Diffusion in Social Networks with Competing Products Krzysztof R. Apt CWI, Amsterdam, the Netherlands , University of Amsterdam based on joint work with E. Markakis Athens University of Economics and Business Diffusion in Social Networks with Competing Products – p. 1/2
Social Networks Facebook, Hyves, LinkedIn, Nasza Klasa (Our Class), . . . Diffusion in Social Networks with Competing Products – p. 2/2
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). Diffusion in Social Networks with Competing Products – p. 3/2
Example 1 (From D. Easley and J. Kleinberg, 2010). Spread of the tuberculosis outbreak. Diffusion in Social Networks with Competing Products – p. 4/2
Example 2 (From D. Easley and J. Kleinberg, 2010). Pattern of e-mail communication among 436 employees of HP Research Lab. Diffusion in Social Networks with Competing Products – p. 5/2
Example 3 (From D. Easley and J. Kleinberg, 2010). Collaboration of mathematicians centered on Paul Erd˝ os. Drawing by Ron Graham. Diffusion in Social Networks with Competing Products – p. 6/2
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. Diffusion in Social Networks with Competing Products – p. 7/2
Our Model Assumptions Weighted directed graph G = ( V, E ) , w ij ∈ [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 ) � = ∅ , � j ∈ N ( i ) w ji ≤ 1 , Threshold function θ : V → (0 , 1] , Finite set P of products. Social network: ( G, P, p, θ ) , where p : V → P ( P ) , with each p ( i ) non-empty. Diffusion in Social Networks with Competing Products – p. 8/2
Reduction Relations → : a binary relation on social networks, → ∗ : reflexive, transitive closure of → . Reduction sequence p → ∗ p ′ is maximal if for no p ′′ , p ′ → p ′′ . Assume an initial social network p . p ′ is reachable (from p ) if p → ∗ p ′ , p ′ is unavoidable (from p ) if for all maximal sequences of reductions p → ∗ p ′′ , p ′ = p ′′ , p has a unique outcome if some social network is unavoidable from p . Diffusion in Social Networks with Competing Products – p. 9/2
Specific Reduction Relation p 1 → p 2 if p 2 � = p 1 , if p 2 ( i ) � = p 1 ( i ) , then | p 1 ( i ) | ≥ 2 ( i had a choice in p 1 ), for some t ∈ p 1 ( i ) p 2 ( i ) = { t } ( i made a choice in p 2 ), � w ji ≥ θ ( i ) j ∈ N ( i ) | p 1 ( j )= { t } (upon influence of its neighbours). If N ( i ) = ∅ , then for all t ∈ p 1 ( i ) , p 2 ( i ) = { t } is allowed. Diffusion in Social Networks with Competing Products – p. 10/2
Adopting a Product Node i in a social network p adopted product t if p ( i ) = { t } , can adopt product t if t ∈ p ( i ) ∧ | p ( i ) | ≥ 2 ∧ � j ∈ N ( i ) | p ( j )= { t } w ji ≥ θ ( i ) . Diffusion in Social Networks with Competing Products – p. 11/2
Comments A node with no neighbours can adopt any product that is a possible choice for it. p 1 → p 2 holds if any node that adopted a product in p 2 either adopted it in p 1 or could adopt it in p 1 , at least one node could adopt a product in p 1 and adopted it in p 2 , the nodes that did not adopt a product in p 2 did not change their product sets. Social network is equitable if each weight w j,i = 1 N ( i ) | . In equitable social networks � j ∈ N ( i ) | p ( j )= { t } w ji ≥ θ ( i ) if at least the fraction θ ( i ) of N ( i ) adopted in p product t . Diffusion in Social Networks with Competing Products – p. 12/2
Examples Mobile phones It is cheaper to choose the provider of our friends. Secondary schools Children prefer to choose a school which their friends will choose, as well. Discussions preceding voting in a club Preferences announced by some members before elections may influence the votes cast by their friends. Common characteristics: Small number of choices (in comparison with the number of agents), Outcome of the adoption process does not need to be unique. Diffusion in Social Networks with Competing Products – p. 13/2
Three Questions What is the complexity of determining that a 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? Diffusion in Social Networks with Competing Products – p. 14/2
