Fairness on the Web: Alternatives to the Power Law

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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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Fairness on the Web: Alternatives to the Power Law

  1. 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. 2. Jérôme Kunegis Fairness on the Web: Alternatives to the Power Lawkunegis@uni-koblenz.de 2 / 19
  3. 3. A IR N F UJérôme Kunegis Fairness on the Web: Alternatives to the Power Lawkunegis@uni-koblenz.de 3 / 19
  4. 4. Edge distribution Degree distribution Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 4 / 19
  5. 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. 6. TODO Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 6 / 19
  7. 7. Flickr Diggtodo Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 7 / 19
  8. 8. The Pareto Principle20% 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. 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. 10. Gini Coefficient – Lorenz Curve Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 10 / 19
  11. 11. todo Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 11 / 19
  12. 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. 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. 14. EntropyJérôme Kunegis Fairness on the Web: Alternatives to the Power Lawkunegis@uni-koblenz.de 14 / 19
  15. 15. Maximum Entropy max He = ln |V|Jérôme Kunegis Fairness on the Web: Alternatives to the Power Lawkunegis@uni-koblenz.de 15 / 19
  16. 16. Normalized Entropy Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 16 / 19
  17. 17. Jérôme Kunegis Fairness on the Web: Alternatives to the Power Lawkunegis@uni-koblenz.de 17 / 19
  18. 18. Comparison Power Law Gini Norm. EntropyGenerality Power-law All networks All networksInterpretation ––– Economy PhysicalRuntime Slow Fast FastCoverage d(u) ≥ dmin All All Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law kunegis@uni-koblenz.de 18 / 19
  19. 19. Thank You!→ 22 - 24 June, Web Science ConferenceDr. Jérôme Kunegis<kunegis@uni-koblenz.de>Institute for Web Science and TechnologiesUniversity of Koblenz–Landau (Campus Koblenz)http://konect.uni-koblenz.de/plots/degree_distributionhttp://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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