Influence Evolution and Competition via a Social Network Users - - PowerPoint PPT Presentation

Influence Evolution and Competition via a Social Network User’s Timeline

Anurag Kumar Joint 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 Model System Model Analysis of Timeline Occupancy Competition for User Attention Numerical Study

3

Infinite Timeline Model System Model Analysis of Influence Evolution Computation of Expected Influence Competition on the Timeline

4

Conclusion

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 2 / 51

1

Problem Motivation

2

Finite Timeline Model System Model Analysis of Timeline Occupancy Competition for User Attention Numerical Study

3

Infinite Timeline Model System Model Analysis of Influence Evolution Computation of Expected Influence Competition on the Timeline

4

Conclusion

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 3 / 51

Social Networks to Content Networks

Popular Online Social Networks (OSN): Facebook, Twitter, Google+ Massive userbase: Facebook (> 1 billion), Google+ (500million), Twitter (300million) Most OSNs are becoming content-centric

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

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 4 / 51

Towards Social Advertising

Advertising: the main revenue stream for OSNs

Traditional online ads: sponsored search slots, featured links, banner ads Consumers are becoming more immune to traditional advertising Ads cannot be shared to our social circle

Advertising on online social networks

Customized suggestions based on personal/social history Brands have their own pages/accounts

• n the social network

Consumers can share or retweet the promotional content

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 5 / 51

Timelines on Social Networks

Facebook, Twitter use a Timeline based social feed

Reverse chronological - latest entries pushing out

• lder entries

Similar to an email inbox

Google+, So.cl(Microsoft) employ parallel timelines Recently, OSNs also sort the entries according to user preference

Priority Inbox, Facebook’s EdgeRank, etc.

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 6 / 51

Limited User Attention

80 % of the users’ viewing time is spent on the contents above the fold True for most web experience Timeline: User attention is limited to the top few items

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 attention Lerman, Ghosh (ICWSM 2010): Empirical study; Twitter and Digg; interpretation in terms of the different network structures Myers, Leskovec (ICDM 2012): Mutual reinforcement or suppression between information cascades Weng, et al. (2012)

OSN structure; users’ limited attention; influence of information spreaders; the intrinsic quality of the information Model a limited “screen” and “user memory;” probabilistic model for new information arrival, user focus, and information sharing

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 8 / 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 attention Lerman, Ghosh (ICWSM 2010): Empirical study; Twitter and Digg; interpretation in terms of the different network structures Myers, Leskovec (ICDM 2012): Mutual reinforcement or suppression between information cascades Weng, et al. (2012)

OSN structure; users’ limited attention; influence of information spreaders; the intrinsic quality of the information Model a limited “screen” and “user memory;” probabilistic model for new 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 of content from different sources, decay of influence with time Performance analysis and competition analysis

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 8 / 51

1

Problem Motivation

2

Finite Timeline Model System Model Analysis of Timeline Occupancy Competition for User Attention Numerical Study

3

Infinite Timeline Model System Model Analysis of Influence Evolution Computation of Expected Influence Competition on the Timeline

4

Conclusion

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 9 / 51

1

Problem Motivation

2

Finite Timeline Model System Model Analysis of Timeline Occupancy Competition for User Attention Numerical Study

3

Infinite Timeline Model System Model Analysis of Influence Evolution Computation of Expected Influence Competition on the Timeline

4

Conclusion

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 10 / 51

Publisher-Subscriber Model

Bipartite graph between C content creators and I users

Content creators do not consume or share competing content

Simplifying assumptions

All users follow/subscribe to all content creators Absence of content sharing among users: publish-subscribe framework Sufficient to restrict attention to an isolated user’s timeline

• Creators

Consumers Mc i νc I C Ni c

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 11 / 51

User’s Timeline

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

c νc, ν−c := c′=c νc′

• Ni

λ1 λc λ|C|

cK(i)−1 c2 c1

1 c |C|

cK(i)

Timeline of a single user i

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 12 / 51

1

Problem Motivation

2

Finite Timeline Model System Model Analysis of Timeline Occupancy Competition for User Attention Numerical Study

3

Infinite Timeline Model System Model Analysis of Influence Evolution Computation of Expected Influence Competition on the Timeline

4

Conclusion

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 13 / 51

Occupancy Distribution of the Timeline

ck−1 = c c1 cK(i) = c ν−c cK(i)−1 cK(i) νc cK(i) = c ck−1 = c cK(i)−1 cnew

1

= cold

2

cnew

1

= cold

2

cK(i)−1 ck = c

Evolution of a single user’s timeline Timeline state, C(t): vector of contents of timeline at time t Continuous time Markov chain (CTMC)

