Diversity Dynamics in Online Networks (HT'2012)
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Web Science and Technologies
University of Koblenz–Landau, Germany
Diversity Dynamics in Online Networks
Jérôme Kunegis¹ Sergej Sizov¹ Felix Schwagereit¹ Damien Fay²
¹ University of Koblenz–Landau, Germany ² University College Cork, Ireland
HT 2012, Milwaukee, WIJune 27, 2012
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Everyone likes good things:
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Or even better: Diversity!
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Structural Diversity
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(1) Length of paths
“Large” world Small world
Diversity No diversity
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90-percentile effective diameter ±0.9
Hop plot
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“The World Gets Smaller”
(Leskovec, Kleinberg & Faloutsos 2007)
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Outline
(A) How can structural diversity be measured?
(B) How does diversity change?
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(A) How to Measure Diversity in a Network?
(1) Length of paths(2) Numbers of neighbors(3) Size of communities(4) Random walks(5) Controllability
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(2) Number of neighbors
Diversity No diversity
d(i) ¼ d(j) d(i) ¿ d(j)
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Gini Coefficient
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“The Rich Get Richer”
(Barabási & Albert 1999)
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(3) Size of communities
Diversity No diversity
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rankF = §k (¸k / ¸1)2 = (kAkF / kAk2)2
Fractional Rank
Spectrum of the graph = f¸1, ¸2, ¸3, . . .g
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Ui1 Uj1 > ¸1 / 2jEj
“Eigenvector Preferential Attachment”
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(4) 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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“Random Walks Arrive Less Often”
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(5) 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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“Networks Get Easier to Control”
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(B) 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
(1) Diameter 12 8 0
(2) Gini coefficient 13 3 5
(3) Fractional rank 10 6 4
(4) Weighted spectral distribution 12 7 1
(5) Controllability 15 5 0
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Jérôme Kunegis @kunegisSergej SizovFelix SchwagereitDamien Fay
University of Koblenz–Landau, GermanyUniversity College Cork, Ireland
konect.uni-koblenz.de
Thank You
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QuestionsDid you try the power law exponent instead of the Gini coefficient?
→ Yes, but see (Kunegis & Preusse 2012)
Did you try the absolute value instead of the square in rankF ? → Yes, it leads to the nuclear norm instead of the Frobenius
norm
Isn't it hard to find a maximal directed 2-matching? → It takes a runtime of O(|V|1/2 |E|)
How is the approximation using only R eigenvalues for the WSD justified?
→ By observing that all eigenvalues shrink
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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.
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Credits
http://www.shewearsshortshorts.com/2012/01/downside.htmlhttps://twitter.com/#!/justinbieberhttp://www.iconspedia.com/icon/nerd-4255.htmlhttp://hk.digikey.com/1/3/index1227.html