influence evolution and competition via a social network user's … · 2014-01-15 · in uence...

Influence Evolution and Competitionvia a Social Network User’s Timeline

Anurag KumarJoint work with Srinivasan Venkatramanan and Eitan Altman

ECE Department, Indian Institute of Science, Bangalore

16 January, 2014

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 1 / 51

Overview

1 Problem Motivation

2 Finite Timeline ModelSystem ModelAnalysis of Timeline OccupancyCompetition for User AttentionNumerical Study

3 Infinite Timeline ModelSystem ModelAnalysis of Influence EvolutionComputation of Expected InfluenceCompetition on the Timeline

4 Conclusion





4 Conclusion


Social Networks to Content Networks

Popular Online Social Networks (OSN):Facebook, Twitter, Google+

Massive userbase: Facebook (> 1billion), Google+ (500million), Twitter(300million)

Most OSNs are becomingcontent-centric

Tool for sharing and discovery of newcontent on the InternetContent: news articles, photos, videos,etc.Users need not own the content


Towards Social Advertising

Advertising: the main revenue streamfor OSNs

Traditional online ads: sponsoredsearch slots, featured links, banner adsConsumers are becoming moreimmune to traditional advertisingAds cannot be shared to our socialcircle

Advertising on online social networks

Customized suggestions based onpersonal/social historyBrands have their own pages/accountson the social networkConsumers can share or retweet thepromotional content


Timelines on Social Networks

Facebook, Twitter use aTimeline based social feed

Reverse chronological -latest entries pushing outolder entriesSimilar to an email inbox

Google+, So.cl(Microsoft)employ parallel timelines

Recently, OSNs also sort theentries according to userpreference

Priority Inbox, Facebook’sEdgeRank, etc.


Limited User Attention

80 % of the users’viewing time is spent onthe contents above thefold

True for most webexperience

Timeline: User attentionis limited to the top fewitems

Source: http://www.nngroup.com/articles/scrolling-and-attention/Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 7 / 51

Literature Survey

Several studies of information flow in online social networks (OSN)and the “dynamics of collective attention”

Wu, Huberman (PNAS 2007): Mutual reinforcement; competition;boredom; show a lognormal distribution for eventual attentionLerman, Ghosh (ICWSM 2010): Empirical study; Twitter and Digg;interpretation in terms of the different network structuresMyers, Leskovec (ICDM 2012): Mutual reinforcement or suppressionbetween information cascadesWeng, et al. (2012)

OSN structure; users’ limited attention; influence of informationspreaders; the intrinsic quality of the informationModel a limited “screen” and “user memory;” probabilistic model fornew information arrival, user focus, and information sharing

We focus on modeling the interaction between publishers on a user’s“timeline”

Incorporating issues such as rates of content arrival, influence ofcontent from different sources, decay of influence with timePerformance analysis and competition analysis





4 Conclusion


Publisher-Subscriber Model

Bipartite graph between Ccontent creators and I users

Content creators do notconsume or share competingcontent

Simplifying assumptions

All users follow/subscribe toall content creatorsAbsence of content sharingamong users:publish-subscribe frameworkSufficient to restrict attentionto an isolated user’s timeline

��

��

CreatorsConsumers

Mc

i

νc

I C

Ni

c


User’s Timeline

Reverse chronological timeline of size K (i)Items of content c generated at points of a Poisson process of rate νcν :=

∑c νc , ν−c :=

∑c ′ 6=c νc ′

��

��

��

��

��

��

��

Ni

λ1 λc λ|C|

User i's timeline

cK(i)−1

c2

c1

1c |C|

cK(i)

Timeline of a single user iAnurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 12 / 51




4 Conclusion


Occupancy Distribution of the Timeline

ck−1 = c

c1

cK(i) 6= c

ν−ccK(i)−1

cK(i)

νc

cK(i) = c

ck−1 = c

cK(i)−1

cnew1 = cold2cnew1 = cold2

cK(i)−1

ck = c

Evolution of a single user’s timeline

Timeline state,C(t): vector ofcontents oftimeline at time t

Continuous timeMarkov chain(CTMC)

Theorem

The stationary probability distribution for the CTMC C(t) is given by,

πc = ΠK(i)k=1

νckν

where ck is the content at the kth position on the timeline.


