Diversity Dynamics in Online Networks (HT'2012)

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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.

Business Technology

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

Everyone likes good things:

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 2

Or even better: Diversity!

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 3

Structural Diversity

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 4

(1) Length of paths

Diversity No diversity

“Large” world Small world

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 5

90-percentile effective diameter ±0.9

Hop plot

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 6

“T
he
Wo
rld
Ge
ts
Sm
a lle
r”

(Leskovec, Kleinberg & Faloutsos 2007)
Jérôme Kunegis et al. Diversity Dynamics in Online Networks 7

Outline

(A) How can structural diversity be measured?
(B) How does diversity change?

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 8

(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

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 9

(2) Number of neighbors

Diversity No diversity

d(i) ¼ d(j) d(i) ¿ d(j)

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 10

Gini Coefficient

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 11

“The Rich Get Richer”

(Barabási & Albert 1999)
Jérôme Kunegis et al. Diversity Dynamics in Online Networks 12

(3) Size of communities
Diversity No diversity

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 13

Fractional Rank

Spectrum of the graph = f¸1, ¸2, ¸3, . . .g

rankF = §k (¸k / ¸1) 2
= (kAkF / kAk2)2

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 14

“Eigenvector Preferential Attachment”

Ui1 Uj1 > ¸1 / 2jEj

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 15

(4) Random walks
Diversity No diversity

Pret(L) large Pret(L) small

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 16

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

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 17

Eigenvalues of N

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 18

“R
an
do
m
Wa
lks
Ar
riv
eL
es
sO
ft e
n”

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 19

(5) Controllability
Diversity No diversity

(Liu, Slotine & Barabási 2011)

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 20

Find a maximal directed 2-matching

#Knobs needed = jVj { max jMj
Jérôme Kunegis et al. Diversity Dynamics in Online Networks 21

“Ne
two
rks
Get
E as
ier
to C
ont
rol”

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 22

(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

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 23

Thank You
Jérôme Kunegis @kunegis
Sergej Sizov
Felix Schwagereit
Damien Fay

University of Koblenz–Landau, Germany
University College Cork, Ireland

konect.uni-koblenz.de

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 24

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

Jérôme Kunegis et al. Diversity Dynamics in Online Networks 25

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

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Diversity Dynamics in Online Networks (HT'2012)

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 Jérôme Kunegis et al. Diversity Dynamics in Online Networks 9

10. (2) Number of neighbors Diversity No diversity d(i) ¼ d(j) d(i) ¿ d(j) Jérôme Kunegis et al. Diversity Dynamics in Online Networks 10

11. Gini Coefficient Jérôme Kunegis et al. Diversity Dynamics in Online Networks 11

12. “The Rich Get Richer” (Barabási & Albert 1999) Jérôme Kunegis et al. Diversity Dynamics in Online Networks 12

13. (3) Size of communities Diversity No diversity Jérôme Kunegis et al. Diversity Dynamics in Online Networks 13

14. Fractional Rank Spectrum of the graph = f¸1, ¸2, ¸3, . . .g rankF = §k (¸k / ¸1) 2 = (kAkF / kAk2)2 Jérôme Kunegis et al. Diversity Dynamics in Online Networks 14

15. “Eigenvector Preferential Attachment” Ui1 Uj1 > ¸1 / 2jEj Jérôme Kunegis et al. Diversity Dynamics in Online Networks 15

16. (4) Random walks Diversity No diversity Pret(L) large Pret(L) small Jérôme Kunegis et al. Diversity Dynamics in Online Networks 16

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 Jérôme Kunegis et al. Diversity Dynamics in Online Networks 17

18. Eigenvalues of N Jérôme Kunegis et al. Diversity Dynamics in Online Networks 18

19. “R an do m Wa lks Ar riv eL es sO ft e n” Jérôme Kunegis et al. Diversity Dynamics in Online Networks 19

20. (5) Controllability Diversity No diversity (Liu, Slotine & Barabási 2011) Jérôme Kunegis et al. Diversity Dynamics in Online Networks 20

21. Find a maximal directed 2-matching #Knobs needed = jVj { max jMj Jérôme Kunegis et al. Diversity Dynamics in Online Networks 21

22. “Ne two rks Get E as ier to C ont rol” Jérôme Kunegis et al. Diversity Dynamics in Online Networks 22

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 Jérôme Kunegis et al. Diversity Dynamics in Online Networks 23

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 Jérôme Kunegis et al. Diversity Dynamics in Online Networks 24

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 Jérôme Kunegis et al. Diversity Dynamics in Online Networks 25

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 Jérôme Kunegis et al. Diversity Dynamics in Online Networks 27

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