Fairness on the Web: Alternatives to the Power Law (WebSci 2012)

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This document is a presentation given by Jérôme Kunegis and Julia Preusse from the University of Koblenz–Landau titled "Fairness on the Web: Alternatives to the Power Law" at the WebSci 2012 conference. It discusses how distributions on the web often follow power laws which can lead to unfairness. It then examines alternatives like the Gini coefficient and normalized entropy that can measure fairness more directly. Finally, it compares the power law, Gini, and normalized entropy approaches in terms of generality, interpretation, computation, and coverage.

Automotive Technology Business

Web Science and Technologies
University of Koblenz–Landau, Germany

Fairness on the Web:
Alternatives to the Power Law
Jérôme Kunegis, Julia Preusse
University of Koblenz–Landau
WebSci 2012, June 24
Evanston, IL

Fairness

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 2 / 20

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 3 / 20

The Pareto Principle

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

(Pareto, 1919)

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 4 / 20

A IR
N F
U

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 5 / 20

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 & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 6 / 20

Degree distribution
C(n) » n¡°

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 7 / 20

TODO

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 8 / 20

Which distribution is fairer?

Flickr friendships OR Digg replies

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 9 / 20

Lorenz Curve – Gini Coefficient

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 10 / 20

Flickr Digg

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 11 / 20

Perfect Power Laws

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 12 / 20

Entropy

High entropy

Low entropy

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 13 / 20

@kunegis @beyonce @ststaab

@noshir @barabasi

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 14 / 20

Entropy

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 15 / 20

max He = ln jVj
Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 16 / 20

Normalized Entropy

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 17 / 20

Gini vs Entropy

½ = ¡0.71
p < 0.001

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 18 / 20

Comparison
Power Law Gini Norm. Entropy
Generality Power-law All networks All networks
Interpretation Many (!) Economy Physical
Runtime Slow Fast Fast
Coverage d(u) ≥ dmin All All

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 19 / 20

Thank You
Jérôme Kunegis @kunegis
Julia Preusse

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

konect.uni-koblenz.de

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 20 / 20

References
Pareto, V. Manuale di economia politica con una introduzione alla
scienza sociale (Manual of political economy). Milano : Societa
Editrice Libraria, 1919.

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 21 / 20

Edge distribution Degree distribution
Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 22 / 20

…

Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law
WebSci 2012 23 / 20

This paper presents several measures of fairness and inequality based on the degree distribution in networks, as alternatives to the well-established power-law exponent. Networks such as social networks, communication networks and the World Wide Web itself are often characterized by their unequal distribution of edges: Few nodes are attached to many edges, while many nodes are attached to only few edges. The inequality of such network structures is typically measured using the power-law exponent, stating that the number of nodes with a given degree is proportional to that degree taken to a certain exponent. However, this approach has several weaknesses, such as its narrow applicability and expensive computational complexity. Beyond the fact that power laws are by far not a universal phenomenon on the Web, the power-law exponent has the surprising property of being negatively correlated with the usual notion of inequality, making it unintuitive as a fairness measure. As alternatives, we propose several measures based on the Lorenz curve, which is used in economics but rarely in networks study, and on the information-theoretical concept of entropy. We show in experiments on a large collection of online networks that these measures do not suffer under the drawbacks of the power-law exponent.

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1. Web Science and Technologies University of Koblenz–Landau, Germany Fairness on the Web: Alternatives to the Power Law Jérôme Kunegis, Julia Preusse University of Koblenz–Landau WebSci 2012, June 24 Evanston, IL

2. Fairness Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 2 / 20

3. Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 3 / 20

4. The Pareto Principle “20% of people own 80% of land.” (In 1910s Italy) (Pareto, 1919) Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 4 / 20

5. A IR N F U Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 5 / 20

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 & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 6 / 20

7. Degree distribution C(n) » n¡° Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 7 / 20

8. TODO Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 8 / 20

9. Which distribution is fairer? Flickr friendships OR Digg replies Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 9 / 20

10. Lorenz Curve – Gini Coefficient Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 10 / 20

11. Flickr Digg Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 11 / 20

12. Perfect Power Laws Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 12 / 20

13. Entropy High entropy Low entropy Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 13 / 20

14. @kunegis @beyonce @ststaab @noshir @barabasi Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 14 / 20

15. Entropy Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 15 / 20

16. max He = ln jVj Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 16 / 20

17. Normalized Entropy Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 17 / 20

18. Gini vs Entropy ½ = ¡0.71 p < 0.001 Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 18 / 20

19. Comparison Power Law Gini Norm. Entropy Generality Power-law All networks All networks Interpretation Many (!) Economy Physical Runtime Slow Fast Fast Coverage d(u) ≥ dmin All All Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 19 / 20

20. Thank You Jérôme Kunegis @kunegis Julia Preusse WeST – Institute for Web Science and Technologies University of Koblenz–Landau, Germany konect.uni-koblenz.de Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 20 / 20

21. References Pareto, V. Manuale di economia politica con una introduzione alla scienza sociale (Manual of political economy). Milano : Societa Editrice Libraria, 1919. Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 21 / 20

22. Edge distribution Degree distribution Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 22 / 20

23. … Jérôme Kunegis & Julia Preusse Fairness on the Web: Alternatives to the Power Law WebSci 2012 23 / 20

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