , j ) , N ( i ) : neighbours of i (nodes from which there is an incoming edge to i ), For each node i such that N ( i ) � = / 0 , ∑ j ∈ N ( i ) w ji ≤ 1 , Finite set P of products, Threshold function θ : V × P → ( 0 , 1 ] . Social network: ( G , P , p , θ ) , where p : V → P ( P ) , with each p ( i ) non-empty. Choosing Products in Social Networks – p. 10/33
Example 1 P = { t 1 , t 2 } , each weight: 1 / #neighbours . Choosing Products in Social Networks – p. 11/33
Product Adoption A node with no neighbours can adopt any product from his product set. Formally: i adopted a product ≡ p ( i ) becomes a singleton. A node with neighbours can adopt any product that sufficiently many of his neighbours adopted. Formally: sufficiently many ≡ ∑ j ∈ N ( i ) | p ( j )= { t } w ji ≥ θ ( i , t ) . Choosing Products in Social Networks – p. 12/33
Example 1 θ ( b , _ ) ≤ 1 2 . Then the network in which each i � = a adopts t 2 is reachable, though not unavoidable. θ ( b , _ ) > 1 2 . Then the network in which each i � = a adopts t 2 is not reachable, the initial network has a unique outcome, node c adopts t 2 iff θ ( c , t 2 ) ≤ 1 2 . Choosing Products in Social Networks – p. 13/33
Example 2 P = { t 1 , t 2 } , each weight: 1 / #neighbours . Choosing Products in Social Networks – p. 14/33
Example 2 Final networks if θ ( x , t ) does not depend on t : θ ( b ) ≤ 1 3 ∧ θ ( c ) ≤ 1 : ( p ( b ) = { t 1 }∨ p ( b ) = { t 2 } ) ∧ 2 ( p ( c ) = { t 1 }∨ p ( c ) = { t 2 } ) θ ( b ) ≤ 1 3 ∧ θ ( c ) > 1 : ( p ( b ) = { t 1 } ∧ p ( c ) = P ) ∨ 2 ( p ( b ) = p ( c ) = { t 2 } ) 3 < θ ( b ) ≤ 2 3 ∧ θ ( c ) ≤ 1 : p ( b ) = p ( c ) = { t 2 } 1 2 3 < θ ( b ) ∧ θ ( c ) > 1 : p ( b ) = p ( c ) = P 1 2 3 < θ ( b ) ∧ θ ( c ) ≤ 1 : p ( b ) = P ∧ p ( c ) = { t 2 } 2 2 Choosing Products in Social Networks – p. 15/33
Three Questions Determine when specific product will possibly be adopted by all nodes, a specific product will necessarily be adopted by all nodes, the adoption process of the products will yield a unique outcome. Choosing Products in Social Networks – p. 16/33
Unique Outcomes (1) Assume an initial network ( G , P , p , θ ) . Node i can switch in p ′ if i adopted in p ′ a product t and for some t ′ � = t t ′ ∈ p ( i ) ∧ ∑ j ∈ N ( i ) | p ′ ( j )= { t ′ } w ji ≥ θ ( i , t ′ ) . p ′ is ambivalent if a node can adopt more than one product, or can switch in p ′ . Choosing Products in Social Networks – p. 17/33
Unique Outcomes (2) Theorem 3 A social network admits a unique outcome iff when reducing it at a fast pace one ends in a non-ambivalent social network. Corollary There exists an O ( n 2 + n | P | ) time algorithm that determines whether a social network admits a unique outcome. Proof Idea. Simulate the reduction at fast pace until ambivalence occurs or a final network is produced. Choosing Products in Social Networks – p. 18/33
Product Adoption (1) Given a social network and a node i determine whether i ADOPTION 1: has to adopt some product in all final networks. ADOPTION 2: has to adopt a given product in all final networks. ADOPTION 3: can adopt some product in some final network. ADOPTION 4: can adopt a given product in some final network. Choosing Products in Social Networks – p. 19/33
Product Adoption (2) Theorem ADOPTION 1 is co-NP-complete. Proof PARTITION problem: given n positive rational numbers ( a 1 ,..., a n ) such that ∑ n i = 1 a i = 1 , is there a set S such that ∑ i ∈ S a i = ∑ i �∈ S a i = 1 2 ? PARTITION problem is NP-complete. Choosing Products in Social Networks – p. 20/33
ADOPTION 1 (ctd) with P = { t , t ′ } and w i , a = w i , b = a i and θ ( c ) = 1 . Claim There exists a solution to instance I of PARTITION iff node c can avoid adopting any product. Choosing Products in Social Networks – p. 21/33
Product Adoption (3) Given a social network and a product top determine MIN-ADOPTION: the minimum number of nodes that adopted top in a final network. MAX-ADOPTION: the maximum number of nodes that adopted top in a final network. Choosing Products in Social Networks – p. 22/33
MIN-ADOPTION (2) Theorem It is NP-hard to approximate MIN-ADOPTION with an approximation ratio better than Ω ( n ) . Proof Using a more complex network and a reduction to the PARTITION problem. Choosing Products in Social Networks – p. 23/33
Choosing Products Basic assumptions: Nodes can choose a product or choose nothing. They make their choices simultaneously. We model it as a strategic game. Choosing Products in Social Networks – p. 24/33
Strategic Games Strategic game for n ≥ 2 players: For each player i : Strategies: a non-empty set S i , Payoff function: p i : S 1 ×···× S n → R , The players choose their strategies simultaneously. Choosing Products in Social Networks – p. 25/33
Main Concepts Notation: s , ( s i , s − i ) ∈ S 1 ×···× S n . s i is a best response to s − i if ∀ s ′ i ∈ S i p i ( s i , s − i ) ≥ p i ( s ′ i , s − i ) . s is a Nash equilibrium if each s i is a best response to s − i . Choosing Products in Social Networks – p. 26/33
Social Network Games Given a social network ( G , P , p , θ ) . Players: the nodes, Strategies for player i : his products and t 0 , Payoff functions: i is a source node: � if s i = t 0 0 p i ( s ) : = c if s i ∈ p ( i ) i is not a source node: p i ( s ) : = if s i = t 0 0 w ji − θ ( i , t ) ∑ if s i = t , for some t ∈ p ( i ) j ∈ N t i ( s ) Choosing Products in Social Networks – p. 27/33
� � � Do Nash Equilibria Always Exist? A social network with no Nash equilibrium: { t 1 } w 1 � ����� { t 1 , t 2 } 1 � � � w 2 w 2 � � � � � � � � { t 2 , t 3 } { t 1 , t 3 } � � � � � � � { t 2 } { t 3 } � � � � � � � � ����������������� � ������ 3 2 w 1 w 2 w 1 where θ < w 1 < w 2 . A check: ( t 1 , t 1 , t 2 ) , ( t 1 , t 1 , t 3 ) , ( t 1 , t 3 , t 2 ) , ( t 1 , t 3 , t 3 ) , ( t 2 , t 1 , t 2 ) , ( t 2 , t 1 , t 3 ) , ( t 2 , t 3 , t 2 ) , ( t 2 , t 3 , t 3 ) . Choosing Products in Social Networks – p. 28/33
Existence of a Nash Equilibrium Theorem Deciding whether a social network game has a Nash equilibrium is NP-complete. Proof Construct a social network from the ‘PARTITION network’ and two copies of the network with no Nash equilibrium. Choosing Products in Social Networks – p. 29/33
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