Expected ic-Busy Period

ic-busy period: The duration for which at least one item of c ’s content ispresent in user i ’s timeline after first entering the head of thetimeline

αc := ν−c

ν

Theorem

The expected ic-busy period is given by

E [Tic(K (i))] =1

νc

(1− α−K(i)

c

1− α−1c

)

Proof sketch: Recursive equation for E [Tic(k)], the duration for whichcontent c stays on user i ’s timeline, starting at position k.

E [Tic(0)] = 0,

E [Tic(k + 1)] =1

ν+νcν

E [Tic(K (i))] +ν−cν

E [Tic(k)]


Probability of Finding Content c on a User’s Timeline

pic := The probability of finding content c in user i ’s timeline(fraction of time)

A measure of effectiveness in getting the user’s attention

Using the expected ic busy period (recalling: αc = ν−c

ν )

pic =E [Tic(K (i))]

E [Tic(K (i))] + 1/νc

= 1− αK(i)c

Can also be obtained from the occupancy distribution

Note that for K (i) = 1, pic = νc/ν





4 Conclusion


Publisher Competition over User’s Timeline

Players: Content creators c ∈ C, |C| = N

Strategies: νc ∈ [φ,∞), ∀c ∈ CUtility of player c : pic − γc (νc − φ)

Linear cost (rate γc) for content generation rateCould model a charge imposed by the social network for additionalpromotion

paid advertising, e.g., featured pages, promoted tweets, etc.


Best Response of a Publisher

max 1− αK(i)c − γc (νc − φ)

s.t. νc ≥ φ (≥ 0)

Maximizing a concave objectivesubject to linear inequalityconstraints

Lic(νc , β) = 1−

(1− νc

νc + ν−c

)K(i)

− γc (νc − φ) + βc (νc − φ)

Best response rate for content creator c obtained by solving for νc in

νc = ν

[1−

(γcν

K (i)

) 1K(i)

]

if this solution > φ, else best response is φ

Theorem

In the symmetric game (γc = γ, ∀c ∈ C), there is a symmetric equilibrium,

at which each player sends either νc = Kγ

(N−1)KN(K+1) or φ, whichever is larger.





4 Conclusion


Symmetric Equilibrium: Numerical Study

We have (with N content providers, user timeline size K , cost rate γ)

ν =1

γK

1

N

(1− 1

N

)K

=1

γK

1

Ne(K ln(1− 1

N ))

For large N this is maximisedwhen K

N ≈ 1

When K << N, too muchcompetition ⇒ use small ν

When K >> N, littlecompetition ⇒ use small ν

10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

KSize of timeline

Equili

brium

rate

(pe

r cre

ato

r)

λ γ = 0.5

N=50

N=10

N=100

N=500

N=1000

0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

N Number of competing content creators

E

qu

ilibrium

rate

(per

cre

ato

r)

λ

K=10

K=5

K=1

K=50

K=100

γ = 0.5

Recall that, in this model, a publisher’s objective is to keep an itemanywhere on the timeline


Symmetric Equilibrium: Numerical Study

We have (with N content providers, user timeline size K , cost rate γ)

ν =1

γK

1

N

(1− 1

N

)K

=1

γK

1

Ne(K ln(1− 1

N ))

For large N this is maximisedwhen K

N ≈ 1

When K << N, too muchcompetition ⇒ use small ν

When K >> N, littlecompetition ⇒ use small ν

10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

KSize of timeline

Equili

brium

rate

(per

cre

ato

r)

λ γ = 0.5

N=50

N=10

N=100

N=500

N=1000

0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

N Number of competing content creators

Equili

brium

rate

(per

cre

ato

r)

λ

K=10

K=5

K=1

K=50

K=100

γ = 0.5

Recall that, in this model, a publisher’s objective is to keep an itemanywhere on the timeline





4 Conclusion


Timeline Model: Content Generation and Influence

Timeline size not restricted to K

C: the set of content creators, N = |C|Each content creator is characterized by

content generation rate νcPoisson point process of arrivals

influence weight distribution Bc(·)Item of content at position k isidentified by (ck , bk)

ck - content source, ck ∈ Cbk - influence weight, bk ∼ Bck (·)

