Why Beyoncé Is More Popular Than Me – Fairness, Diversity and Other Measures

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

Why Beyoncé Is More Popular Than Me 2

Fairness





(1) The Pareto Principle

“20% of people own 80% of land.”

(In 1910s Italy)

(Pareto, 1919)



UNFAIR



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.



Degree distribution

C(n) » n¡°



TODO



Which distribution is fairer?

Flickr friendships OR Digg replies



Lorenz Curve – Gini Coefficient



Flickr Digg



Perfect Power Laws



High entropy

Low entropy

(2) Entropy



@kunegis

@noshir

@beyonce

@barabasi

@ststaab



Entropy



max He = ln jVj



Normalized Entropy



Gini vs Entropy

½ = ¡0.71p < 0.001



(3) Random Walks

Pret(L) large Pret(L) small

Diversity No diversity



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



Eigenvalues of N



(4) Controllability

(Liu, Slotine & Barabási 2011)

Diversity No diversity



#Knobs needed = jVj { max jMj

Find a maximal directed 2-matching



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



Network Evolution



“The Rich Get Richer”

(Barabási & Albert 1999)



“Random Walks Arrive Less Often”



“Networks Get Easier to Control”



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

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



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



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.

Technology

Why Beyoncé Is More Popular Than Me – Fairness, Diversity and Other Measures