Why Beyoncé Is More Popular Than Me – Fairness, Diversity and Other Measures

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Jérôme KUNEGIS

Technology Entertainment & Humor

Web Science and Technologies
University of Koblenz–Landau, Germany

Why Beyoncé Is More Popular Than Me
Fairness, Diversity and Other Measures

Jérôme Kunegis

Albert-Ludwigs-Universität Freiburg

July 27, 2012

Fairness

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 2

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 3

(1) The Pareto Principle

“20% of people own 80% of land.”
(In 1910s Italy)

(Pareto, 1919)

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 4

A IR
N F
U

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
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8.9% of bands make up 91.1% of plays on Last.fm.
14.6% of user groups account for 75.4% of group memberships on Flickr.
17.4% of movies receive 82.6% of ratings on MovieLens.
17.7% of proﬁles receive 82.3% of ratings on Czech dating site Libimseti.cz.
19.7% of all cats receive 80.3% of all friendships on Catster.com.
20.3% of all users receive 79.7% of “friend” and “foe” links on Slashdot.
21.3% of users receive 78.7% of wall posts on Facebook.
22.9% of users make up 77.1% of all “@” mentions on Twitter.
23.1% of projects make up 76.9% of project memberships on Github.
27.3% of users receive 72.7% of replies on Digg.
27.6% of all hamsters receive 72.4% of all friendships on Hamsterster.com.
35.7% of all Twitter users receive 64.3% of all follows.
Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 6

Degree distribution
C(n) » n¡°

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 7

TODO

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 8

Which distribution is fairer?

Flickr friendships OR Digg replies

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 9

Lorenz Curve – Gini Coefficient

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
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Flickr Digg

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
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Perfect Power Laws

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
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(2) Entropy

High entropy

Low entropy

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
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@kunegis @beyonce @ststaab

@noshir @barabasi

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 14

Entropy

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
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max He = ln jVj
Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 16

Normalized Entropy

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 17

Gini vs Entropy

½ = ¡0.71
p < 0.001

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
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(3) Random Walks
Diversity No diversity

Pret(L) large Pret(L) small
Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 19

Weighted Spectral Distribution
Pret(L) = §(i, j, . . . k) (d(i) d(j) . . . d(k)){1
= tr(NL)
= § k ¸k L

where ¸k are eigenvalues of N = D{1/2 A D{1/2.

Here: Use L = 4 and k · R

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 20

Eigenvalues of N

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 21

(4) Controllability
Diversity No diversity

(Liu, Slotine & Barabási 2011)

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 22

Find a maximal directed 2-matching

#Knobs needed = jVj { max jMj
Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 23

Comparison
Power Law Gini Norm. Entropy Random walks Controllability

Generality Power-law All networks All networks All networks All networks

Interpretation Many (!) Economy Physical Simulation Control theory

Runtime Slow Fast Fast Medium Fast

Coverage d(u) ≥ dmin All All All All

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 24

Network Evolution

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 25

“The Rich Get Richer”

(Barabási & Albert 1999)
Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 26

“R
an
do
m
Wa
lks
Ar
riv
eL
es
sO
ft e
n”

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 27

“Ne
two
rks
Get
E as
ier
to C
ont
rol”

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 28

Experiments
20 networks from konect.uni-koblenz.de

9 authorship, 3 communication, 3 social, 3
interaction, 1 rating, 1 physical
Measure Diversity No diversity Diversity
decreasing trend increasing
Diameter 12 8 0
Gini coefficient 13 3 5
Fractional rank 10 6 4
Weighted spectral distribution 12 7 1
Controllability 15 5 0

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 29

Thank You
Jérôme Kunegis @kunegis

Work in collaboration with Julia Preusse, Sergej Sizov, Damien
Fay & Felix Schwagereit

WeST – Institute for Web Science and Technologies
University of Koblenz–Landau, Germany

konect.uni-koblenz.de

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
Albert-Ludwigs-Universität Freiburg 30

References
J. Kunegis, S. Sizov, F. Schwagereit, D. Fay. Diversity Dynamics in Online
Networks. Proc. Conf. on Hypertext and Social Media, 2012.
J. Kunegis, J. Preusse. Fairness on the Web: Alternatives to the Power Law.
Proc. Web Science Conf., 2012.
Y.-Y. Liu, J.-J. Slotine, A.-L. Barabási. Controllability of Complex Networks.
Nature, 473:167–173, May 2011.
J. Leskovec, J. Kleinberg, C. Faloutsos. Graph Evolution: Densification and
Shrinking Diameters. ACM Trans. Knowledge Discovery from Data, 1(1):1–40,
2007.
A.-L. Barabási, R. Albert. Emergence of Scaling in Random Networks.
Science, 286(5439):509–512, 1999.
V. Pareto. Manuale di economia politica con una introduzione alla scienza
sociale (Manual of political economy). Milano : Societa Editrice Libraria, 1919.

