Diversity is an important characterization aspect for online social networks that usually denotes the homogeneity of a network's content and structure. This paper addresses the fundamental question of diversity evolution in large-scale online communities over time. In doing so, we study different established notions of network diversity, based on paths in the network, degree distributions, eigenvalues, cycle distributions, and control models. This leads to five appropriate characteristic network statistics that capture corresponding aspects of network diversity: effective diameter, Gini coefficient, fractional network rank, weighted spectral distribution, and number of driver nodes of a network. Consequently, we present and discuss comprehensive experiments with a broad range of directed, undirected, and bipartite networks from several different network categories~-- including hyperlink, interaction, and social networks. An important general observation is that network diversity shrinks over time. From the conceptual perspective, our work generalizes previous work on shrinking network diameters, putting it in the context of network diversity. We explain our observations by means of established network models and introduce the novel notion of eigenvalue centrality preferential attachment.

1. 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, WI
June 27, 2012

2. Everyone likes good things:
Jérôme Kunegis et al. Diversity Dynamics in Online Networks 2

3. Or even better: Diversity!
Jérôme Kunegis et al. Diversity Dynamics in Online Networks 3

4. Structural Diversity
Jérôme Kunegis et al. Diversity Dynamics in Online Networks 4

5. (1) Length of paths
Diversity No diversity
“Large” world Small world
Jérôme Kunegis et al. Diversity Dynamics in Online Networks 5

6. 90-percentile effective diameter ±0.9
Hop plot
Jérôme Kunegis et al. Diversity Dynamics in Online Networks 6

7. “T
he
Wo
rld
Ge
ts
Sm
a lle
r”
(Leskovec, Kleinberg & Faloutsos 2007)
Jérôme Kunegis et al. Diversity Dynamics in Online Networks 7

8. Outline
(A) How can structural diversity be measured?
(B) How does diversity change?
Jérôme Kunegis et al. Diversity Dynamics in Online Networks 8

9. (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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10. (2) Number of neighbors
Diversity No diversity
d(i) ¼ d(j) d(i) ¿ d(j)
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11. Gini Coefficient
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12. “The Rich Get Richer”
(Barabási & Albert 1999)
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13. (3) Size of communities
Diversity No diversity
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14. Fractional Rank
Spectrum of the graph = f¸1, ¸2, ¸3, . . .g
rankF = §k (¸k / ¸1) 2
= (kAkF / kAk2)2
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15. “Eigenvector Preferential Attachment”
Ui1 Uj1 > ¸1 / 2jEj
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16. (4) Random walks
Diversity No diversity
Pret(L) large Pret(L) small
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17. Weighted Spectral Distribution
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
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18. Eigenvalues of N
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19. “R
an
do
m
Wa
lks
Ar
riv
eL
es
sO
ft e
n”
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20. (5) Controllability
Diversity No diversity
(Liu, Slotine & Barabási 2011)
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21. Find a maximal directed 2-matching
#Knobs needed = jVj { max jMj
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22. “Ne
two
rks
Get
E as
ier
to C
ont
rol”
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23. (B) Experiments
20 networks from konect.uni-koblenz.de
9 authorship, 3 communication, 3 social, 3
interaction, 1 rating, 1 physical
Measure Diversity No diversity Diversity
decreasing trend 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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24. Thank You
Jérôme Kunegis @kunegis
Sergej Sizov
Felix Schwagereit
Damien Fay
University of Koblenz–Landau, Germany
University College Cork, Ireland
konect.uni-koblenz.de
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25. Questions
Did 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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26. References
J. 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.
Jérôme Kunegis et al. Diversity Dynamics in Online Networks 26

27. Credits
http://www.shewearsshortshorts.com/2012/01/downside.html
https://twitter.com/#!/justinbieber
http://www.iconspedia.com/icon/nerd-4255.html
http://hk.digikey.com/1/3/index1227.html
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