Social Influence & Popularity
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Transcript of 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
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Experiment of Salganik et al. Models Results Conclusions
BA-model
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Experiment of Salganik et al. Models Results Conclusions
BA-model
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Experiment of Salganik et al. Models Results Conclusions
BA-model
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Experiment of Salganik et al. Models Results Conclusions
BA-model
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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?