��

��

��

��

��

��

��

��

A(t)µ

(ci+1, bi+1)

(ci, bi)

(c1, b1)

(ν|C|, B|C|(·))(νc, Bc(·))

(c2, b2)

(ν1, B1(·))


Timeline Model: User Interaction

User visits timeline at points of aPoisson process of rate µ (countingprocess A(t))

On each visit, user scans the timelinebeginning at the top and terminatingafter a random number of posts

Number of posts seen isgeometrically distributed

stops at position j , j ≥ 1 with prob.αj−1(1− α)j = 1 denotes the top of the timeline

��

��

��

��

��

��

��

��

A(t)µ

(ci+1, bi+1)

(ci, bi)

(c1, b1)

(ν|C|, B|C|(·))(νc, Bc(·))

(c2, b2)

(ν1, B1(·))


Expected Influence

Expected influence during a visit isv =

∑∞j=1 bjα

j−1

Expectation is over the number oftimeline entries seen by the userThus α serves as a discount factor

We will use this measure to quantifythe influence of the timeline V (t) atany given time t

When V (t) = v , arrival of an item withinfluence b

V (t+) = b + αv

��

��

��

��

��

��

��

��

A(t)µ

(ci+1, bi+1)

(ci, bi)

(c1, b1)

(ν|C|, B|C|(·))(νc, Bc(·))

(c2, b2)

(ν1, B1(·))

The timeline potentially influences the user to take an action

e.g., purchasing a product, sharing the content with friends, etc.

We do not model user actions here, but study the level of influenceimparted by the timeline





4 Conclusion


Evolution of Expected Influence, V (t)

Focus on singlecontent case

Will be extendedto the multiplecontent case

Note that jumps inV (t) take place onlyat the content arrivalinstants Tk , k ≥ 1

V (t)

W2

V2

T1 T2 T3 T4t

V1

W1

V3

W3

W4

V4

Between these instants we assume that the value of V (t) decreases ata constant rate (1 after appropriate scaling of time)

models the decreasing value of information with time


Evolution of Expected Influence

User’s visits do not affect the process V (t), but these visits result inthe user being influenced by the contents of the timelineFor instance, average influence on the user over the visits to thetimeline

limt→∞

1

A(t)

∫ t

0V (u)dA(u)

Note that V (t) is a piecewise deterministic Markov process, i.e.,Markov processes with deterministic trajectories between randomjumpsIf V (t) is asymptotically stationary and ergodic, the above limit canbe obtained a.s. as the expectation w.r.t the stationary distributionThe analysis is in two main steps:

Establish the existence of the stationary distribution F (·)Obtain an integral equation to characterise F (·)Use successive approximation for numerical calculations from theintegral equation




limt→∞

1

A(t)

∫ t

0V (u)dA(u)

Note that V (t) is a piecewise deterministic Markov process, i.e.,Markov processes with deterministic trajectories between randomjumpsIf V (t) is asymptotically stationary and ergodic, the above limit canbe obtained a.s. as the expectation w.r.t the stationary distribution

The analysis is in two main steps:Establish the existence of the stationary distribution F (·)Obtain an integral equation to characterise F (·)Use successive approximation for numerical calculations from theintegral equation




limt→∞

1

A(t)

∫ t

0V (u)dA(u)

Note that V (t) is a piecewise deterministic Markov process, i.e.,Markov processes with deterministic trajectories between randomjumpsIf V (t) is asymptotically stationary and ergodic, the above limit canbe obtained a.s. as the expectation w.r.t the stationary distributionThe analysis is in two main steps:

Establish the existence of the stationary distribution F (·)Obtain an integral equation to characterise F (·)Use successive approximation for numerical calculations from theintegral equation


Existence of Stationary Distribution, F (·)

Theorem

Let 0 ≤ α < 1 and B(·) be supported on [0, bmax]. Then, for the processV (t), there exists a probability distribution F (·) on [0, bmax

1−α), such that,

∀v ∈ [0, bmax1−α)

