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