Cascade Model Slides

7/23/2019 Cascade Model Slides

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Modelling Cascades Over Time

in Microblogs

Wei Xie, Feida Zhu, Siyuan Liu and Ke Wang*Living Analytics Research Centre

Singapore Management University

* Ke Wang is from Simon Fraser University, and this work was done when the author was visiting

Living Analytics Research Centre in Singapore Management University.



Motivation

• Business applications such as viral marketinghave driven a lot of research effort predicting

whether a cascade will go viral.

• In real life, there are very few truly viralcascades.

• Previous research work* shows that temporal

features are the key predictor of cascade size.

* Justin Cheng, Lada A. Adamic, P. Alex Dow, Jon M. Kleinberg, Jure Leskovec:

Can cascades be predicted? WWW 2014: 925-936



Time-aware Cascade Model

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P (C(t+ dt)) = P (C(t+ dt)|C(t)) ⋅ P (C(t))

P (C( )) = 1t0

P (C(t+ dt)|C(t)) = (t) ⋅ (1 − (t))∏∈ (t)u i X

(1)

P i ∏∈ (t)u i′ X

(2)

P i ′

(t) = (t, { ;Θ) ⋅ dt i i t j} ∈Followe (t)u j e(i)




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users who have re-shared

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P (C( )) = 1t0

P (C(t+ dt)|C(t)) = (t) ⋅ (1 − (t))∏∈ (t)u i X

(1)

P i ∏∈ (t)u i′ X

(2)

P i ′





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users who have re-shared users who haven’t yet

⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪


P (C( )) = 1t0

P (C(t+ dt)|C(t)) = (t) ⋅ (1 − (t))∏∈ (t)u i X

(1)

P i ∏∈ (t)u i′ X

(2)

P i ′




Observations in Twitter

Observation 1. Only the first re-sharer matters.

(t) = (t, ;Θ) ⋅ dt P i h i t j⋆

where = argmi { | ∈ Followe (t)} j⋆ n j t j u j e(i)



Observations in Twitter

Observation 1. Only the first re-sharer matters.

(t) = (t, ;Θ) ⋅ dt P i h i t j⋆

where = argmi { | ∈ Followe (t)} j⋆ n j t j u j e(i)

Observation 2. The chance of a tweet to beretweeted decreases as time goes by.

(t

) = (τ

;Θ

) ⋅ dt

P i h i

where and is a decreasing function. τ = t − t j⋆ (τ )h i



Hazard Function Design

h(t) = =limdt→0

P (t < T ≤ t+ dt|T > t)

dt

f (t)

1− F (t)




h(t) = =limdt→0

P (t < T ≤ t+ dt|T > t)

dt

f (t)

1− F (t)

H (t) = h(u)du = du = −log(1 − F (u)) = −log(1 − F (t))∫ 0

t

∫ 0

t

(u)F ′

1 − F (u)| t0




h(t) = =limdt→0

P (t < T ≤ t+ dt|T > t)

dt

f (t)

1− F (t)


t

∫ 0

t

(u)F ′

1 − F (u)| t0

F (t) = 1−

e−H (t)




h(t) = =limdt→0

P (t < T ≤ t+ dt|T > t)

dt

f (t)

1− F (t)


t

∫ 0

t

(u)F ′

1 − F (u)| t0

F (t) = 1−

e−H (t)

H (t) =t

λ

⇒ F (t) = 1− e

−

t

λ Exponential distribution




h(t) = =limdt→0

P (t < T ≤ t+ dt|T > t)

dt

f (t)

1− F (t)


t

∫ 0

t

(u)F ′

1 − F (u)| t0

F (t) = 1−

e−H (t)