Theorem

The stationary probability distribution for the CTMC C(t) is given by, πc = ΠK(i)

k=1

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

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 14 / 51

Expected ic-Busy Period

ic-busy period: The duration for which at least one item of c’s content is present in user i’s timeline after first entering the head of the timeline 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−1

c

• Proof sketch: Recursive equation for E[Tic(k)], the duration for which

content c stays on user i’s timeline, starting at position k. E[Tic(0)] = 0, E[Tic(k + 1)] = 1 ν + νc ν E[Tic(K(i))] + ν−c ν E[Tic(k)]

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 15 / 51

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

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

1

Problem Motivation

2

Finite Timeline Model System Model Analysis of Timeline Occupancy Competition for User Attention Numerical Study

3

Infinite Timeline Model System Model Analysis of Influence Evolution Computation of Expected Influence Competition on the Timeline

4

Conclusion

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 17 / 51

Publisher Competition over User’s Timeline

Players: Content creators c ∈ C, |C| = N Strategies: νc ∈ [φ, ∞), ∀c ∈ C Utility of player c: pic − γc (νc − φ)

Linear cost (rate γc) for content generation rate Could model a charge imposed by the social network for additional promotion

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

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 18 / 51

Best Response of a Publisher

max 1 − αK(i)

c

− γc (νc − φ) s.t. νc ≥ φ (≥ 0) Maximizing a concave objective subject to linear inequality constraints

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 19 / 51

Best Response of a Publisher

max 1 − αK(i)

c

− γc (νc − φ) s.t. νc ≥ φ (≥ 0) Maximizing a concave objective subject to linear inequality constraints Li

c(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)

• 1

K(i)

• if this solution > φ, else best response is φ

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 19 / 51

Best Response of a Publisher

max 1 − αK(i)

c

− γc (νc − φ) s.t. νc ≥ φ (≥ 0) Maximizing a concave objective subject to linear inequality constraints Li

c(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)

• 1

K(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 = K

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

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 19 / 51

1

Problem Motivation

2

Finite Timeline Model System Model Analysis of Timeline Occupancy Competition for User Attention Numerical Study

3

Infinite Timeline Model System Model Analysis of Influence Evolution Computation of Expected Influence Competition on the Timeline

4

Conclusion

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 20 / 51

Symmetric Equilibrium: Numerical Study

We have (with N content providers, user timeline size K, cost rate γ) ν = 1 γ K 1 N

• 1 − 1

N K = 1 γ K 1 N e(K ln(1− 1

N ))

For large N this is maximised when K

N ≈ 1

When K << N, too much competition ⇒ use small ν When K >> N, little competition ⇒ use small ν

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 21 / 51

Symmetric Equilibrium: Numerical Study

We have (with N content providers, user timeline size K, cost rate γ) ν = 1 γ K 1 N

• 1 − 1

N K = 1 γ K 1 N e(K ln(1− 1

N ))

For large N this is maximised when K

N ≈ 1

When K << N, too much competition ⇒ use small ν When K >> N, little competition ⇒ use small ν

10 20 30 40 50 60 70 80 90 100 0.2 0.4 0.6 0.8 K Size of timeline Equilibrium rate (per creator) λ γ = 0.5 N=50 N=10 N=100 N=500 N=1000 200 400 600 800 1000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 N Number of competing content creators Equilibrium rate (per creator) λ K=10 K=5 K=1 K=50 K=100 γ = 0.5

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 21 / 51

Symmetric Equilibrium: Numerical Study

We have (with N content providers, user timeline size K, cost rate γ) ν = 1 γ K 1 N

• 1 − 1

N K = 1 γ K 1 N e(K ln(1− 1

N ))

For large N this is maximised when K

N ≈ 1

When K << N, too much competition ⇒ use small ν When K >> N, little competition ⇒ use small ν

10 20 30 40 50 60 70 80 90 100 0.2 0.4 0.6 0.8 K Size of timeline Equilibrium rate (per creator) λ γ = 0.5 N=50 N=10 N=100 N=500 N=1000 200 400 600 800 1000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 N Number of competing content creators Equilibrium rate (per creator) 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 item anywhere on the timeline

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 21 / 51

1

Problem Motivation

2

Finite Timeline Model System Model Analysis of Timeline Occupancy Competition for User Attention Numerical Study

3

Infinite Timeline Model System Model Analysis of Influence Evolution Computation of Expected Influence Competition on the Timeline

4

Conclusion

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 22 / 51

1

Problem Motivation

2

Finite Timeline Model System Model Analysis of Timeline Occupancy Competition for User Attention Numerical Study

