Social Networks with Multiple Products Krzysztof R. Apt CWI and University of Amsterdam Based on joint works with Evangelos Markakis and Sunil Simon
Social Networks Facebook, Hyves, LinkedIn, Nasza Klasa, . . . Krzysztof R. Apt Social Networks with Multiple Products
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). Krzysztof R. Apt Social Networks with Multiple Products
Example 1 (From D. Easley and J. Kleinberg, 2010). Spread of the tuberculosis outbreak. Krzysztof R. Apt Social Networks with Multiple Products
Example 2 (From D. Easley and J. Kleinberg, 2010). Pattern of e-mail communication among 436 employees of HP Research Lab. Krzysztof R. Apt Social Networks with Multiple Products
Example 3 (From D. Easley and J. Kleinberg, 2010). Collaboration of mathematicians centered on Paul Erd˝ os. Drawing by Ron Graham. Krzysztof R. Apt Social Networks with Multiple Products
Some 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. Krzysztof R. Apt Social Networks with Multiple Products
Some Research Topics Spread of a disease. Viral marketing. Possible impact of a product. Krzysztof R. Apt Social Networks with Multiple Products
Our model Social network ([Apt, Markakis ’11, ’14]) Weighted directed graph: G = ( V , → , w ), where V : a finite set of agents, w ij ∈ (0 , 1]: weight of the edge i → j . Products: A finite set of products P . Product assignment: P : V → 2 P \ {∅} ; assigns to each agent a non-empty set of products. Threshold function: θ ( i , t ) ∈ (0 , 1], for each agent i and product t ∈ P ( i ). Neighbours of node i : { j ∈ V | j → i } . Source nodes: Agents with no neighbours. Krzysztof R. Apt Social Networks with Multiple Products
Diffusion Initially no node adopted any product. Source nodes can adopt any product from their product sets. A non-source node i can adopt some product t if � w ji > θ ( i , t ) , j ∈N t i where N t i : the set of neighbours of i who already adopted the product t . At each stage one or more nodes can adopt a product. The adopted choices are final. Krzysztof R. Apt Social Networks with Multiple Products
Example {•} 4 0.4 1 {• , •} 0.5 0.5 {• , •} 3 2 {• , •} 0.5 0.4 0.4 6 5 {•} {•} Threshold is 0 . 3 for all the players. P = {• , • , •} Krzysztof R. Apt Social Networks with Multiple Products
Example {•} 4 0.4 1 {• , •} 0.5 0.5 {• , •} 3 2 {• , •} 0.5 0.4 0.4 6 5 {•} {•} Threshold is 0 . 3 for all the players. P = {• , • , •} This diffusion process can take between 3 and 6 steps. Krzysztof R. Apt Social Networks with Multiple Products
Some Research Questions (1) Given an initial network. FINAL: Determine whether a final network exists in which every node adopted some product. Given 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. Krzysztof R. Apt Social Networks with Multiple Products
Results Theorem FINAL is NP-complete. ADOPTION 1 (some/all) is co-NP-complete. ADOPTION 2 (given/all) for 2 products can be solved in O ( n 2 ) time. ADOPTION 2 is co-NP-complete for at least 3 products. ADOPTION 3 (some/some) can be solved in O ( n 2 | P | ) time. ADOPTION 4 (given/some) can be solved in O ( n 2 ) time. Krzysztof R. Apt Social Networks with Multiple Products
Some Research Questions (2) Given an initial network and a product top . MAX-ADOPTION: What is the maximum number of nodes that adopted top in a final network. MIN-ADOPTION: What is the minimum number of nodes that adopted top in a final network. Krzysztof R. Apt Social Networks with Multiple Products
Results Theorem MAX-ADOPTION can be solved in O ( n 2 ) time. MIN-ADOPTION for 2 products can be solved in O ( n 2 ) time. For at least 3 products it is NP-hard to approximate MIN-ADOPTION with an approximation ratio better than Ω( n ). Krzysztof R. Apt Social Networks with Multiple Products