Reachable Outcomes Theorem 1 Assume ( G, P, p, θ ) and a product top ∈ P . There is an O ( n 2 ) time algorithm that determines whether the social network ( G, P, [ top ] , θ ) is reachable. Diffusion in Social Networks with Competing Products – p. 15/2
Proof Idea (1) Definition Weighted directed graph is θ -well-structured if for some level : V → N for all i such that N ( i ) � = ∅ � w ji ≥ θ ( i ) . j ∈ N ( i ) | level ( j ) <level ( i ) Example (all weights 1 / 2 , θ ( i ) = 1 / 2 ) Diffusion in Social Networks with Competing Products – p. 16/2
Proof Idea (2) Given ( G, P, p, θ ) and t ∈ P . G p,t : weighted directed graph obtained by removing from G all edges to nodes i with p ( i ) = { t } . So in G p,t for all such nodes i , N ( i ) = ∅ . Lemma Assume ( G, P, p, θ ) and a product top ∈ P . A social network ( G, P, [ top ] , θ ) is reachable iff for all i , top ∈ p ( i ) , G p,top is θ -well-structured. Diffusion in Social Networks with Competing Products – p. 17/2
Proof Idea (3) Lemma Assume a weighted directed graph G and θ . There is an O ( n 2 ) time algorithm that determines whether G is θ -well-structured. Algorithm Assign level 0 to all nodes with in-degree 0 . If no such node exists, then output No. At step i , assign level i to each node for which the θ -well-structuredness condition holds when considering only its neighbours with assigned levels 0 , . . ., i − 1 . If by iterating all nodes are assigned a level, then output Yes and otherwise output No. Diffusion in Social Networks with Competing Products – p. 18/2
Unavoidable Outcomes Theorem 2 Assume ( G, P, p, θ ) and a product top ∈ P . There is an O ( n 2 ) time algorithm that determines whether the social network ( G, P, [ top ] , θ ) is unavoidable. Lemma Assume ( G, P, p, θ ) and a product top ∈ P . A social network ( G, P, [ top ] , θ ) is unavoidable iff for all i , if N ( i ) = ∅ , then p ( i ) = { top } , for all i , top ∈ p ( i ) , G p,top is θ -well-structured. Diffusion in Social Networks with Competing Products – p. 19/2
Unique Outcomes (1) Theorem 3 There exists an O ( n 2 + n | P | ) time algorithm that determines whether a social network admits a unique outcome. Diffusion in Social Networks with Competing Products – p. 20/2
Proof Idea Node i can switch in p ′ given 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 ) . p ′ is ambivalent given p if a node can adopt more than one product or can switch in p ′ given p . Contraction sequence: the unique reduction sequence p → ∗ p ′ such that each of its reduction steps is fast, either p → ∗ p ′ is maximal or p ′ is the first network in p → ∗ p ′ that is ambivalent given p . Lemma A social network admits a unique outcome iff its contraction sequence ends in a non-ambivalent social network. Diffusion in Social Networks with Competing Products – p. 21/2
Algorithm Produce the representation with a list of outgoing edges for each node; for i ∈ V do set p ( i ) to be the initial list of products available to node i end for for j ∈ V, t ∈ p ( j ) do S j,t := 0 ; / counts total weight of incoming edges to j from nodes that have adopted t ; end for if ∃ i ∈ V with N ( i ) = ∅ and | p ( i ) | ≥ 2 then return "No unique outcome"; end if L := { i ∈ V : | p ( i ) | = 1 } ; Diffusion in Social Networks with Competing Products – p. 22/2
Algorithm, ctd while L � = ∅ do R := ∅ ; for i ∈ L and j such that ( i, j ) ∈ E do if i has adopted t and t ∈ p ( j ) then S j,t := S j,t + w ij end if ; R := R ∪ { j } ; / nodes we need to check for ambivalence end for for j ∈ R do Compute |{ t : S j,t ≥ θ ( j ) }| ; if |{ t : S j,t ≥ θ ( j ) }| ≥ 2 return "No unique outcome" endif ; if |{ t : S j,t ≥ θ ( j ) }| = 1 and j has not yet adopted t then node j adopts product t ; else R := R \ { j } ; / j does not adopt any product; end if end for L := R end while return "Unique outcome" Diffusion in Social Networks with Competing Products – p. 23/2
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