3

Infinite Timeline Model System Model Analysis of Influence Evolution Computation of Expected Influence Competition on the Timeline

4

Conclusion

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 23 / 51

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 νc

Poisson point process of arrivals

influence weight distribution Bc(·)

Item of content at position k is identified by (ck, bk)

ck - content source, ck ∈ C bk - 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(·))

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 24 / 51

Timeline Model: User Interaction

User visits timeline at points of a Poisson process of rate µ (counting process A(t)) On each visit, user scans the timeline beginning at the top and terminating after a random number of posts Number of posts seen is geometrically 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(·))

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 25 / 51

Expected Influence

Expected influence during a visit is v = ∞

j=1 bjαj−1

Expectation is over the number of timeline entries seen by the user Thus α serves as a discount factor

We will use this measure to quantify the influence of the timeline V (t) at any given time t When V (t) = v, arrival of an item with influence 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(·))

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 26 / 51

Expected Influence

Expected influence during a visit is v = ∞

j=1 bjαj−1

Expectation is over the number of timeline entries seen by the user Thus α serves as a discount factor

We will use this measure to quantify the influence of the timeline V (t) at any given time t When V (t) = v, arrival of an item with influence 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 influence imparted by the timeline

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 26 / 51

1

Problem Motivation

2

Finite Timeline Model System Model Analysis of Timeline Occupancy Competition for User Attention Numerical Study

3

Infinite Timeline Model System Model Analysis of Influence Evolution Computation of Expected Influence Competition on the Timeline

4

Conclusion

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 27 / 51

Evolution of Expected Influence, V (t)

Focus on single content case

Will be extended to the multiple content case

Note that jumps in V (t) take place only at the content arrival instants Tk, k ≥ 1

V (t) W2 V2 T1 T2 T3 T4 t V1 W1 V3 W3 W4 V4

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

models the decreasing value of information with time

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 28 / 51

Evolution of Expected Influence

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

t→∞

1 A(t) t V (u)dA(u)

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 29 / 51

Evolution of Expected Influence

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

t→∞

1 A(t) t V (u)dA(u) Note that V (t) is a piecewise deterministic Markov process, i.e., Markov processes with deterministic trajectories between random jumps If V (t) is asymptotically stationary and ergodic, the above limit can be obtained a.s. as the expectation w.r.t the stationary distribution

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 29 / 51

Evolution of Expected Influence

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

t→∞

1 A(t) t V (u)dA(u) Note that V (t) is a piecewise deterministic Markov process, i.e., Markov processes with deterministic trajectories between random jumps If V (t) is asymptotically stationary and ergodic, the above limit can be 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 the integral equation

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 29 / 51

Existence of Stationary Distribution, F(·)

Theorem

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

1−α), such that,

∀v ∈ [0, bmax

1−α)

1 Almost surely,

lim

t→∞

1 t t I{V (u)≤v}du = F(v)

2 Almost surely,

lim

t→∞

1 t t V (u)du = ∞ (1 − F(u))du

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 30 / 51

Sketch of Proof

For k ≥ 1, define Wk = V (Tk−)

V (t) W2 V2 T1 T2 T3 T4 t V1 W1 V3 W3 W4 V4 Wk Vk Vk+1 Wk+1 Bk Bk+1 Zk+1 Tk Tk+1

Wk+1 = (αWk + Bk − Zk+1)+ Show that Wk is φ-irreducible and positive Harris recurrent

We use a Foster-Lyapunov criterion (Meyn and Tweedie’s book)

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

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 31 / 51

Quick Review: Why Harris Recurrence?

For a Markov chain Xk, k ≥ 0, on countable X

If there exists an invariant probability mass function πx, x ∈ X If the transition structure is irreducible and aperiodic Then, for all x ∈ X, limk→∞ p(k)

x,y = πy

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 32 / 51

Quick Review: Why Harris Recurrence?

For a Markov chain Xk, k ≥ 0, on countable X

If there exists an invariant probability mass function πx, x ∈ X If 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 state space X

Definition: A Markov chain is φ-irreducible if ∃ a nonzero σ-finite measure 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 aperiodic Let π be an invariant measure on X, i.e., for all Borel sets A in X π(A) =

• X

π(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 G C from which convergence fails

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 32 / 51

Quick Review: Why Harris Recurrence? An Example2

Define G C = {1

2, 1 3, 1 4, . . .}

This chain has stationary distribution π(.) = Uniform[0, 1] and it is φ-irreducible (w.r.t. π) and aperiodic But if X0 = 1

m, m ≥ 2, then

P[Xn =

1 (m+n)∀n] > 0

Thus ||Pn(x, ·) − π(·)|| → 0 fails from the set G C

2Roberts, Gareth O., and Jeffrey S. Rosenthal. ”Harris recurrence of

Metropolis-within-Gibbs and trans-dimensional Markov chains.” The Annals of Applied Probability 16.4 (2006): 2123-2139.

