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

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

Mohammad MahdianMohammad MahdianYahoo! ResearchYahoo! Research

Joint work with N. Immorlica, J. Kleinberg, and T. Joint work with N. Immorlica, J. Kleinberg, and T. WexlerWexler

Social networksSocial networks

Representation of Representation of underlying underlying connections between connections between peoplepeople

Basic structure upon Basic structure upon which which communication communication happens and happens and information spreadsinformation spreads

co-authorship graph

DiffusionDiffusion

Underlying topology can make or Underlying topology can make or break the influence of particular break the influence of particular ideas and technologies byideas and technologies by– spreading informationspreading information

e.g.: word of mouth, viral marketinge.g.: word of mouth, viral marketing

– creating value through creating value through communicationcommunication

e.g.: human languages, telephone, instant e.g.: human languages, telephone, instant messaging, Xbox livemessaging, Xbox live

CompatibilityCompatibility

coexistence of multiple technologies, coexistence of multiple technologies, with varying degrees of compatibilitywith varying degrees of compatibility

Examples:Examples:– Human communication requires common Human communication requires common

language/culture.language/culture.– Telephone system in the early 20Telephone system in the early 20thth century century– Cell phone companies: cheaper M2M callsCell phone companies: cheaper M2M calls– Instant messaging technologies: Yahoo! Instant messaging technologies: Yahoo!

messenger, MSN messenger, Google talk, messenger, MSN messenger, Google talk, AIMAIM

In this talk…In this talk…

a game-theoretic model of diffusiona game-theoretic model of diffusion

Question:Question: can a new technology spread can a new technology spread through a network where almost everyone through a network where almost everyone is initially using another technology?is initially using another technology?

allowing limited compatibilityallowing limited compatibility examplesexamples epidemic regions in general graphsepidemic regions in general graphs

The modelThe model

People are represented by nodes in a graphPeople are represented by nodes in a graph Links represent friendships or social Links represent friendships or social

interactionsinteractions

Coordination gameCoordination game

Players:Players: node of the social node of the social networknetwork

Strategies:Strategies: Each player chooses Each player chooses which technology to use (e.g.: which technology to use (e.g.: A=Y!Msgr, B=MSN Msgr)A=Y!Msgr, B=MSN Msgr)

Payoff:Payoff: Players gain from every Players gain from every neighbor who uses the same neighbor who uses the same technology.technology.

Coordination game, Coordination game, cont’dcont’d Payoff for each edge uv:Payoff for each edge uv:

Payoff of a node is the sum over all Payoff of a node is the sum over all incident edges.incident edges.

An An equilibriumequilibrium is a strategy profile is a strategy profile where no player can gain by changing where no player can gain by changing strategies.strategies.

AA

BB

A BA B1-q,1-q1-q,1-q 0,00,0

0,00,0 q,qq,q

Example Example

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

= 2= 2

A

A A

A

A

B

BB

q = 1/3

u

EquilibriaEquilibria

This game has many equilibria, e.g. This game has many equilibria, e.g. an all-A and an all-B equilibrium.an all-A and an all-B equilibrium.

Q:Q: starting from an all-B starting from an all-B equilibrium, can a “small” equilibrium, can a “small” perturbation causes a cascading perturbation causes a cascading sequence of nodes to switch to A, sequence of nodes to switch to A, resulting in an all-A equilibrium?resulting in an all-A equilibrium?

Steve Morris, 2000.Steve Morris, 2000.

Morris’s modelMorris’s model

Assumptions:Assumptions:– graph is infinitegraph is infinite– finite degree. Further, assume finite degree. Further, assume --

regular.regular.

Starting from an Starting from an all-Ball-B equilibrium, is equilibrium, is it possible to change the strategy of it possible to change the strategy of a a finitefinite set set of nodes to A and let of nodes to A and let nodes play nodes play best responsebest response, so that we , so that we converge to an converge to an all-Aall-A equilibrium? equilibrium?

Limited compatibilityLimited compatibility

Assume, we allow a player to use both Assume, we allow a player to use both technologies (e.g., install two IM software), technologies (e.g., install two IM software), but at an additional cost of but at an additional cost of c=rc=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?

AA

BB

ABAB

A B ABA B AB1-q,1-q1-q,1-q 0,00,0 1-q,1-q1-q,1-q-r-r

0,00,0 q,qq,q q,qq,q-r-r

1-q1-q-r-r,1-,1-qq

qq-r-r,q,q max(q,1-q)max(q,1-q)-r-r,max(q,1-,max(q,1-q)q)-r-r

Example Example

Endow group 0 with strategy A.Endow group 0 with strategy A. Morris’s model: vertices of group 1 Morris’s model: vertices of group 1

have utility of have utility of 3q3q with the strategy B, with the strategy B, and and 3(1-q)3(1-q) if they switch to A. if they switch to A.

A spreads iff A spreads iff q q · · ½½..

0 1 2-1

Example Example

Our model: vertices of group 1 have utility:Our model: vertices of group 1 have utility:3q3q with the strategy B, with the strategy B, 3(1-q)3(1-q) if they switch to A, and if they switch to A, and 3q+3(1-q)-6r 3q+3(1-q)-6r if the switch to AB.if the switch to AB.

If If qq··½½ and and 2r2r¸̧qq, group 1 switches to A, …, group 1 switches to A, … If If qq··½½ and and 2r2r··qq, group 1 switches to AB. But , group 1 switches to AB. But

then group 2 might not switch!then group 2 might not switch!

0 1 2-1

ExampleExample

Technology A can spread if Technology A can spread if qq··½ ½ and and either either q+rq+r··½½ or or 2r2r¸̧qq..

B can defend against A by adopting a B can defend against A by adopting a limited level of compatibility.limited level of compatibility.

1/2

1/2

1

1 q

r

*

Other examplesOther examples

Infinite tree 2-d grid

Formal definitionFormal definition

Infinite Infinite -regular graph G-regular graph G Strategy profile: s: V(G)Strategy profile: s: V(G)!!{A,B,AB}{A,B,AB} ss!!s’s’ if s’ is obtained from s by letting v if s’ is obtained from s by letting v

play her best response.play her best response. Similar defn for a finite seq of verticesSimilar defn for a finite seq of vertices T infinite seq, TT infinite seq, Tkk first k elements of T first k elements of T ss!!s’s’ if for every u, there is k if for every u, there is k00(u) such (u) such

that for every k>kthat for every k>k00(u), s (u), s !! a profile that a profile that assigns s’(u) to u.assigns s’(u) to u.

v

Tk

T

Definition, cont’dDefinition, cont’d

For XFor X½½V(G), V(G), ssXX is the profile that is the profile that assigns A to X and B to V(G)\X.assigns A to X and B to V(G)\X.

A can become A can become epidemicepidemic in (G,q,r) in (G,q,r) if there is if there is – a finite set X, and a finite set X, and – sequence T of V(G)\X sequence T of V(G)\X

such that ssuch that sXX !! (all-A). (all-A).T

Basic factsBasic facts

Lemma. The only possible changes Lemma. The only possible changes in the strategy of a vertex arein the strategy of a vertex are– from B to Afrom B to A– from B to ABfrom B to AB– from AB to A.from AB to A.

Corollary. For every set X and Corollary. For every set X and sequence T of V(G)\X, there is sequence T of V(G)\X, there is unique s such that sunique s such that sXX ! ! s.s.T

Order independenceOrder independence

Theorem. If for a set X and some Theorem. If for a set X and some sequence T of V(G)\X, ssequence T of V(G)\X, sXX ! ! (all-A), (all-A), then for then for everyevery sequence T’ that sequence T’ that contains every vertex of V(G)\X an contains every vertex of V(G)\X an infinite # of times, sinfinite # of times, sXX ! ! (all-A).(all-A).

Pf idea. T is a subseq of T’. Extra moves Pf idea. T is a subseq of T’. Extra moves make it only more likely to reach all-A.make it only more likely to reach all-A.

T

T’

General graphsGeneral graphs

Q. What are the possible values of Q. What are the possible values of (q,r) where A can become epidemic (q,r) where A can become epidemic in in somesome graph? graph?

Theorem. A cannot become Theorem. A cannot become epidemic in any game (G,q,r) with epidemic in any game (G,q,r) with q>½.q>½.

This generalizes Morris’s result.This generalizes Morris’s result.

General graphs, cont’dGeneral graphs, cont’d

Theorem. A cannot become epidemic in Theorem. A cannot become epidemic in any game (G,q,r) with q>½.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 finiteit is initially finite– decreases with every best-response movedecreases with every best-response move

The following potential function works:The following potential function works:q(# A-B edges) + c(# AB vertices)q(# A-B edges) + c(# AB vertices)

General graphs, cont’dGeneral graphs, cont’d

Can A become epidemic for every (q,r) Can A become epidemic for every (q,r) with q<½?with q<½?

Not quite!Not quite! Theorem. For every Theorem. For every , there is q<½ , there is q<½

and r such that A cannot become and r such that A cannot become epidemic in any (G,q,r).epidemic in any (G,q,r).

Pf idea. Use same potential function, Pf idea. Use same potential function, show after a while the potential fn show after a while the potential fn stays constant and vertices on the stays constant and vertices on the boundary switch to AB…boundary switch to AB…

Variants/extensionsVariants/extensions

Alternative model for limited compatibility:Alternative model for limited compatibility:– Assume a player using A derives a utility of Assume a player using A derives a utility of

qqABAB··min(q,1-q) from communicating with a player min(q,1-q) from communicating with a player using B (and vice versa).using B (and vice versa).

– Example: users of Y! Messenger can send msgs Example: users of Y! Messenger can send msgs (but not files) to users of MSN Messenger.(but not files) to users of MSN Messenger.

Results:Results:– 2 technologies: better technology always benefits.2 technologies: better technology always benefits.– 3 technologies: two inferior technologies might 3 technologies: two inferior technologies might

benefit from forming a benefit from forming a strategic alliancestrategic alliance..

ConclusionConclusion

Simple, mathematically tractable Simple, mathematically tractable modelmodel

yet rich enough to explain certain yet rich enough to explain certain phenomenaphenomena

Useful for understanding the role Useful for understanding the role of network effects and strategic of network effects and strategic incompatibilityincompatibility

Open questionsOpen questions

More realistic models – e.g.: can More realistic models – e.g.: can we predict which games become we predict which games become popular on Xbox live based on popular on Xbox live based on early activities?early activities?

How does the diffusion process How does the diffusion process influence the graph formation?influence the graph formation?