The associated strategic game Interaction between agents: Each agent i can adopt a product from the set P ( i ) or choose not to adopt any product ( t 0 ). Social network games Players: Agents in the network. Strategies: Set of strategies for player i is P ( i ) ∪ { t 0 } . Payoff: Fix c > 0. Given a joint strategy s and an agent i , Krzysztof R. Apt Social Networks with Multiple Products
The associated strategic game Interaction between agents: Each agent i can adopt a product from the set P ( i ) or choose not to adopt any product ( t 0 ). Social network games Players: Agents in the network. Strategies: Set of strategies for player i is P ( i ) ∪ { t 0 } . Payoff: Fix c > 0. Given a joint strategy s and an agent i , � 0 if s i = t 0 ◮ if i ∈ source ( S ), p i ( s ) = if s i ∈ P ( i ) c Krzysztof R. Apt Social Networks with Multiple Products
The associated strategic game Interaction between agents: Each agent i can adopt a product from the set P ( i ) or choose not to adopt any product ( t 0 ). Social network games Players: Agents in the network. Strategies: Set of strategies for player i is P ( i ) ∪ { t 0 } . Payoff: Fix c > 0. Given a joint strategy s and an agent i , � 0 if s i = t 0 ◮ if i ∈ source ( S ), p i ( s ) = if s i ∈ P ( i ) c ◮ if i �∈ source ( S ), p i ( s ) = 0 if s i = t 0 � w ji − θ ( i , t ) if s i = t , for some t ∈ P ( i ) j ∈N t i ( s ) N t i ( s ): the set of neighbours of i who adopted in s the product t . Krzysztof R. Apt Social Networks with Multiple Products
Example {•} 4 0.4 {• , •} 1 0.5 0.5 {• , •} {• , •} 3 2 0.5 0.4 0.4 6 5 {•} {•} Threshold is 0 . 3 for all the players. P = {• , • , •} Krzysztof R. Apt Social Networks with Multiple Products
Example {•} 4 0.4 {• , •} 1 0.5 0.5 Payoff: p 4 ( s ) = p 5 ( s ) = p 6 ( s ) = c {• , •} {• , •} 3 2 0.5 0.4 0.4 6 5 {•} {•} Threshold is 0 . 3 for all the players. P = {• , • , •} Krzysztof R. Apt Social Networks with Multiple Products
Example {•} 4 0.4 {• , •} 1 0.5 0.5 Payoff: p 4 ( s ) = p 5 ( s ) = p 6 ( s ) = c {• , •} {• , •} 3 2 0.5 p 1 ( s ) = 0 . 4 − 0 . 3 = 0 . 1 0.4 0.4 6 5 {•} {•} Threshold is 0 . 3 for all the players. P = {• , • , •} Krzysztof R. Apt Social Networks with Multiple Products
Example {•} 4 0.4 {• , •} 1 0.5 0.5 Payoff: p 4 ( s ) = p 5 ( s ) = p 6 ( s ) = c {• , •} {• , •} 3 2 0.5 p 1 ( s ) = 0 . 4 − 0 . 3 = 0 . 1 0.4 0.4 p 2 ( s ) = 0 . 5 − 0 . 3 = 0 . 2 6 5 p 3 ( s ) = 0 . 4 − 0 . 3 = 0 . 1 {•} {•} Threshold is 0 . 3 for all the players. P = {• , • , •} Krzysztof R. Apt Social Networks with Multiple Products
Social network games Properties Graphical game: The payoff for each player depends only on the choices made by his neighbours. Join the crowd property: The payoff of each player weakly increases if more players choose the same strategy. Krzysztof R. Apt Social Networks with Multiple Products
Does Nash equilibrium always exist? {•} 4 0.4 {• , •} 1 0.5 0.5 {• , •} {• , •} 3 2 0.5 0.4 0.4 6 5 {•} {•} Threshold is 0 . 3 for all the players. Krzysztof R. Apt Social Networks with Multiple Products
Does Nash equilibrium always exist? {•} 4 0.4 {• , •} 1 Observation: No player has the 0.5 0.5 incentive to choose t 0 . {• , •} {• , •} 3 2 Source nodes can ensure a 0.5 payoff of c > 0. 0.4 0.4 6 5 Each player on the cycle can ensure a payoff of at least 0 . 1. {•} {•} Threshold is 0 . 3 for all the players. Krzysztof R. Apt Social Networks with Multiple Products
Does Nash equilibrium always exist? {•} ( • , • , • ) 4 0.4 {• , •} 1 Observation: No player has the 0.5 0.5 incentive to choose t 0 . {• , •} {• , •} 3 2 Source nodes can ensure a 0.5 payoff of c > 0. 0.4 0.4 6 5 Each player on the cycle can ensure a payoff of at least 0 . 1. {•} {•} Threshold is 0 . 3 for all the players. Krzysztof R. Apt Social Networks with Multiple Products
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