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 33 / 51

Quick Review: Why Harris Recurrence? An Example2

Define G C = {1

2, 1 3, 1 4, . . .}

This chain has stationary distribution π(.) = Uniform[0, 1] and it is φ-irreducible (w.r.t. π) and aperiodic But if X0 = 1

m, m ≥ 2, then

P[Xn =

1 (m+n)∀n] > 0

Thus ||Pn(x, ·) − π(·)|| → 0 fails from the set G C Harris recurrence eliminates such cases Harris Recurrence: A φ-irreducible Markov chain with stationary distribution π(·) is Harris recurrent if ∀A ⊆ X with π(A) > 0, and all x ∈ X, we have P(τA < ∞|X0 = x) = 1.

2Roberts, Gareth O., and Jeffrey S. Rosenthal. ”Harris recurrence of

Metropolis-within-Gibbs and trans-dimensional Markov chains.” The Annals of Applied 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

• x

α

x

B(x − αy)f (y)dy = νp0Bc(x) + ν x Bc(x − αy)f (y)dy

• 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, 1977

Anurag 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

• x

α

x

B(x − αy)f (y)dy = νp0Bc(x) + ν x Bc(x − αy)f (y)dy LHS: Downcrossing rate of level x

unit rate of decay with time arrival of new content with influence that does not compensate for the α discount

• 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, 1977

Anurag 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

• x

α

x

B(x − αy)f (y)dy = νp0Bc(x) + ν x Bc(x − αy)f (y)dy LHS: Downcrossing rate of level x

unit rate of decay with time arrival of new content with influence that does not compensate for the α discount

RHS: Upcrossing rate of level x

arrival of new content which sees the influence process at level 0 (with prob p0), or at level y, so that the incoming influence large enough to cause an upcrossing of x

• 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, 1977

Anurag 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

• x

α

x

B(x − αy)f (y)dy = νp0Bc(x) + ν x Bc(x − αy)f (y)dy LHS: Downcrossing rate of level x

unit rate of decay with time arrival of new content with influence that does not compensate for the α discount

RHS: Upcrossing rate of level x

arrival of new content which sees the influence process at level 0 (with prob p0), or at level y, so that the incoming influence large enough to cause an upcrossing of x

Taking Laplace transforms across the integral equation yields ˜ f (s)(s − ν) + ν˜ b(s)˜ f (αs) = νp0(1 − ˜ b(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, 1977

Anurag 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 Taylor expansion 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(·)

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 35 / 51

Moments of the Stationary Distribution

Differentiation the Laplace transform equation, writing out the Taylor expansion 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)

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 35 / 51

Moments of the Stationary Distribution

Differentiation the Laplace transform equation, writing out the Taylor expansion 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 an M/G/1 queue with immediate discount at arrivals

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 35 / 51

1

Problem Motivation

2

Finite Timeline Model System Model Analysis of Timeline Occupancy Competition for User Attention Numerical Study

3

Infinite Timeline Model System Model Analysis of Influence Evolution Computation of Expected Influence Competition on the Timeline

4

Conclusion

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 36 / 51

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)EV Average 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 rate balance equation to compute EV numerically

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Successive Approximation

p(0) = max(0, 1 − νEB), f (0) = (1 − p(0)

0 )U[0, b 1−α]

We obtain (p(k+1) , 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) = 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 iterations

• f (p0, f ) is sufficiently small

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 38 / 51

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 distributed number of timeline entries seen by the user on each visit

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 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, thus increasing the expected influence of the content

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 39 / 51

Expected Influence EV - Effect of b

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 0.5 1 1.5 2 2.5 3 3.5 b EV Simulation of V(t) process Analysis of Integral Equation α = 0.8 ν = 3

1 2 3 4 5 6 7 8 9 10 5 10 15 20 25 30 35 40 45 50 b EV Analysis of Integral Equation Simulation of V(t) process α = 0.8 ν = 3

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

Increasing b offsets the effect of decrease of influence with time

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

1 1−α)

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Expected Influence EV - Effect of ν

1 2 3 4 5 6 7 8 9 10 0.5 1 1.5 2 2.5 3 3.5 4

ν EV

Simulation of V(t) process Analysis of Integral Equation

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

EB 1−α as ν → ∞

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Expected Influence: Multiple Content Case

