COMPETITION OVER POPULARITY IN SOCIAL NETWORKS

Eitan AltmanMay, 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

Integrating, we get:

Finally.

EXAMPLES WITH X(0) = 0.0001, 0.01, 0.3 k=1 k=3

UNPOPULAR VIDEOWITH MANY VIEWS

President Barack Obama 2009 Inauguration and Address 3 years. Concave? Epidemic?

PROPAGATION MODELS WITHOUT VIRALITY, WITH MAX POPULATION SIZE

Consider the model: This models a constant rate M at which a non-infected node becomes infected. An infected node does not infect others. This gives

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 badPage rank.

DYNAMIC GAME MODELS FOR POPULARITY

Markov Decision Processes: We are given a1. State space2. Action space3. Transition probabilities4. Immediate costs/utilities5. We define information and strategies6. 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 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) +

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

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: YOUTUBEAD 1

AD 3

AD 2

{}

EXAMPLE: YOUTUBEAD 1

AD 3

AD 2

{}

Recomgraph

A LIST CONTAINING OTHER AD EVENTS:SHARING AND EMBEDDING

SNOWBALL EPIDEMIC EFFECTS

Other accelerationFactors:

• Other publishersEmbed content• Comments andResponses increasevisibility

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).

Model

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 AGGREGATIONPossible 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

This is a differential game with a compact state space.

The case of no information

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))

Results

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

The case of no information

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.

Semi-dynamic case (policies constant in time): explicit expressions for the state evolution and the utility.

The state is proportional to

Results

GOOD FIT!

MOBILE SOCIAL NETWORKSInstead 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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