[PPT] - Modeling the structure and evolution of online discussion cascades PowerPoint Presentation

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Introduction Likelihood-based framework Conclusions

Modeling the structure and evolution of online discussion cascades

Andreas Kaltenbrunner

Social Media Research Group, Barcelona Media, Barcelona, Spain

School of advanced sciences of Luchon, July 4th, 2014

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions

Outline

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Introduction Motivation Datasets

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Likelihood-based framework Model definition Parameter estimation Validation

3

Conclusions

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

Agenda

Structure and evolution of online discussion cascades

Gómez V., Kappen H. J., Litvak N., and Kaltenbrunner, A. (2012). A likelihood-based framework for the analysis of discussion threads. World Wide Web Journal, vol. 16, no. 5-6, pages 645–675, 2013. Gómez V., Kappen H. J., and Kaltenbrunner, A. (2011). Modelling the Structure and Evolution of Discussion Cascades. In HT2011 22nd ACM Conference on Hypertext and Hypermedia, , Eindhoven, The Netherlands.

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

Outline

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Introduction Motivation Datasets

2

Likelihood-based framework Model definition Parameter estimation Validation

3

Conclusions

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

Motivation

Example of online discussion (from Slashdot) Title: "Can Ordinary PC Users Ditch Windows for Linux?. Online conversations as networks: nodes correspond to comments, edges represent a reply action.

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

Motivation - Online discussion threads

Scientific questions What are the structural patterns governing these responses? What determines the growth of a conversation? Is there a generative model that captures their statistical properties? Can we use the model parameters to characterize websites, user behaviour, discussions? Implications / Applications Understanding communication in large webspaces that comprise many-to-many interaction. Understanding diffusion of news and opinion in social networks. Community management, forum design/maintenance, ...

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

Outline

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Introduction Motivation Datasets

2

Likelihood-based framework Model definition Parameter estimation Validation

3

Conclusions

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

Online discussion threads

Datasets

We collected data from the following sources: Slashdot (SL) : Technological news aggregator. 473, 065 discussions, 2 · 106 comments, 93 · 103 users Barrapunto (BP) : Spanish version of Slashdot. 44, 208 discussions, 4 · 105 comments, 50 · 103 users Meneame (MN) : Spanish Digg clone (general news aggregator) 58, 613 discussions, 2.1 · 106 comments, 5, 4 · 104 users. Wikipedia (WK) : discussion pages related to every article. 871, 485 discussions, ≈ 107 comments, 3.5 · 105 users.

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

Motivation

Example of discussion in Slashdot (post):

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

Motivation

Example of discussion in Slashdot (comments):

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

Motivation

Example of discussion in Barrapunto (comments):

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

Motivation

Example of discussion in Meneame:

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

Motivation

Example of discussion in Wikipedia (I)

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

Motivation

Example of discussion in Wikipedia (II)

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

How to measure the complexity of a Discussion?

Using the h-index of a discussion introduced in [Gómez 2008]

A balanced depth measure maximal number h: at least h comments at level (depth) h, but not h + 1 comments at level h + 1. There are h sub- threads of depth at least h. Example: h-index=3 Other possibility: number of chains consecutive replies between two users example chain of length 4: A ← B ← A ← B

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

Most discussed Wikipedia articles

In parenthesis: rank according to the corresponding variable

1Climatic Research Unit Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

Temporal patterns. [Kaltenbrunner 2007]

Time series of total number of comments

24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552 576 600 624 648 672 696 720 200 400 600 800 1000

hours num of comments September 2005 Thursday 01/09/2005 Friday 30/09/2005 (a)

24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552 576 600 624 648 672 696 720 1 2 3 4 5

hours number of posts (b)

0 1224 84 168 5 10 15 x 10

10

Period (hours) FFT comments (c) Comments all year

0 1224 84 168 5 10 15 x 10

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Period (hours) FFT posts (d) Posts all year

"Sustained" activity coupled with the circadian rhythm.

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

Temporal patterns. [Kaltenbrunner 2007]

Single post level analysis

60 120 180 240 300 5 10 15 20 time num comments (a) data 1−LN approx 2−LN approx 60 120 180 240 300 2 4 6 8 time num comments (b) data 1−LN approx 2−LN approx 10 10

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0.2 0.4 0.6 0.8 1 time cdf post id : 0532219 pvalue 1LN : 0.07 pvalue 2LN : 0.57 (c) 10 10

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0.2 0.4 0.6 0.8 1 time cdf post id : 1231259 pvalue 1LN : 0.84 pvalue 2LN : 0.95 (d)

Posts create cascades of comments which propagate over the network. All posts show a stereotyped behaviour. Response times can be described using a log-normal distribution.