1 Almost surely,

limt→∞

1

t

∫ t

0I{V (u)≤v}du = F (v)

2 Almost surely,

limt→∞

1

t

∫ t

0V (u)du =

∫ ∞0

(1− F (u))du


Sketch of Proof

For k ≥ 1, define Wk = V (Tk−)

V (t)

W2

V2

T1 T2 T3 T4t

V1

W1

V3

W3

W4

V4

Wk Vk Vk+1Wk+1

Bk Bk+1

Zk+1

Tk Tk+1

t

Wk+1 = (αWk + Bk − Zk+1)+

Show that Wk is φ-irreducible and positive Harris recurrentWe use a Foster-Lyapunov criterion (Meyn and Tweedie’s book)

This implies mean time of return to {0} in {Wk} is finiteThus V (t) is a nonnegative regenerative process with finite meanregeneration timeThe theorem follows from results on regenerative processes


Quick Review: Why Harris Recurrence?

For a Markov chain Xk , k ≥ 0, on countable XIf there exists an invariant probability mass function πx , x ∈ XIf the transition structure is irreducible and aperiodic

Then, for all x ∈ X , limk→∞ p(k)x,y = πy

Consider, however, a Markov chain Xk , k ≥ 0, on a continuous statespace X

Definition: A Markov chain is φ-irreducible if ∃ a nonzero σ-finitemeasure ψ(·) on (X ,F) s.t. P[τA <∞|X0 = x ] > 0,∀x ∈ X and∀A ∈ F with ψ(A) > 0.Suppose Xk is φ-irreducible and aperiodicLet π be an invariant measure on X , i.e., for all Borel sets A in X

π(A) =

∫Xπ(dx)P(x ,A)

Let G be the set of x ∈ X such that limn→∞ ||Pn(x , ·)− π(·)|| = 0;then π(G ) = 1

This allows the possibility of a null set GC from which convergencefails


Quick Review: Why Harris Recurrence? An Example2

Define GC = {12 , 13 , 14 , . . .}This chain has stationary distributionπ(.) = Uniform[0, 1] and it isφ-irreducible (w.r.t. π) and aperiodic

But if X0 = 1m ,m ≥ 2, then

P[Xn = 1(m+n)∀n] > 0

Thus ||Pn(x , ·)− π(·)|| → 0 fails fromthe set GC

Harris recurrence eliminates such cases

Harris Recurrence: A φ-irreducible Markov chain with stationarydistribution π(·) is Harris recurrent if ∀A ⊆ X with π(A) > 0, and allx ∈ X , we have P(τA <∞|X0 = x) = 1.

2Roberts, Gareth O., and Jeffrey S. Rosenthal. ”Harris recurrence ofMetropolis-within-Gibbs and trans-dimensional Markov chains.” The Annals ofApplied Probability 16.4 (2006): 2123-2139.Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 33 / 51

Level Crossing Analysis for F (·)We established existence of F (·): point mass p0; density f (x), x ∈ [0, bmax

1−α)

Using level-crossing analysis3, we obtain

f (x) + ν

∫ xα

xB(x − αy)f (y)dy = νp0Bc(x) + ν

∫ x

0Bc(x − αy)f (y)dy

LHS: Downcrossing rate of level xunit rate of decay with timearrival of new content with influence that does not compensate for theα discount

RHS: Upcrossing rate of level xarrival of new content which sees the influence process at level 0 (withprob p0), orat level y , so that the incoming influence large enough to cause anupcrossing of x

Taking Laplace transforms across the integral equation yields

f̃ (s)(s − ν) + νb̃(s)f̃ (αs) = νp0(1− b̃(s))

3P. H. Brill and M. J. M. Posner, “Level Crossings in Point Processes Appliedto Queues: Single-Server Case,” Operations Research, Vol. 25, No. 4, 1977Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 34 / 51

Level Crossing Analysis for F (·)We established existence of F (·): point mass p0; density f (x), x ∈ [0, bmax

1−α)

Using level-crossing analysis3, we obtain

f (x) + ν

∫ xα

xB(x − αy)f (y)dy = νp0Bc(x) + ν

∫ x

0Bc(x − αy)f (y)dy

LHS: Downcrossing rate of level xunit rate of decay with timearrival of new content with influence that does not compensate for theα discount

