Web Science and Technologies
University of Koblenz–Landau, Germany
Why Beyoncé Is More Popular Than MeFairness, Diversity and Other Measures
Jérôme Kunegis
Albert-Ludwigs-Universität Freiburg
July 27, 2012
Jérôme KunegisAlbert-Ludwigs-Universität Freiburg
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Fairness
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(1) The Pareto Principle
“20% of people own 80% of land.”
(In 1910s Italy)
(Pareto, 1919)
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UNFAIR
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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.
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Degree distribution
C(n) » n¡°
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TODO
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Which distribution is fairer?
Flickr friendships OR Digg replies
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Lorenz Curve – Gini Coefficient
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Flickr Digg
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Perfect Power Laws
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High entropy
Low entropy
(2) Entropy
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@kunegis
@noshir
@beyonce
@barabasi
@ststaab
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Entropy
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max He = ln jVj
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Normalized Entropy
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Gini vs Entropy
½ = ¡0.71p < 0.001
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(3) Random Walks
Pret(L) large Pret(L) small
Diversity No diversity
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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
Weighted Spectral Distribution
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Eigenvalues of N
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(4) Controllability
(Liu, Slotine & Barabási 2011)
Diversity No diversity
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#Knobs needed = jVj { max jMj
Find a maximal directed 2-matching
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Generality
Interpretation
Runtime
Coverage
Power Law
Power-law
Many (!)
Slow
d(u) ≥ dmin
Gini
All networks
Economy
Fast
All
Norm. Entropy
All networks
Physical
Fast
All
ComparisonRandom walks
All networks
Simulation
Medium
All
Controllability
All networks
Control theory
Fast
All
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Network Evolution
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“The Rich Get Richer”
(Barabási & Albert 1999)
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“Random Walks Arrive Less Often”
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“Networks Get Easier to Control”
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Experiments20 networks from konect.uni-koblenz.de
9 authorship, 3 communication, 3 social, 3 interaction, 1 rating, 1 physical
Measure Diversity decreasing
No diversity trend
Diversity 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
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Thank YouJérôme Kunegis @kunegis
Work in collaboration with Julia Preusse, Sergej Sizov, Damien Fay & Felix Schwagereit
WeST – Institute for Web Science and TechnologiesUniversity of Koblenz–Landau, Germany
konect.uni-koblenz.de
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ReferencesJ. 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.