Transcript Competition over popularity in social networks

COMPETITION OVER
POPULARITY IN SOCIAL
NETWORKS
Eitan Altman
May, 2013
INTRODUCTION
WHAT MAKES A CONTENT POPULAR?
Cultural, Social, Artistic reasons can make a
content a potential success
 We are interested in understanding how
Information technology can contribute to the
dissemination of content

QUESTIONS WE WISH TO ANSWER
Who are the actors related to dissemination of
content?
 What are the tools for dissemination of content?
 How efficient are they?
 Can data analysis be used to understand why a
given content is successful?
 When and how much should we invest in
promoting content?

OUTLINE OF TALK
The actors and their strategic choices
 Zoom: In what content to specialize
 Tools for accelerating dissemination
 Zoom: Analyzing the role of recommendation lists
 Dissemination models
 Dynamic game models for competition over
popularity: tools for the solution, results
 Start with classification of models for content
dissemination

WHAT IS THERE IN COMMON BETWEEN THE
FOLLOWING VIDEOS?
The most popular video
with more than 1.5
billion viewers on
youtube
A POPULAR MUSIC VIDEO
WHAT IS COMMON? WHAT IS DIFFERENT?
Difference in potential interested audience size
 Both exhibit viral behavior

DETERMINISTIC EPIDEMIC MODELS:

Fraction of infected:

Solution:

Hence
1
𝑥 1−𝑥
1
𝑥
= +
𝑑𝑥
𝑑𝑡
= 𝑘𝑥 1 − 𝑥
1
1−𝑥
𝑑𝑥 1
1
+
=𝑘
𝑥𝑡 𝑥 1 − 𝑥
Integrating, we get:
log 𝑥 − log 1 − 𝑥 = 𝑘𝑡
Finally.
𝑥(𝑡) =
1
1
1+ 𝑥(0)−1 exp(−𝑘𝑡)
EXAMPLES WITH X(0) = 0.0001, 0.01, 0.3

k=1
k=3
UNPOPULAR VIDEO
WITH MANY VIEWS

President Barack Obama 2009 Inauguration
and Address 3 years. Concave?
Epidemic?
PROPAGATION MODELS WITHOUT VIRALITY,
WITH MAX POPULATION SIZE

Consider the model:
𝑑𝑥
𝑑𝑡
= 𝑀(1 − 𝑥)
This models a constant rate M at which a noninfected node becomes infected. An infected node
does not infect others.
This gives
𝑥 𝑡 = 𝑥 0 + (1 − 𝑥 0 )(1 − exp −𝑀𝑡 )
This is a negative exponential model that converges
to a constant
CURVES WITH DIFFERENT X(0)

Converge to 1
DECISION MAKING IN SOCIAL NETWORKS

Involved decision makers: Social network provider
(SNP), content provider (CP), content creators
(CCr) consumers of content (CoCo).
Goal of SNP, CP, CCr: maximize visibility of content.
Higher visibility (more views) allows SNP, CP and CCr
to receive more advertisements money. The content
itself can be an advertisement which the CCr wishes
to be visible.
ACTORS AND ACTIONS:

SNP: what type of services to offer.

CP: what type of content to specializes in

CCr: have actions available by the SNR (share,
like, embed)

CoCo: can decide what to consume based on
available information (recommendation lists)
A STATIC GAME PROBLEM
R resources (eg content types), M players.
Cost C(ji) for player i to associate with
resource j
The cost depends C(ji) depends on the
number n(j) connected to j. Nondecreasing.
Application: Each of M content providers
has to decide in which type of content to
specialize.
SOLUTION: MAP TO CROWDING GAMES
2. SPLITABLE CASE
A CP can diversify its content
 MAPS to splittable routing games by [ORS]
 The utility is a decreasing function in the total
amount of competing content.

Need to revise the whole routing
game basic results.
YOUTUBE DATA FOR RECOMMENDATIONS
Each video has a recommendation list: set of
recommended videos
 Size of the list N: depends on the screen size.
 Define a weighted recommendation graph.
 Nodes: videos.
 Weight of a node: number of views, or age etc.
 Direct link between A and B if B is in the
recommendation list of A.