RHS: Upcrossing rate of level xarrival of new content which sees the influence process at level 0 (withprob p0), orat level y , so that the incoming influence large enough to cause anupcrossing of x

Taking Laplace transforms across the integral equation yields

f̃ (s)(s − ν) + νb̃(s)f̃ (αs) = νp0(1− b̃(s))3P. H. Brill and M. J. M. Posner, “Level Crossings in Point Processes Applied

to Queues: Single-Server Case,” Operations Research, Vol. 25, No. 4, 1977Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 34 / 51

Moments of the Stationary Distribution

Differentiation the Laplace transform equation, writing out the Taylorexpansion at s = 0, and setting s = 0, we get:

p0 = 1− νEB + ν(1− α)EV

= 1− ν(1− α)

(EB

1− α − EV

)where EV is the expectation of F (·)

Similarly, we can get

EV =νEB2 − ν(1− α2)EV 2

2(1− ανEB)(1)

When α = 1, the process V (t) is the work-in-system of an M/G/1 queue

p0 is the probability of finding the queue empty

EV is the expected work-in-system

With 0 ≤ α < 1,V (t) can be thought of as work-in-system of anM/G/1 queue with immediate discount at arrivals


Moments of the Stationary Distribution

Differentiation the Laplace transform equation, writing out the Taylorexpansion at s = 0, and setting s = 0, we get:

p0 = 1− νEB + ν(1− α)EV

= 1− ν(1− α)

(EB

1− α − EV

)where EV is the expectation of F (·)Similarly, we can get

EV =νEB2 − ν(1− α2)EV 2

2(1− ανEB)(1)

When α = 1, the process V (t) is the work-in-system of an M/G/1 queue

p0 is the probability of finding the queue empty

EV is the expected work-in-system

With 0 ≤ α < 1,V (t) can be thought of as work-in-system of anM/G/1 queue with immediate discount at arrivals





4 Conclusion


Computation of EV

Recall: p0 = 1− ν(EB − (1− α)EV )

Each arrival (occuring at rate ν) brings in an average influence of(EB + αEV )− EV = EB − (1− α)EVAverage arriving workload depends on the existing workload

Also, we note that computation of EV requires EV 2

No closed form expressions possible for EV as in M/G/1 analysis

Successive approximation to numerically obtain EV

We use expressions for p0, definition of EV and the level crossing ratebalance equation to compute EV numerically


Successive Approximation

p(0)0 = max(0, 1− νEB), f (0) = (1− p

(0)0 )U [0, b

1−α ]

We obtain (p(k+1)0 , f (k+1)) from (p

(k)0 , f (k)) via EV (k+1) as follows:

Obtain EV (1) from (p(0)0 , f (0))

p(1)0 = 1− νEB + ν(1− α)EV (1)

We can then obtain f (1) by using (p(1)0 , f (0)) using the rate balance

equation

Iterate until the total variation distance between successive iterationsof (p0, f ) is sufficiently small


Expected Influence EV - Effect of α

Single publisher; content generation rate ν

Each item brings a fixed influence b

Recall that α is the parameter of the geometrically distributednumber of timeline entries seen by the user on each visit

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

2

4

6

8

10

α

EV

Simulation of V(t) process

Analysis of Integral Equation

b = 0.8

ν = 3

For α close to zero, only the top entry’s influence matters

As α increases, the number of entries seen by the user increases, thusincreasing the expected influence of the content


Expected Influence EV - Effect of b

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

0.5

1

1.5

2

2.5

3

3.5

b

EV



α = 0.8

ν = 3

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

50

b

EV



α = 0.8

ν = 3

b, the influence weight is indicative of quality (perhaps, includingreputation)Increasing marginal returns for small ranges of b

Increasing b offsets the effect of decrease of influence with time

As b increases, p0 → 0As p0 approaches 0, the effect of b on EV becomes linear (with slope1

1−α)


