Social Influence & Popularity

Experiment of Salganik et al. Models Results Conclusions

Social Influence & Popularity

V.A. Traag

March 26, 2009

Experiment of Salganik et al. Models Results Conclusions

Outline

1 Experiment of Salganik et al.

2 Models

3 Results

4 Conclusions

Experiment of Salganik et al. Models Results Conclusions

Introduction

• What items (e.g. movies, books) become popular?

• Quality leads to popularity? (Harry Potter, Da Vinci code,Pirandello)

• Idea emerged from web based experiment of Salganik et al.(Science, 2006)

Experiment of Salganik et al. Models Results Conclusions

Experiment of Salganik et al.

• Study inequality and unpredictability experimentally.

• Set up a website with various songs which could bedownloaded.

• Vary some conditions to study the effect of social influence.

• Use multiple realisations to study unpredictability.

Experiment of Salganik et al. Models Results Conclusions

Experimental design

More social influence 1...

More social influence 8

Social influence 1...

Social influence 8

No social influence 1...

No social influence 8

User arrival

Experiment of Salganik et al. Models Results Conclusions

Screenshots of website

Experiment of Salganik et al. Models Results Conclusions

Screenshots of website

Experiment of Salganik et al. Models Results Conclusions

Main conclusions

• Inequality rises with social influence.

• Unpredictability rises with social influence.

• Unpredictability also rises with ’quality’.

• Result of a rich-get-richer effect?

Experiment of Salganik et al. Models Results Conclusions

BA-model

• Model for links from websites to websites.

• Start out with some small number of websites.

• At each time step add a new website, and add some links.

• Web sites (items) attract links (votes) proportional to thenumber of links (votes) (rich-get-richer effect).

Experiment of Salganik et al. Models Results Conclusions

BA-model

01

2

Experiment of Salganik et al. Models Results Conclusions

BA-model

01

2

3

Experiment of Salganik et al. Models Results Conclusions

BA-model

01

2

3

4

Experiment of Salganik et al. Models Results Conclusions

BA-model

01

2

3

4

5

Experiment of Salganik et al. Models Results Conclusions

BA-model

01

2

3

4

5

6

Experiment of Salganik et al. Models Results Conclusions

BA-model

01

2

3

4

5

6

7

Experiment of Salganik et al. Models Results Conclusions

BA-model

Experiment of Salganik et al. Models Results Conclusions

BA-model

5 10 20 50

0.00

50.

020

0.05

00.

200

0.50

0

k

Pr(

X>

k)

Experiment of Salganik et al. Models Results Conclusions

BA-model

• We can formalise this process with mathematics.

• Web sites (items) attract links (votes) proportional to thenumber of links (votes).

k̇i = mki

∑

j kj

• Yields stationary power law degree distribution.

Pr(X = k) = 2m2k−3

Experiment of Salganik et al. Models Results Conclusions

Social influence

• Add a base-line effect of quality.

• Introduce quality φ ≥ 0 with mean quality µ and variance σ.

• Balance quality and popularity through parameter 0 ≤ λ ≤ 1.

• Additional good-get-richer effect.

• New differential equation

k̇i = m

[

(1 − λ)φi

∑

j φj

+ λki

∑

j kj

]

.

Experiment of Salganik et al. Models Results Conclusions

Theoretical results

0

200

400

600

800

1000

1200

0 100 200 300 400 500

Low QualityHigh Quality

High Social Influence

k

t

Time dependent results:

• Votes increase with time

• Older items obtain morevotes

• Better items obtain morevotes

• Changing social influencechanges growth pattern

Experiment of Salganik et al. Models Results Conclusions

Theoretical results

Results for items with a given quality

• Mean popularity and variance

E (X |φ) =mφ

µand Var(X |φ) =

E (X |φ)2

1 − 2λ.

• Expected number of votes rise with quality

• Uncertainty rises with quality and with social influence

• In congruence with experiment from Salganik et al.

Experiment of Salganik et al. Models Results Conclusions

Theoretical results

Results for items

• Quality distribution is ρ(φ) with mean µ and variance σ.

• In general, mean popularity and variance is

E (X ) = m and Var(X ) =m

2(2σ(1 − λ) + µ2)

µ2(1 − 2λ).

• Inequality in popularity increases with inequality in quality

• Inequality rises with social influence

• Again in congruence with experiment from Salganik et al.

Experiment of Salganik et al. Models Results Conclusions

Theoretical results

10-30

10-25

10-20

10-15

10-10

10-5

100

100 101 102 103 104 105 106 107 108 109

k

Pr(

X=

k)

λ = 0λ = 0.1λ = 0.5

λ = 0.99

Experiment of Salganik et al. Models Results Conclusions

Empirical results

• Quality usually a problem, how to estimate it?

• Workaround: assume a quality distribution (e.g. Dirac,Exponential).

• Compare empirical popularity distribution (#views, #sales) totheoretical distribution.

• Estimate social influence parameter λ using MLE.

Experiment of Salganik et al. Models Results Conclusions

10-4

10-3

10-2

10-1

100

10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

HollywoodYouTube

Fit (Hollywood)Fit (YouTube)

k

Pr(

X>

k)

YouTube1 λ ≈ 0.878

Hollywood1 λ ≈ 0.663 (0.843 for Dirac)

1Assuming an exponential distribution

Experiment of Salganik et al. Models Results Conclusions

Conclusions

Empirical conclusions.

• YouTube shows higher social influence.

• Perhaps a broader distinction (traditional/online)?

• Suggests popular thesis that the Internet individualises isincorrect.

• With massive choices, following others not a bad heuristic?

Experiment of Salganik et al. Models Results Conclusions

Conclusions

Conclusions for model

• Qualitatively congruent with experiment from Salganik.

• Quantitatively not supported by data.

• First rough approximation for modelling the amount of socialinfluence.

• Might be used for getting rough estimates of social influence.

Experiment of Salganik et al. Models Results Conclusions

Questions?