MEASUREMENTS AND CURVE FITTING
We take 1000 random videos
 Draw a curve where
 X=number of views of a
video
 Y=average no. of views of
its recommended list.


Not a good fit
THE LOG OF NUMBER OF VIEWS
Horizontal axis: a function f of number of views of
a video
 Vertical axis: average of a function f of the
average no. of
views of videos in its
recommended list.

Good linear fit
 Average(Log(y))= a log(x) + b,

a>1, b>0 for N<5
MARKOV ANALYSIS

Consider a random walk over the
recommendation graph. At time n+1 it visits at
random (uniform probability) one of the videos
recommended at time n.

State x(n )=number of views of a video at step n

Assume: x(n) is Markov.
STABILITY ANALYSIS:

F can serve as a Lyapunov function
E[f(x(n+1)- x(n)|x(n))> (a-1)f(x(n))+b
For N<5 since a>1, the Markov chain is instable
(not positive recurrent).
Therefore the expected time to return to a given
video is infinite. Hence small screen means bad
Page rank.
DYNAMIC GAME MODELS FOR POPULARITY
Markov Decision Processes: We are given a
1. State space
2. Action space
3. Transition probabilities
4. Immediate costs/utilities
5. We define information and strategies
6. Cost criterion to minimize, or payoff to
maximize over a subset of policies V(x,t,u)
7. x- is initial state, t is the horizon, u is the policy

STATES

The state at time T contain all the information
that determines the future evolution for given
choices of control after T
Optimality principle: Let V(x,t) be the optimal
value starting at time 0 at state x till some time
t. Then
V(x,t) = Max E[V(x,s,u)+V(X(s),t-s)]
This is Dynamic Programming PRINCIPLE

CRITERIA

Total cost (reward): E
[ 𝑇−1
+ 𝑔(𝑋 𝑇 )]
𝑡=1 𝑟 𝑋 𝑡 , 𝐴 𝑡
T can be a stopping time. Running reward ( r)
and final reward (g).
 Other criterion: Risk sensitive cost
 Sample path criteria. E.G. sample path total
cost (without expectation). Denote by R(x,t).

DISCRETE TIME TOTAL PAYOFF CRITERION

The optimality principle implies:

V(x,t+1)=Max_a [ r(x,a) +
𝑦𝑃
𝑥, 𝑎, 𝑦 𝑉 𝑦, 𝑡 ]
Total cost: V(x,t) does not depend on t.
 Finite spaces: V is the unique solution of the DP

RISK SENSITIVE COST

Define
J(x,t,u)=Eu [exp ( - a R(x,t) ]
The standard optimality principle does not hold. Instead,
V(x,t) = Max E[V(x,s,u) x V(X(s),t-s)]
We obtain a multiplicative dynamic programming.
Dynammic programming transforms optimization over
strategies to one over actions. In games: NE over
strategies transforms to a set of fixed point equations:
NE over actions.
CONTINUOUS TIME CONTROL: MARKOV CASE

Assume one can go from state x to any state y in
the set S(x). The time T(x,a,y) till a transition occurs
to state y if an action a is used, is exponentially
distributed with parameter L(x,a,y).

Then the next transition from state occurs at a
time T that is the minimum over all y of T(x,a,y). It
is exponentially distributed with parameter
𝐿 𝑥, 𝑎 = 𝑦 𝐿(𝑥, 𝑎, 𝑦)

The next transition is to state y w.p. L(x,a,y)/L(y)
UNIFORMIZATION

We may view this as if there are different exponential
timers in different states.

We may wish to have a single one.

