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The role of compatibility in the diffusion of technologies in social networks Mohammad Mahdian Yahoo! Research Joint work with N. Immorlica, J. Kleinberg, and T. Wexler

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Social networks Representation of underlying connections between people Representation of underlying connections between people Basic structure upon which communication happens and information spreads Basic structure upon which communication happens and information spreads co-authorship graph

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Diffusion Underlying topology can make or break the influence of particular ideas and technologies by Underlying topology can make or break the influence of particular ideas and technologies by –spreading information e.g.: word of mouth, viral marketing –creating value through communication e.g.: human languages, telephone, instant messaging, Xbox live

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Compatibility coexistence of multiple technologies, with varying degrees of compatibility coexistence of multiple technologies, with varying degrees of compatibility Examples: Examples: –Human communication requires common language/culture. –Telephone system in the early 20 th century –Cell phone companies: cheaper M2M calls –Instant messaging technologies: Yahoo! messenger, MSN messenger, Google talk, AIM

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In this talk… a game-theoretic model of diffusion a game-theoretic model of diffusion Question: can a new technology spread through a network where almost everyone is initially using another technology? allowing limited compatibility allowing limited compatibility examples examples epidemic regions in general graphs epidemic regions in general graphs

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The model People are represented by nodes in a graph People are represented by nodes in a graph Links represent friendships or social interactions Links represent friendships or social interactions

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Coordination game Players: node of the social network Players: node of the social network Strategies: Each player chooses which technology to use (e.g.: A=Y!Msgr, B=MSN Msgr) Strategies: Each player chooses which technology to use (e.g.: A=Y!Msgr, B=MSN Msgr) Payoff: Players gain from every neighbor who uses the same technology. Payoff: Players gain from every neighbor who uses the same technology.

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Coordination game, contd Payoff for each edge uv: Payoff for each edge uv: Payoff of a node is the sum over all incident edges. Payoff of a node is the sum over all incident edges. An equilibrium is a strategy profile where no player can gain by changing strategies. An equilibrium is a strategy profile where no player can gain by changing strategies. AB A B A B 1-q,1-q0,0 0,0q,q

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Example Payoff of u = 2q = 2/3 Payoff of u = 2q = 2/3 If it switches to A: payoff = 3(1-q) = 2 If it switches to A: payoff = 3(1-q) = 2 A A A A A B B B q = 1/3 u

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Equilibria This game has many equilibria, e.g. an all-A and an all-B equilibrium. This game has many equilibria, e.g. an all-A and an all-B equilibrium. Q: starting from an all-B equilibrium, can a small perturbation causes a cascading sequence of nodes to switch to A, resulting in an all-A equilibrium? Q: starting from an all-B equilibrium, can a small perturbation causes a cascading sequence of nodes to switch to A, resulting in an all-A equilibrium? Steve Morris, Steve Morris, 2000.

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Morriss model Assumptions: Assumptions: –graph is infinite –finite degree. Further, assume -regular. Starting from an all-B equilibrium, is it possible to change the strategy of a finite set of nodes to A and let nodes play best response, so that we converge to an all-A equilibrium? Starting from an all-B equilibrium, is it possible to change the strategy of a finite set of nodes to A and let nodes play best response, so that we converge to an all-A equilibrium?

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Limited compatibility Assume, we allow a player to use both technologies (e.g., install two IM software), but at an additional cost of c=r. Assume, we allow a player to use both technologies (e.g., install two IM software), but at an additional cost of c=r. Payoff on an edge is computed as follows: Payoff on an edge is computed as follows: For which values of (q,r) new tech can spread? For which values of (q,r) new tech can spread? ABAB A B AB A B AB 1-q,1-q0,0 1-q,1-q-r 0,0q,q q,q-r 1-q-r,1-q q-r,q max(q,1-q)-r,max(q,1-q)-r

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Example Endow group 0 with strategy A. Endow group 0 with strategy A. Morriss model: vertices of group 1 have utility of 3q with the strategy B, and 3(1-q) if they switch to A. Morriss model: vertices of group 1 have utility of 3q with the strategy B, and 3(1-q) if they switch to A. A spreads iff q · ½. A spreads iff q · ½. 012

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Example Our model: vertices of group 1 have utility: Our model: vertices of group 1 have utility: 3q with the strategy B, 3(1-q) if they switch to A, and 3q+3(1-q)-6r if the switch to AB. If q · ½ and 2r ¸ q, group 1 switches to A, … If q · ½ and 2r ¸ q, group 1 switches to A, … If q · ½ and 2r · q, group 1 switches to AB. But then group 2 might not switch! If q · ½ and 2r · q, group 1 switches to AB. But then group 2 might not switch! 012