Jérôme Kunegis Why Beyoncé Is More Popular Than Me
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Why Beyoncé Is More Popular Than Me – Fairness, Diversity and Other Measures

1. Web Science and Technologies University of Koblenz–Landau, Germany Why Beyoncé Is More Popular Than Me Fairness, Diversity and Other Measures Jérôme Kunegis Albert-Ludwigs-Universität Freiburg July 27, 2012

2. Fairness Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 2

3. Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 3

4. (1) The Pareto Principle “20% of people own 80% of land.” (In 1910s Italy) (Pareto, 1919) Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 4

5. A IR N F U Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 5

6. 8.9% of bands make up 91.1% of plays on Last.fm. 14.6% of user groups account for 75.4% of group memberships on Flickr. 17.4% of movies receive 82.6% of ratings on MovieLens. 17.7% of proﬁles receive 82.3% of ratings on Czech dating site Libimseti.cz. 19.7% of all cats receive 80.3% of all friendships on Catster.com. 20.3% of all users receive 79.7% of “friend” and “foe” links on Slashdot. 21.3% of users receive 78.7% of wall posts on Facebook. 22.9% of users make up 77.1% of all “@” mentions on Twitter. 23.1% of projects make up 76.9% of project memberships on Github. 27.3% of users receive 72.7% of replies on Digg. 27.6% of all hamsters receive 72.4% of all friendships on Hamsterster.com. 35.7% of all Twitter users receive 64.3% of all follows. Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 6

7. Degree distribution C(n) » n¡° Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 7

8. TODO Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 8

9. Which distribution is fairer? Flickr friendships OR Digg replies Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 9

10. Lorenz Curve – Gini Coefficient Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 10

11. Flickr Digg Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 11

12. Perfect Power Laws Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 12

13. (2) Entropy High entropy Low entropy Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 13

14. @kunegis @beyonce @ststaab @noshir @barabasi Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 14

15. Entropy Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 15

16. max He = ln jVj Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 16

17. Normalized Entropy Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 17

18. Gini vs Entropy ½ = ¡0.71 p < 0.001 Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 18

19. (3) Random Walks Diversity No diversity Pret(L) large Pret(L) small Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 19

20. Weighted Spectral Distribution Pret(L) = §(i, j, . . . k) (d(i) d(j) . . . d(k)){1 = tr(NL) = § k ¸k L where ¸k are eigenvalues of N = D{1/2 A D{1/2. Here: Use L = 4 and k · R Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 20

21. Eigenvalues of N Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 21

22. (4) Controllability Diversity No diversity (Liu, Slotine & Barabási 2011) Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 22

23. Find a maximal directed 2-matching #Knobs needed = jVj { max jMj Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 23

24. Comparison Power Law Gini Norm. Entropy Random walks Controllability Generality Power-law All networks All networks All networks All networks Interpretation Many (!) Economy Physical Simulation Control theory Runtime Slow Fast Fast Medium Fast Coverage d(u) ≥ dmin All All All All Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 24

25. Network Evolution Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 25

26. “The Rich Get Richer” (Barabási & Albert 1999) Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 26

27. “R an do m Wa lks Ar riv eL es sO ft e n” Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 27

28. “Ne two rks Get E as ier to C ont rol” Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 28

29. Experiments 20 networks from konect.uni-koblenz.de 9 authorship, 3 communication, 3 social, 3 interaction, 1 rating, 1 physical Measure Diversity No diversity Diversity decreasing trend increasing Diameter 12 8 0 Gini coefficient 13 3 5 Fractional rank 10 6 4 Weighted spectral distribution 12 7 1 Controllability 15 5 0 Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 29

30. Thank You Jérôme Kunegis @kunegis Work in collaboration with Julia Preusse, Sergej Sizov, Damien Fay & Felix Schwagereit WeST – Institute for Web Science and Technologies University of Koblenz–Landau, Germany konect.uni-koblenz.de Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 30

31. References J. Kunegis, S. Sizov, F. Schwagereit, D. Fay. Diversity Dynamics in Online Networks. Proc. Conf. on Hypertext and Social Media, 2012. J. Kunegis, J. Preusse. Fairness on the Web: Alternatives to the Power Law. Proc. Web Science Conf., 2012. Y.-Y. Liu, J.-J. Slotine, A.-L. Barabási. Controllability of Complex Networks. Nature, 473:167–173, May 2011. J. Leskovec, J. Kleinberg, C. Faloutsos. Graph Evolution: Densification and Shrinking Diameters. ACM Trans. Knowledge Discovery from Data, 1(1):1–40, 2007. A.-L. Barabási, R. Albert. Emergence of Scaling in Random Networks. Science, 286(5439):509–512, 1999. V. Pareto. Manuale di economia politica con una introduzione alla scienza sociale (Manual of political economy). Milano : Societa Editrice Libraria, 1919. Jérôme Kunegis Why Beyoncé Is More Popular Than Me Albert-Ludwigs-Universität Freiburg 31

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Why Beyoncé Is More Popular Than Me – Fairness, Diversity and Other Measures