Idea: Assume we have rate L(1) at state 1 and rate
L(2)>L(1) at state 2. Let p=L(1)/L(2). We shall now use
the same rate of transition L(2) in both states, but at
state 1 we shall also allow the possibility of transitions
from state 1 to state 1 which occur with probability 1-p .
These are called fictitious transitions. Only a fraction p
of the transitions ae to othe states, which occur with
rate L(2)p = L(1)
PROBLEM 3:
COMPETING OVER POPULARITY OF CONTENT:
Individuals who wish to disseminate content through
a social network. Goal: visibility, popularity
 Social network provider (SNP) interested in
maximizing the amount of downloads
 Has tools to accelerate the dissemination of popular
content. Example: Recommendation graph
 The SNP can give priority in the recommendation
graph to someone who pays

EXAMPLE: YOUTUBE
EXAMPLE: YOUTUBE
AD 1
AD 2
AD 3
EXAMPLE: YOUTUBE
AD 1
AD 2
AD 3
Recom
graph
A LIST CONTAINING OTHER AD EVENTS:
SHARING AND EMBEDDING
SNOWBALL EPIDEMIC EFFECTS
Other acceleration
Factors:
• Other publishers
Embed content
• Comments and
Responses increase
visibility
Model
N content creators (seeds)– players
 M potential destination
 A destination m is interested in the first content
that it will be aware of.
 Information on content n arrives at a destination
after a time exponentially distributed with
parameter λ(n).
 The goal of a seed: maximize the number of
destinations Xi(T) at time T (T large) that have its
content (dissemination utility).

Player n can accelerate its information process by
a constant a at a cost c(a)
 Uniformization: let

= total utility for player i if at time 0
the system is at state x, player j takes action aj
and the utility to go for player i from the next
transition onwards is v(y) if the state after the
next transition is y.
 Define dessimination utility of player i to be g(xi)
and
ζi (xi) = g(xi+1) – g(xi)


We solve the DP
Fixed Point Eq:

For linear dissemination utility, we can reduce the
state space to the number of destinations that
have some content. 1-dimensional!

Solution: formulate explicit M matrix games, the
equilibrium at matrix m is the equilibrium of the
original game at state m

If Ci(a)=Gi (a-1) (linear in a) then the equilibrium
policy for player I is a threshold (Gi/λi)
STATE AGGREGATION
Possible to aggregate set of states S1, S2, … , Sr
into states if states within Si are not
distinguishable:
 Same transition probabilities from any x in Si to
any Sj
 Same immediate rewards/costs for any x in Si
 Same available actions
The case of no information
This is a differential game with a compact state
space.
Results
Again state space collapce to dimension 1

Equilibrium at state m obtained as equilibrium of
m-th matrix game. Now m is a real number

For linear acceleration cost – same threshold
policies
Results

Semi-dynamic case (policies constant in time):
explicit expressions for the state evolution and
the utility.
Taking the sum, we get: dx/dt = C(M-x)
Hence X(t) = M(1-exp(-Ct))
The case of no information
Let Xi be lim Xi(t) as t-> infinity. Then starting at
X(0)=0, we get
Xi = Ci/(C1+ … + Cn)
Where Ci = lambda(i) w(i)
Assume symmetry
KELLY PROBLEM:

Player I chooses w(i)
Pays g w(i)
 Earns
Ui ( M w(i)/( w(1) + … + w(n) )


There exists a unique equilibrium. Can be
computed using a convex optimization problem.
Results
Semi-dynamic case (policies constant in time):
explicit expressions for the state evolution and
the utility.
 The state is proportional to

GOOD FIT!
MOBILE SOCIAL NETWORKS
Instead of M wireline destinations, consider relay
destinations where
 A mobile relay stores at most one copy of content.
 Mobile users get the content from the relays. An
end user is interested only in a single copy of the
content (e.g. list of open restaurants)
 Only the first content received in a relay is stored
The sources compete over (distributed) memory
(relays) and on visibility space
POWER CONTROL MODEL
The dissemination rate to mobile end users
depend on how many relays have the copy of a
content.
 To reach more relays each of N (mobile)
sources has to transmit with larger power
 The power determines the rate of contacts
between a source and the relays

THE COST
EXPIRATION PROBABILITIES
OBBJECTIVE FUNCTIONS:
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