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Example Technology A can spread if q · ½ and either q+r · ½ or 2r ¸ q. Technology A can spread if q · ½ and either q+r · ½ or 2r ¸ q. B can defend against A by adopting a limited level of compatibility. B can defend against A by adopting a limited level of compatibility. 1/2 1 1 q r *

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Other examples Infinite tree 2-d grid

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Formal definition Infinite -regular graph G Infinite -regular graph G Strategy profile: s: V(G) ! {A,B,AB} Strategy profile: s: V(G) ! {A,B,AB} s ! s if s is obtained from s by letting v play her best response. s ! s if s is obtained from s by letting v play her best response. Similar defn for a finite seq of vertices Similar defn for a finite seq of vertices T infinite seq, T k first k elements of T T infinite seq, T k first k elements of T s ! s if for every u, there is k 0 (u) such that for every k>k 0 (u), s ! a profile that assigns s(u) to u. s ! s if for every u, there is k 0 (u) such that for every k>k 0 (u), s ! a profile that assigns s(u) to u. v TkTk T

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Definition, contd For X ½ V(G), s X is the profile that assigns A to X and B to V(G)\X. For X ½ V(G), s X is the profile that assigns A to X and B to V(G)\X. A can become epidemic in (G,q,r) if there is A can become epidemic in (G,q,r) if there is –a finite set X, and –sequence T of V(G)\X such that s X ! (all-A). T

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Basic facts Lemma. The only possible changes in the strategy of a vertex are Lemma. The only possible changes in the strategy of a vertex are –from B to A –from B to AB –from AB to A. Corollary. For every set X and sequence T of V(G)\X, there is unique s such that s X ! s. Corollary. For every set X and sequence T of V(G)\X, there is unique s such that s X ! s. T

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Order independence Theorem. If for a set X and some sequence T of V(G)\X, s X ! (all-A), then for every sequence T that contains every vertex of V(G)\X an infinite # of times, s X ! (all-A). Theorem. If for a set X and some sequence T of V(G)\X, s X ! (all-A), then for every sequence T that contains every vertex of V(G)\X an infinite # of times, s X ! (all-A). Pf idea. T is a subseq of T. Extra moves make it only more likely to reach all-A. Pf idea. T is a subseq of T. Extra moves make it only more likely to reach all-A. T T

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General graphs Q. What are the possible values of (q,r) where A can become epidemic in some graph? Q. What are the possible values of (q,r) where A can become epidemic in some graph? Theorem. A cannot become epidemic in any game (G,q,r) with q>½. Theorem. A cannot become epidemic in any game (G,q,r) with q>½. This generalizes Morriss result. This generalizes Morriss result.

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General graphs, contd Theorem. A cannot become epidemic in any game (G,q,r) with q>½. Theorem. A cannot become epidemic in any game (G,q,r) with q>½. Pf idea. Define potential function s.t. Pf idea. Define potential function s.t. –it is initially finite –decreases with every best-response move The following potential function works: The following potential function works: q(# A-B edges) + c(# AB vertices)

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General graphs, contd Can A become epidemic for every (q,r) with q<½? Can A become epidemic for every (q,r) with q<½? Not quite! Not quite! Theorem. For every, there is q<½ and r such that A cannot become epidemic in any (G,q,r). Theorem. For every, there is q<½ and r such that A cannot become epidemic in any (G,q,r). Pf idea. Use same potential function, show after a while the potential fn stays constant and vertices on the boundary switch to AB… Pf idea. Use same potential function, show after a while the potential fn stays constant and vertices on the boundary switch to AB…

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Variants/extensions Alternative model for limited compatibility: Alternative model for limited compatibility: –Assume a player using A derives a utility of q AB · min(q,1-q) from communicating with a player using B (and vice versa). –Example: users of Y! Messenger can send msgs (but not files) to users of MSN Messenger. Results: Results: –2 technologies: better technology always benefits. –3 technologies: two inferior technologies might benefit from forming a strategic alliance.

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Conclusion Simple, mathematically tractable model Simple, mathematically tractable model yet rich enough to explain certain phenomena yet rich enough to explain certain phenomena Useful for understanding the role of network effects and strategic incompatibility Useful for understanding the role of network effects and strategic incompatibility

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Open questions More realistic models – e.g.: can we predict which games become popular on Xbox live based on early activities? More realistic models – e.g.: can we predict which games become popular on Xbox live based on early activities? How does the diffusion process influence the graph formation? How does the diffusion process influence the graph formation?

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