C, the set of content creators with rates νc Define ν :=

c∈C νc and ν−c = ν − νc

Vc(t) : the average influence process for content c

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Expected Influence: Multiple Content Case

C, the set of content creators with rates νc Define ν :=

c∈C νc and ν−c = ν − νc

Vc(t) : the average influence process for content c Whenever a content c′ = c arrives, the net influence of content c is scaled down by α

From the perspective of content c, this is equivalent to an arrival with 0 influence Reduce the analysis to the single content case, by introducing point mass at 0 in the arrival influence distribution

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 42 / 51

Expected Influence: Multiple Content Case

C, the set of content creators with rates νc Define ν :=

c∈C νc and ν−c = ν − νc

Vc(t) : the average influence process for content c Whenever a content c′ = c arrives, the net influence of content c is scaled down by α

From the perspective of content c, this is equivalent to an arrival with 0 influence Reduce the analysis to the single content case, by introducing point mass at 0 in the arrival influence distribution

If bc(x) is the probability density function of arrival influence of content c, then the modified distribution would be ν−c ν δ(x) + νc ν bc(x) where δ(x) indicates point mass at 0

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Level Crossing Analysis: Multiple Content Case

Integral equation for fc(·) fc(x) + ν−c

• x

α

x

fc(y)dy + νc

• x

α

x

Bc(x − αy)f (y)dy = νcpc,0Bc

c (x) + νc

x Bc

c (x − αy)fc(y)dy

On taking Laplace transform: ˜ fc(s)(s − ν) + (νc˜ bc(s) + ν−c)˜ f (αs) = νcpc,0(1 − ˜ bc(s)) Expressions for pc,0 and EVc pc,0 = 1 − νcEBc + ν(1 − α)EVc EVc = νcEB2

c − ν(1 − α2)EV 2 c

2(1 − ανcEBc)

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 43 / 51

1

Problem Motivation

2

Finite Timeline Model System Model Analysis of Timeline Occupancy Competition for User Attention Numerical Study

3

Infinite Timeline Model System Model Analysis of Influence Evolution Computation of Expected Influence Competition on the Timeline

4

Conclusion

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 44 / 51

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 νi In real world systems, the influence generated, b, is usually a function

• f established reputation

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

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Symmetric Game between Publishers

Consider ηi = η, bi = b, ∀i ∈ C

N-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 equilibria

• btained by exhaustive search

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N-Player Symmetric Game: Equilibrium Rate vs. N

1 3 5 7 9 11 13 15 0.5 1 1.5 2 2.5 3 N νopt 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 objective function), where the symmetric equilibrium rate peaked at N ≈ K, the timeline size

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Two-Player Rate Competition: Examples of Equilibria

0.25 0.5 0.75 1 1.25 1.5 1.75 2 0.25 0.5 0.75 1 1.25 1.5 1.75 2 ν

1

ν2

η

1 = 1

b

1 =1

η

2 = 1

b

2 = 0.9

η

2 = 1

b

2 = 1

η

2 = 0.8

b

2 = 1

U1 = EV1 − η1b1ν1 U2 = EV2 − η2b2ν2 Best response curves of player 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}

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Two-Player Rate Competition: Examples of Equilibria

0.25 0.5 0.75 1 1.25 1.5 1.75 2 0.25 0.5 0.75 1 1.25 1.5 1.75 2 ν

1

ν2

η

1 = 1

b

1 =1

η

2 = 1

b

2 = 0.9

η

2 = 1

b

2 = 1

η

2 = 0.8

b

2 = 1

U1 = EV1 − η1b1ν1 U2 = EV2 − η2b2ν2 Best response curves of player 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∗ 2

Decrease 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 ν2

Permits player 1 to use lower content generation rate ν1 at equilibrium

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 48 / 51

1

Problem Motivation

2

Finite Timeline Model System Model Analysis of Timeline Occupancy Competition for User Attention Numerical Study

3

Infinite Timeline Model System Model Analysis of Influence Evolution Computation of Expected Influence Competition on the Timeline

4

Conclusion

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 49 / 51

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 timeline Rate (of content generation) game betweem publishers

Infinite size timeline:

Contents carry influence levels Linear model for influence decay with time Modeled the expected influence process Studied competition among publishers

Possible future work

Network effects: content sharing among users Reinforcement between different content types Multiple timelines Experimentation with real data

Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 50 / 51

Acknowledgements CEFIPRA (IFCPAR) Project GANESH (funded by INRIA, France) Department of Science and Technology (DST), Govt. of India

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