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.
Call Girls In Panjim North Goa 9971646499 Genuine Service
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
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
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
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