H (t) =t

λ

⇒ F (t) = 1− e

−

t

λ Exponential distribution

H (t) = (t

α)β

⇒ F (t) = 1− e

−( t

α)β Weibull distribution




H (t) = (t

α)β

H (t) =t

λ

⇒

⇒

F (t) = 1− e−

t

λ

F (t) = 1− e−( t

α)β

Exponential distribution

Weibull distribution




H (t) = (t

α)β

H (t) =t

λ

⇒

⇒

F (t) = 1− e−

t

λ

F (t) = 1− e−( t

α)β



H (∞) = ∞ ⇒ F (∞) = 1 − ⇒ F (∞) = 1e−∞




H (τ ) = λ ⋅ (1 − ( + 1 )τ α

)−β

h(τ ) = = λ ⋅ ⋅ ( + 1dH (τ )

dτ

β

α

τ

α)−(β +1)




H (τ ) = λ⋅(1

−( + 1 )τ α

)−β




H (τ ) = λ⋅(1

−( + 1 )τ α

)−β

scale parameter




H (τ ) = λ⋅(1

−( + 1 )τ α

)−β

scale parametershape parameter

H d F i D i




H (τ ) = λ⋅(1

−( + 1 )τ α

)−β

F (∞) ≈ H (∞) = λ


H d F i D i




H (τ ) = λ⋅(1

−( + 1 )τ α

)−β

F (∞) ≈ H (∞) = λ


describes the eventual re-tweeting probability

H d R Ill i



Hazard Rate Illustration

H d R t Ill t ti




0

4

8

12

16

20

tC 60

R e t w e e t i n g R

a t e

Time (Minute)

H d R t Ill t ti




0

4

8

12

16

20

tC 60

R e t w e e t i n g R

a t e

Time (Minute)

0

4e-4

8e-4

12e-4

16e-4

0 10 20 30 40 50 60

H a

z a r d R a t e

Time (Minute)

Emperical Rate

Estimated Rate

D t t



Dataset

From a Singapore based Twitter data set, we get all the

retweets to construct retweeting cascades. In all we get2,425,348 cascades.

P b bili ti M d l Fitti



Probabilistic Model Fitting

• TMt Threshold Model

where

• TCM-CH Constant Hazard

• TCM-EH Exponential Hazard

• TCM-LH Long tail Hazard (our proposed)

(t) = λ⋅ s(|Followe (t)|)h i e

(i)

s(x) =1

1 + e−a(x−b)

H (τ ) = λ ⋅ τ h(τ ) = = λ

dH (τ )

dτ

H (τ ) = λ ⋅ (1 − ( + 1 ) h(τ ) = = λ ⋅ ⋅ ( + 1τ

α)−β dH (τ )

dτ

β

α

τ

α)−(β +1)

H (τ ) = λ ⋅ (1 − ) h(τ ) = = λ ⋅ k ⋅ e−k⋅τ

dH (τ )

dτ e−k⋅τ





For each cascade, observe its development in first for

training, and the next for testing.∆T

T 0





P di ti C d G th



Predicting Cascade Growth

Virality Prediction



Virality Prediction



Thanks



Our work is based on previouscascade models

• J. Goldenberg, B. Libai, and E. Muller. Talk of the network: A complex systems look at theunderlying process of word-of- mouth. Marketing letters, 12(3):211–223, 2001.

• M.Gomez-Rodriguez,D.Balduzzi,andB.Scho "lkopf.Uncovering the temporal dynamics ofdiffusion networks. In Proceedings of the 28th International Conference on Machine Learning,ICML 2011, Bellevue, Washington, USA, June 28 - July 2, 2011, pages 561–568, 2011.

• S. A. Myers, C. Zhu, and J. Leskovec. Information diffusion and external influence in networks.In The 18th ACM SIGKDD Inter- national Conference on Knowledge Discovery and Data Mining,KDD ’12, Beijing, China, August 12-16, 2012, pages 33–41, 2012.

• M. Gomez-Rodriguez, J. Leskovec, and B. Scho "lkopf. Modeling information propagation with

survival theory. In ICML (3), pages 666–674, 2013.

• N. Du, L. Song, M. Gomez-Rodriguez, and H. Zha. Scalable influence estimation in continuous-time diffusion networks. In Advances in Neural Information Processing Systems 26: 27thAnnual Conference on Neural Information Processing Systems 2013. Proceedings of a meetingheld December 5-8, 2013, Lake Tahoe, Nevada, United States., pages 3147–3155, 2013.

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