Modelling Cascades Over Time in Microblogs Wei Xie , Feida Zhu, - - PowerPoint PPT Presentation

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 marketing

have driven a lot of research effort predicting whether a cascade will go viral.

• In real life, there are very few truly viral

cascades.

• 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

u0 u1 u2 u3 u5 u4 t1 t0 t3 t2

t t + dt

u0 u1 u2 u3 u5 u4 t1 t0 t3 t2 t4

Time-aware Cascade Model

u0 u1 u2 u3 u5 u4 t1 t0 t3 t2

t t + dt

u0 u1 u2 u3 u5 u4 t1 t0 t3 t2 t4

Time-aware Cascade Model

u0 u1 u2 u3 u5 u4 t1 t0 t3 t2

t t + dt

u0 u1 u2 u3 u5 u4 t1 t0 t3 t2 t4

⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ P(C(t + dt)) = P(C(t + dt)|C(t)) ⋅ P(C(t)) P(C( )) = 1 t0 P(C(t + dt)|C(t)) = (t) ⋅ (1 − (t)) ∏

∈ (t) ui X(1)

Pi ∏

∈ (t) ui′ X(2)

Pi′

(t) = (t, { ; Θ) ⋅ dt Pi hi tj}

∈Followe (t) uj e(i)

Time-aware Cascade Model

u0 u1 u2 u3 u5 u4 t1 t0 t3 t2

t t + dt

u0 u1 u2 u3 u5 u4 t1 t0 t3 t2 t4

users who have re-shared

⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ P(C(t + dt)) = P(C(t + dt)|C(t)) ⋅ P(C(t)) P(C( )) = 1 t0 P(C(t + dt)|C(t)) = (t) ⋅ (1 − (t)) ∏

∈ (t) ui X(1)

Pi ∏

∈ (t) ui′ X(2)

Pi′

(t) = (t, { ; Θ) ⋅ dt Pi hi tj}

∈Followe (t) uj e(i)

Time-aware Cascade Model

u0 u1 u2 u3 u5 u4 t1 t0 t3 t2

t t + dt

u0 u1 u2 u3 u5 u4 t1 t0 t3 t2 t4

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

⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ P(C(t + dt)) = P(C(t + dt)|C(t)) ⋅ P(C(t)) P(C( )) = 1 t0 P(C(t + dt)|C(t)) = (t) ⋅ (1 − (t)) ∏

∈ (t) ui X(1)

Pi ∏

∈ (t) ui′ X(2)

Pi′

(t) = (t, { ; Θ) ⋅ dt Pi hi tj}

∈Followe (t) uj e(i)

Time-aware Cascade Model

u0 u1 u2 u3 u5 u4 t1 t0 t3 t2

t t + dt

u0 u1 u2 u3 u5 u4 t1 t0 t3 t2 t4

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

⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ P(C(t + dt)) = P(C(t + dt)|C(t)) ⋅ P(C(t)) P(C( )) = 1 t0 P(C(t + dt)|C(t)) = (t) ⋅ (1 − (t)) ∏

∈ (t) ui X(1)

Pi ∏

∈ (t) ui′ X(2)

Pi′

(t) = (t, { ; Θ) ⋅ dt Pi hi tj}

∈Followe (t) uj e(i)

Observations in Twitter

Observation 1. Only the first re-sharer matters.

(t) = (t, ; Θ) ⋅ dt Pi hi tj⋆

where

= argmi { | ∈ Followe (t)} j⋆ nj tj uj e(i)

Observations in Twitter

Observation 1. Only the first re-sharer matters.

(t) = (t, ; Θ) ⋅ dt Pi hi tj⋆

where

= argmi { | ∈ Followe (t)} j⋆ nj tj uj e(i)

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

(t) = (τ ; Θ) ⋅ dt Pi hi

where and is a decreasing function. τ = t − tj⋆

(τ) hi

Hazard Function Design

h(t) = = lim

dt→0

P(t < T ≤ t + dt|T > t) dt f(t) 1 − F(t)

Hazard Function Design

h(t) = = lim

dt→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)) ∫

t

∫

t

(u) F′ 1 − F(u) |t

Hazard Function Design

h(t) = = lim

dt→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)) ∫

t

∫

t

(u) F′ 1 − F(u) |t

F(t) = 1 − e−H(t)

Hazard Function Design

h(t) = = lim

dt→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)) ∫

t

∫

t

(u) F′ 1 − F(u) |t

F(t) = 1 − e−H(t)

H(t) = t λ

⇒

F(t) = 1 − e− t

λ

Exponential distribution

Hazard Function Design

h(t) = = lim

dt→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)) ∫

t

∫

t

(u) F′ 1 − F(u) |t

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

Hazard Function Design

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

Hazard Function Design

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(∞) = ∞ ⇒ F(∞) = 1 − ⇒ F(∞) = 1 e−∞

Hazard Function Design

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(∞) = ∞ ⇒ F(∞) = 1 − ⇒ F(∞) = 1 e−∞

Hazard Function Design

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(∞) = ∞ ⇒ F(∞) = 1 − ⇒ F(∞) = 1 e−∞

Hazard Function Design

H(τ) = λ ⋅ (1 − ( + 1 ) τ α )−β

h(τ) = = λ ⋅ ⋅ ( + 1 dH(τ) dτ β α τ α )−(β+1)

Hazard Function Design

H(τ) = λ ⋅ (1 − ( + 1 ) τ α )−β

Hazard Function Design

H(τ) = λ ⋅ (1 − ( + 1 ) τ α )−β

scale parameter

Hazard Function Design

H(τ) = λ ⋅ (1 − ( + 1 ) τ α )−β

scale parameter shape parameter

Hazard Function Design

H(τ) = λ ⋅ (1 − ( + 1 ) τ α )−β

F(∞) ≈ H(∞) = λ

scale parameter shape parameter

Hazard Function Design

H(τ) = λ ⋅ (1 − ( + 1 ) τ α )−β

F(∞) ≈ H(∞) = λ

scale parameter shape parameter describes the eventual re-tweeting probability

Hazard Rate Illustration

Hazard Rate Illustration

4 8 12 16 20 tC 60 Retweeting Rate Time (Minute)

Hazard Rate Illustration

4 8 12 16 20 tC 60 Retweeting Rate Time (Minute)

4e-4 8e-4 12e-4 16e-4 10 20 30 40 50 60 Hazard Rate Time (Minute) Emperical Rate Estimated Rate

Dataset

From a Singapore based Twitter data set, we get all the retweets to construct retweeting cascades. In all we get 2,425,348 cascades.

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)|) hi 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 ⋅ ⋅ ( + 1 τ 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

Probabilistic Model Fitting

For each cascade, observe its development in first for
 training, and the next for testing.

∆T

T0

Probabilistic Model Fitting

Predicting Cascade Growth

Virality Prediction

Thanks

Our work is based on previous cascade models

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