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

Online discussion threads

Examples of real discussions Typical cascades for each website: Degrees Slashdot:

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Motivation Datasets

Online discussion threads

Global analysis

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# threads thread size 10 10

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10 complementary cdf thread size SL BP MN WK SL BP MN WK

SL, BP and MN present a distribution with a defined scale. Cascade sizes in Wikipedia "seem to be" scale-free.

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Outline

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Introduction Motivation Datasets

2

Likelihood-based framework Model definition Parameter estimation Validation

3

Conclusions

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Model definition

Our approach: The model must reproduce:

The statistical structure of threads. Their evolution.

No content involved. No authorship. Essentially "Which comment is going to be replied next?" Empirical facts: Popular comments receive more replies: preferential attachment. New comments are more attractive than old ones. Replies to the post behave different than replies to comments.

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Model definition

Thread representation: vector of parent nodes π, where πt denotes the parent of the node with id t + 1 added at time-step t. π0 = () π1 = (1) . . .

1 8 3 2 4 7 5 6 9 10 ?

π = π π

1 2 1 5 2 1 6 ? 1

time t = 0 t = 1 t = 2 t = 3 t = 6 t = 5 t = 8 t = 7 t = 9 t = 4

Parameters of the model At time t, the popularity of node k is its degree: dk,t(π(1:t−1)) =

1 + t−1

m=2 δkπm

for k ∈ {1, . . . , t}

therwise,

(dk,t weighted by α) At time t, the novelty of node k is nk,t = τ t−k+1, τ ∈ [0, 1]. Root bias: The bias of a node k is is either zero or β for the root: bk = β, for k = 1, and 0 otherwise.

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Model definition

We define a model by means of its associated attractiveness function φ(·), which is defined for each of the nodes. At time t + 1, a new node is linked to node k with probability: p(πt = k|π(1:t−1)) = φ(k) Zt , Zt =

t

l=1

φ(l), Different model variants:

Model Attractiveness funct. φ(·) Parameters θ Constraint Full model (FM) αdk,t + bk + τ t−k+1 {α, τ, β} without popularity (NO-α) bk + τ t−k+1 {τ, β} α = 0 without novelty (NO-τ) αdk,t + bk + 1 {α, β} τ = 1 without bias (NO-bias) αdk,t + τ t−k+1 {α, τ} β = 0

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Outline

1

Introduction Motivation Datasets

2

Likelihood-based framework Model definition Parameter estimation Validation

3

Conclusions

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Parameter estimation

Maximum likelihood

We can compute the likelihood of the full model The likelihood of a set Π := {π1, . . . πN} of N trees with respective sizes |πi|, i ∈ {1, . . . N}, given the values of θ can be written as: L(Π|θ) =

N

i=1

p(πi|θ) =

N

i=1

|πi|

t=2

p(πt,i|π(1:t−1),i, θ) =

N

i=1

|πi|

t=2

φ(πt,i) Zt,i We minimise the negative of the log-likelihood function: − log L(Π|θ) = −

N

i=1

|πi|

t=2

φ(πt,i) − log Zt,i.

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Parameter estimation

Validation

For each model: Choose θ∗ randomly. Generate N threads. Find estimates ˆ θ. Compute residuals θ∗ − ˆ θ. Repeat for 100 times. Estimation is unbiased. Good estimates can be

btained using N = 500.

−0.5 0.5 β α τ N = 50 Residual

FM

β α τ N = 500 β α τ N = 5000 β α τ N = 50000 −0.5 0.5 β τ Residual

no−α

β τ β τ β τ −0.5 0.5 β α Residual

no−τ

β α β α β α −0.5 0.5 α τ Residual

no−bias

α τ α τ α τ

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Parameter estimation

Model Comparison

For each dataset: Select N = 5 · 104 threads randomly with replacement. Find estimates ˆ θ. Compute likelihoods. Repeat for 100 times. Model comparison based on likelihoods for each dataset.

