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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 2 / 51
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
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 (> 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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 4 / 51
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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 5 / 51
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.
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 6 / 51
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
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 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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 8 / 51
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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 9 / 51
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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 10 / 51
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
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 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
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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 13 / 51
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.
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 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)]
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 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
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 ∈ 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.
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 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.
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 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.
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 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.
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 19 / 51
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
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
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
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
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
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
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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 21 / 51
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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 22 / 51
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
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 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(·))
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 24 / 51
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(·))
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 25 / 51
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
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��
��
����
����
��
��
����������������
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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 26 / 51
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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 26 / 51
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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 27 / 51
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
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 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
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 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 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
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 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
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 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
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 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
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 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
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 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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 32 / 51
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
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 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 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
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 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
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 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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 35 / 51
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
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)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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 37 / 51
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
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 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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 39 / 51
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
Simulation of V(t) process
Analysis of Integral Equation
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
Analysis of Integral Equation
Simulation of V(t) process
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)
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 40 / 51
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
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 EB1−α as ν →∞
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 41 / 51
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
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 − 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
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 − 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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 42 / 51
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)
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 43 / 51
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
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 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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 45 / 51
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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 46 / 51
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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 47 / 51
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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 48 / 51
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
Anurag Kumar (ECE, IISc, Bangalore) Competition over Timeline 16 January, 2014 48 / 51
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
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 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
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