The role of compatibility in the diffusion of technologies in social networks

1. 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

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

3. Diffusion • 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

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

5. In this talk… • 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 • examples • epidemic regions in general graphs

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

7. Coordination game • Players: node of the social network • 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.

8. Coordination game, cont’d • Payoff for each edge uv: • 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.

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

10. Equilibria • 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? • Steve Morris, 2000.

11. Morris’s model • 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?

12. 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. • Payoff on an edge is computed as follows: • For which values of (q,r) new tech can spread?

13. Example • Endow group 0 with strategy A. • Morris’s 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. -1 0 1 2

14. Example • 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 AB. But then group 2 might not switch! -1 0 1 2

15. Example r • 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. 1 1/2 * q 1 1/2

16. Other examples 2-d grid Infinite tree

17. Formal definition • Infinite -regular graph G • Strategy profile: s: V(G)!{A,B,AB} • s!s’ if s’ is obtained from s by letting v play her best response. • Similar defn for a finite seq of vertices • T infinite seq, Tk first k elements of T • s!s’ if for every u, there is k0(u) such that for every k>k0(u), s ! a profile that assigns s’(u) to u. v T Tk

18. Definition, cont’d • For X½V(G), sX 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 finite set X, and • sequence T of V(G)\X such that sX! (all-A). T

19. Basic facts • 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 sX! s. T

20. Order independence • Theorem. If for a set X and some sequence T of V(G)\X, sX! (all-A), then for every sequence T’ that contains every vertex of V(G)\X an infinite # of times, sX! (all-A). • Pf idea. T is a subseq of T’. Extra moves make it only more likely to reach all-A. T T’

21. General graphs • 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. • This generalizes Morris’s result.

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

23. General graphs, cont’d • Can A become epidemic for every (q,r) with q<½? • Not quite! • 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…

24. Variants/extensions • Alternative model for limited compatibility: • Assume a player using A derives a utility of qAB·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: • 2 technologies: better technology always benefits. • 3 technologies: two inferior technologies might benefit from forming a strategic alliance.

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

26. Open questions • 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?