4.15 4.2 4.25 4.3 4.35 x 10

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Slashdot Mean negative log−likelihood

no−bias no−τ no−α FM

7 7.1 7.2 7.3 7.4 x 10

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Barrapunto Mean negative log−likelihood

no−bias no−τ no−α FM

2.8 3 3.2 3.4 3.6 3.8 x 10

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Meneame Mean negative log−likelihood

no−bias no−τ no−α FM

0.8 1 1.2 1.4 1.6 x 10

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Wikipedia Mean negative log−likelihood

no−bias no−τ no−α FM

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Parameter estimation

Parameter estimates for the different datasets

Dataset log β α τ N = 50 SL 2.39 (0.17) 0.31 (0.02) 0.98 (0.02) BP 0.93 (0.12) 0.08 (0.04) 0.92 (0.00) MN 1.66 (0.16) 0.03 (0.01) 0.72 (0.04) WK −0.21 (0.81) 0.00 (0.00) 0.40 (0.19) N = 5000 SL 2.39 (0.01) 0.31 (0.01) 0.98 (0.00) BP 0.96 (0.02) 0.08 (0.00) 0.92 (0.00) MN 1.69 (0.03) 0.02 (0.00) 0.74 (0.01) WK 0.39 (0.22) 0.00 (0.00) 0.60 (0.01) Bootstrap with N = 50 threads already gives good estimates.

−0.1 0.1 0.2 0.3 0.4 0.5 1 1.5 α τ Dataset parameters SL BP MN WK Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Outline

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Introduction Motivation Datasets

2

Likelihood-based framework Model definition Parameter estimation Validation

3

Conclusions

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Growing tree model for discussion threads

Validation of the model We calculate the following quantities from the empirical data and from the synthetic threads produced by the model: Degrees distribution. Subtree sizes distribution. Mean node depth versus size. Node depths distribution. Size of the discussion N is drawn from the empirical distribution. We use model NO-BIAS for comparison [Kumar 2010].

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Barrapunto dataset

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size depth 5 10 15 20 0.1 0.2 0.3 depth probability data no−bias FM

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Slashdot dataset

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Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Meneame dataset

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Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Wikipedia dataset

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Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Growing tree model for discussion threads

Real cascades: Synthetic cascades:

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Evolution of mean depths and mean widths

FULL MODEL:

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2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 time depth FULL MODEL: mean depth over time SL: data SL: model BP: data BP: model MN: data MN: model WK: data WK: model 10 10

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NO-BIAS model:

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10 20 30 40 50 60 70 time width NO BIAS: mean width over time SL: data SL: model BP: data BP: model MN: data MN: model WK: data WK: model

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Theoretical Result

Asymptotics for degree distribution in FM

It can be shown that ... the degree distribution follows (asymptotically) a power-law with exponent 3. The parameter τ does not affect the power-law exponent but ... formally c1x−2 ≤ P(degree ≥ x) ≤ c2x−2, 0 ≤ c1 ≤ c2 with τ affecting c2 which is bounded by exp(

τ 1−τ ).

Thus τ can affect the fraction of nodes with a degree larger than x by several orders of magnitude.

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Related work

Galton-Watson branching process

Idea Tree grows level by level. Nodes at level i receive a random number of child-nodes at level i + 1 (according to a probability distribution). Pros The model is simple. Explains chain letter trees (combined with a selection bias)

[Golub 2010].

Cons Not a generative model. Does not capture the order of message creation.

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Model definition Parameter estimation Validation

Related work II

T-MODEL [Kumar 2010]

Features Equivalent to model NO-bias with an extra parameter to model the death of a discussion. Model is illustrated on USENET. Authorship model (TI-model). Both are independent of the structure. Could be build on top of other structural models as well. T-model re-creates a power-law relation in the data between size and depth of the discussions, but is not the best model.

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions

Conclusions and future work

Conclusions Framework which allows to re-create discussions with similar structural features as real instances. Likelihood-based optimisation on the entire cascade evolution. Large datasets are not necessary. Parameters allow to characterize audience and platform:

Same platform : differences between SL and BP . Influence of the interface: MN (flat) characterised by bias. Main difference between news media and WK: popularity.

Future work Include prior authorship structure in model and analysis Application to other types of information cascades.

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions

Bibliography I

B. Golub & M.O. Jackson.

Using selection bias to explain the observed structure of Internet diffusions. Proceedings of the National Academy of Sciences, vol. 107, no. 24, page 10833, 2010. Vicenç Gómez, Andreas Kaltenbrunner & Vicente López. Statistical analysis of the social network and discussion threads in Slashdot. In WWW ’08: Proceeding of the 17th international conference on World Wide Web, pages 645–654, New York, NY, USA, 2008. ACM.

V. Gómez, H. J. Kappen & A. Kaltenbrunner.

Modeling the structure and evolution of discussion cascades. In HT ’11, Eindhoven, The Netherlands, 2011. ACM.

