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.

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
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3. A IR
N F
U
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4. Edge distribution Degree distribution
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5. Power-law degree distribution
C(n) ~ n−γ
Jérôme Kunegis Fairness on the Web: Alternatives to the Power Law
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6. TODO
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7. Flickr Digg
todo
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8. The Pareto Principle
20% of people own 80% of land.
(In 1920 Italy)
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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
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10. Gini Coefficient – Lorenz Curve
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11. todo
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12. Zeta distribution C(n) = n−γ / ζ(γ)
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13. Random Edge→Vertex
P(u) = d(u) / 2|E|
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14. Entropy
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15. Maximum Entropy
max He = ln |V|
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16. Normalized Entropy
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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
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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
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