Expected Influence EV - Effect of ν

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

ν

EV



b = 0.8

α = 0.8

ν, the content generation rate is indicative of quantity

Diminishing marginal returns for increasing values of ν

EV approaches EB1−α as ν →∞


Expected Influence: Multiple Content Case

C, the set of content creators with rates νc

Define ν :=∑

c∈C νc and ν−c = ν − νcVc(t) : the average influence process for content c

Whenever a content c ′ 6= c arrives, the net influence of content c isscaled down by α

From the perspective of content c , this is equivalent to an arrival with0 influenceReduce the analysis to the single content case, by introducing pointmass at 0 in the arrival influence distribution

If bc(x) is the probability density function of arrival influence ofcontent c , then the modified distribution would be

ν−cνδ(x) +

νcν

bc(x)

where δ(x) indicates point mass at 0


Level Crossing Analysis: Multiple Content Case

Integral equation for fc(·)

fc(x) + ν−c

∫ xα

xfc(y)dy + νc

∫ xα

xBc(x − αy)f (y)dy

= νcpc,0Bcc (x) + νc

∫ x

0Bcc (x − αy)fc(y)dy

On taking Laplace transform:

f̃c(s)(s − ν) + (νc b̃c(s) + ν−c)f̃ (αs) = νcpc,0(1− b̃c(s))

Expressions for pc,0 and EVc

pc,0 = 1− νcEBc + ν(1− α)EVc

EVc =νcEB2

c − ν(1− α2)EV 2c

2(1− ανcEBc)





4 Conclusion


A Payoff Model for Publishers

Model for payoff for publisher i , with content influence bi

(deterministic)Ui = EVi − ηibiνi

where ηi is the cost parameter

Assume that the players have fixed bi and optimize only over νiIn real world systems, the influence generated, b, is usually a functionof established reputation

cannot be changed as easily as changing the content generation rate ν


Symmetric Game between Publishers

Consider ηi = η, bi = b, ∀i ∈ CN-player game with symmetric costs

Suppose Publisher 1 uses rate λ and all others use ν

Write EV1 = v(λ, ν), and

u(λ, ν) = v(λ, ν)− ηbλ

Symmetric equilibrium: we need a ν such that, for all λ,u(λ, ν) ≤ u(ν, ν)

Since EVi can only be obtained numerically, symmetric equilibriaobtained by exhaustive search


N-Player Symmetric Game: Equilibrium Rate vs. N

1 3 5 7 9 11 13 150

0.5

1

1.5

2

2.5

3

N

νo

pt

b = 0.3

α = 0.9

η = 0.095

Equilibrium rate decreases monotonically with number of players N

Contrast with the finite timeline case (with a different objectivefunction), where the symmetric equilibrium rate peaked at N ≈ K ,the timeline size


Two-Player Rate Competition: Examples of Equilibria

0 0.25 0.5 0.75 1 1.25 1.5 1.75 20

0.25

0.5

0.75

1

1.25

1.5

1.75

2

ν1

ν2

η1 = 1

b1 =1

η2 = 1

b2 = 0.9

η2 = 1

b2 = 1

η2 = 0.8

b2 = 1

U1 = EV1 − η1b1ν1

U2 = EV2 − η2b2ν2

Best response curves ofplayer 1 (blue) and player 2(red) for various values of(η2, b2) and (η1 = 1, b1 = 1)

In this example,{νmin, νmax} = {0.1, 2}

If η1 = η2 and b1 = b2, the equilibrium is symmetric ν∗1 = ν∗2Decrease in 2’s cost rate, η2, allows use of larger ν2

Causes player 1 to use higher content generation rate ν1 at equilibrium

Decrease in 2’s influence, b2, causes player 2 to use lower ν2Permits player 1 to use lower content generation rate ν1 at equilibrium





4 Conclusion


Summary and Future Work

Analysed both finite size and infinite size versions of the timeline

Finite size timeline:

Probability of content being anywhere on the timelineRate (of content generation) game betweem publishers

Infinite size timeline:

Contents carry influence levelsLinear model for influence decay with timeModeled the expected influence processStudied competition among publishers

Possible future work

Network effects: content sharing among usersReinforcement between different content typesMultiple timelinesExperimentation with real data


Acknowledgements

CEFIPRA (IFCPAR)

Project GANESH (funded by INRIA, France)

Department of Science and Technology (DST), Govt. of India


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