Kaltenbrunner A. Online discussion threads

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Introduction Likelihood-based framework Conclusions

Bibliography II

V. Gómez, H. J. Kappen, N. Litvak & A. Kaltenbrunner.

A likelihood-based framework for the analysis of discussion threads. World Wide Web, vol. 16, no. 5-6, pages 645–675, 2013. Andreas Kaltenbrunner, Vicenç Gómez & Vicente López. Description and Prediction of Slashdot Activity. In Proceedings of the 5th Latin American Web Congress (LA-WEB 2007), Santiago de Chile, 2007. IEEE Computer Society.

R. Kumar, M. Mahdian & M. McGlohon.

Dynamics of conversations. In SIGKDD ’10, pages 553–562, New York, USA, 2010. ACM.

D. Laniado, R. Tasso, Y. Volkovich & A. Kaltenbrunner.

When the Wikipedians talk: Network and tree structure of Wikipedia discussion pages. In ICWSM-11 - 5th International AAAI Conference on Weblogs and Social Media. The AAAI Press, 2011.

Kaltenbrunner A. Online discussion threads

Modeling the structure and evolution of online discussion cascades

Andreas Kaltenbrunner

Social Media Research Group, Barcelona Media, Barcelona, Spain

School of advanced sciences of Luchon, July 4th, 2014

Outline

1

Introduction Motivation Datasets

2

Likelihood-based framework Model definition Parameter estimation Validation

3

Conclusions

Agenda

Structure and evolution of online discussion cascades

Outline

1

Introduction Motivation Datasets

2

Likelihood-based framework Model definition Parameter estimation Validation

3

Conclusions

Motivation

Example of online discussion (from Slashdot) Title: "Can Ordinary PC Users Ditch Windows for Linux?. Online conversations as networks: nodes correspond to comments, edges represent a reply action.

Motivation - Online discussion threads

Outline

1

Introduction Motivation Datasets

2

Likelihood-based framework Model definition Parameter estimation Validation

3

Conclusions

Online discussion threads

Datasets

Motivation

Example of discussion in Slashdot (post):

Motivation

Example of discussion in Slashdot (comments):

Motivation

Example of discussion in Barrapunto (comments):

Motivation

Example of discussion in Meneame:

Motivation

Example of discussion in Wikipedia (I)

Motivation

Example of discussion in Wikipedia (II)

How to measure the complexity of a Discussion?

Using the h-index of a discussion introduced in [Gómez 2008]

Most discussed Wikipedia articles

Top 20 articles ordered by number of chains in the discussion [Laniado 2011]

In parenthesis: rank according to the corresponding variable

Temporal patterns. [Kaltenbrunner 2007]

Time series of total number of comments

"Sustained" activity coupled with the circadian rhythm.

Temporal patterns. [Kaltenbrunner 2007]

Single post level analysis

Posts create cascades of comments which propagate over the network. All posts show a stereotyped behaviour. Response times can be described using a log-normal distribution.

Online discussion threads

Examples of real discussions Typical cascades for each website: Degrees Slashdot:

Online discussion threads

Global analysis

SL, BP and MN present a distribution with a defined scale. Cascade sizes in Wikipedia "seem to be" scale-free.

Outline

1

Introduction Motivation Datasets

2

Likelihood-based framework Model definition Parameter estimation Validation

3

Conclusions

Model definition

Our approach: The model must reproduce:

The statistical structure of threads. Their evolution.

No content involved. No authorship. Essentially "Which comment is going to be replied next?" Empirical facts: Popular comments receive more replies: preferential attachment. New comments are more attractive than old ones. Replies to the post behave different than replies to comments.

Model definition

Thread representation: vector of parent nodes π, where πt denotes the parent of the node with id t + 1 added at time-step t. π0 = () π1 = (1) . . .

Parameters of the model At time t, the popularity of node k is its degree: dk,t(π(1:t−1)) =

for k ∈ {1, . . . , t}

(dk,t weighted by α) At time t, the novelty of node k is nk,t = τ t−k+1, τ ∈ [0, 1]. Root bias: The bias of a node k is is either zero or β for the root: bk = β, for k = 1, and 0 otherwise.

Model definition

We define a model by means of its associated attractiveness function φ(·), which is defined for each of the nodes. At time t + 1, a new node is linked to node k with probability: p(πt = k|π(1:t−1)) = φ(k) Zt , Zt =

t

φ(l), Different model variants: