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
Social networks

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Representation of
underlying connections
between people
Basic structure upon
which communication
happens and
information spreads
co-authorship graph
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
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
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
The model

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People are represented by nodes in a graph
Links represent friendships or social
interactions
Coordination game
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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.
Coordination game, cont’d

Payoff for each edge uv:
A
B


A
B
1-q,1-q
0,0
0,0
q,q
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.
Example
B
q = 1/3
B
B
A
u
A


A
A
A
Payoff of u = 2q = 2/3
If it switches to A: payoff = 3(1-q) = 2
Equilibria


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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.
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?
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:
A
B
AB

A
B
AB
1-q,1-q
0,0
1-q,1-q-r
0,0
q,q
q,q-r
1-q-r,1-q
q-r,q
max(q,1-q)-r,max(q,1-q)-r
For which values of (q,r) new tech can spread?
Example
-1
0
1
2

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.

Example
-1

0
1
2
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!
Example
r
1
1/2
*
1/2


1
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.
Other examples
Infinite tree
2-d grid
Formal definition
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Infinite -regular graph G
Strategy profile: s: V(G)!{A,B,AB}
v
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
T
s!s’
if for every u, there is k0(u) such that
T
for every k>k0(u), s ! a profile that assigns
s’(u) to u.
k
Definition, cont’d
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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
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
T
s such that sX !
s.
Order independence
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Theorem. If for a set X and some
T
sequence T of V(G)\X, sX !
(all-A),
then for every sequence T’ that
contains every vertex of V(G)\X an
T’
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.
General graphs
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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.
General graphs, cont’d
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
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)
General graphs, cont’d
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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…
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.
Conclusion
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Simple, mathematically tractable
model
yet rich enough to explain certain
phenomena
Useful for understanding the role of
network effects and strategic
incompatibility
Open questions
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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?