Fairness on the Web: Alternatives to the Power Law

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This document discusses alternatives to power law distributions for modeling networks. It presents the zeta distribution and normalized entropy as alternatives that can model networks with similar accuracy to power laws but are faster to compute. It also discusses using the Gini coefficient to measure inequality in networks as an alternative to power law fitting. These alternative methods are evaluated based on their generality, interpretability, runtime and ability to model all nodes in a network.

Bildung Business

Web Science and Technologies
University of Koblenz–Landau, Germany

Fairness on the Web:
Alternatives to the Power Law
Jérôme Kunegis
University of Koblenz–Landau
(with Julia Preusse)
GESIS Cologne, Mai 2012

Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 2 / 19

A IR
N F
U
Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 3 / 19

Edge distribution Degree distribution
Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 4 / 19

Power-law degree distribution
C(n) ~ n−γ

Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 5 / 19

TODO

Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 6 / 19

Flickr Digg

todo

Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 7 / 19

The Pareto Principle

20% of people own 80% of land.
(In 1920 Italy)

Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 8 / 19

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.

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 receive 76.9% of project memberships on Github.

27.3% of users receive 72.7% of replies on Digg.

30.4% of all sites receive 69.6% of all hyperlinks on the Web.

Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 9 / 19

Gini Coefficient – Lorenz Curve

Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 10 / 19

todo

Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 11 / 19

Zeta distribution C(n) = n−γ / ζ(γ)

Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 12 / 19

Random Edge→Vertex

P(u) = d(u) / 2|E|

Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 13 / 19

Entropy

Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 14 / 19

Maximum Entropy

max He = ln |V|

Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 15 / 19

Normalized Entropy

Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 16 / 19

Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 17 / 19

Comparison
Power Law Gini Norm. Entropy

Generality Power-law All networks All networks

Interpretation ––– Economy Physical

Runtime Slow Fast Fast

Coverage d(u) ≥ dmin All All

Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 18 / 19

Thank You!
→ 22 - 24 June, Web Science Conference

Dr. Jérôme Kunegis
<kunegis@uni-koblenz.de>
Institute for Web Science and Technologies
University of Koblenz–Landau (Campus Koblenz)

http://konect.uni-koblenz.de/plots/degree_distribution

http://konect.uni-koblenz.de/plots/lorenz_curve

Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
kunegis@uni-koblenz.de 19 / 19

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Fairness on the Web: Alternatives to the Power Law

1. Web Science and Technologies University of Koblenz–Landau, Germany Fairness on the Web: Alternatives to the Power Law Jérôme Kunegis University of Koblenz–Landau (with Julia Preusse) GESIS Cologne, Mai 2012

2. Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 2 / 19

3. A IR N F U Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 3 / 19

4. Edge distribution Degree distribution Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 4 / 19

5. Power-law degree distribution C(n) ~ n−γ Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 5 / 19

6. TODO Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 6 / 19

7. Flickr Digg todo Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 7 / 19

8. The Pareto Principle 20% of people own 80% of land. (In 1920 Italy) Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 8 / 19

9. 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. 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 receive 76.9% of project memberships on Github. 27.3% of users receive 72.7% of replies on Digg. 30.4% of all sites receive 69.6% of all hyperlinks on the Web. Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 9 / 19

10. Gini Coefficient – Lorenz Curve Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 10 / 19

11. todo Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 11 / 19

12. Zeta distribution C(n) = n−γ / ζ(γ) Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 12 / 19

13. Random Edge→Vertex P(u) = d(u) / 2|E| Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 13 / 19

14. Entropy Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 14 / 19

15. Maximum Entropy max He = ln |V| Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 15 / 19

16. Normalized Entropy Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 16 / 19

17. Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 17 / 19

18. Comparison Power Law Gini Norm. Entropy Generality Power-law All networks All networks Interpretation ––– Economy Physical Runtime Slow Fast Fast Coverage d(u) ≥ dmin All All Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 18 / 19

19. Thank You! → 22 - 24 June, Web Science Conference Dr. Jérôme Kunegis <kunegis@uni-koblenz.de> Institute for Web Science and Technologies University of Koblenz–Landau (Campus Koblenz) http://konect.uni-koblenz.de/plots/degree_distribution http://konect.uni-koblenz.de/plots/lorenz_curve Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 19 / 19