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Web Science and Technologies                                          University of Koblenz–Landau, Germany    Diversity D...
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                              S...
90-percentile effective diameter ±0.9                                   Hop plot       Jérôme Kunegis et al.   Diversity D...
“T                                                  he                                                         Wo         ...
Outline(A) How can structural diversity be measured?(B) How does diversity change?         Jérôme Kunegis et al.   Diversi...
(A) How to Measure Diversity in a Network?      (1) Length of paths      (2) Numbers of neighbors      (3) Size of communi...
(2) Number of neighborsDiversity                                    No diversityd(i) ¼ d(j)                               ...
Gini CoefficientJérôme Kunegis et al.   Diversity Dynamics in Online Networks   11
“The Rich Get Richer”(Barabási & Albert 1999)                       Jérôme Kunegis et al.   Diversity Dynamics in Online N...
(3) Size of communitiesDiversity                                    No diversity    Jérôme Kunegis et al.   Diversity Dyna...
Fractional Rank Spectrum of the graph = f¸1, ¸2, ¸3, . . .grankF =   §k (¸k / ¸1)              2                          ...
“Eigenvector Preferential Attachment”                        Ui1 Uj1 > ¸1 / 2jEjJérôme Kunegis et al.   Diversity Dynamics...
(4) Random walks Diversity                                     No diversityPret(L) large                                Pr...
Weighted Spectral DistributionPret(L) = §(i, j, . . . k) (d(i) d(j) . . . d(k)){1                  = tr(NL)               ...
Eigenvalues of N  Jérôme Kunegis et al.   Diversity Dynamics in Online Networks   18
“R          an            do                        m                            Wa                                 lks   ...
(5) ControllabilityDiversity                                    No diversity                                  (Liu, Slotin...
Find a maximal directed 2-matching   #Knobs needed = jVj { max jMj       Jérôme Kunegis et al.   Diversity Dynamics in Onl...
“Ne                        two                           rks                                 Get                          ...
(B) Experiments       20 networks from konect.uni-koblenz.de       9 authorship, 3 communication, 3 social, 3       intera...
Thank YouJérôme Kunegis                                               @kunegisSergej SizovFelix SchwagereitDamien FayUnive...
QuestionsDid you try the power law exponent instead of the Gini coefficient?    → Yes, but see (Kunegis & Preusse 2012)Did...
ReferencesJ. Kunegis, S. Sizov, F. Schwagereit, D. Fay. Diversity Dynamics inOnline Networks. Proc. Conf. on Hypertext and...
Creditshttp://www.shewearsshortshorts.com/2012/01/downside.htmlhttps://twitter.com/#!/justinbieberhttp://www.iconspedia.co...
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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.

Publié dans : Business, Technologie
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Diversity Dynamics in Online Networks (HT'2012)

  1. 1. Web Science and Technologies University of Koblenz–Landau, Germany Diversity Dynamics in Online Networks Jérôme Kunegis¹ Sergej Sizov¹ Felix Schwagereit¹ Damien Fay²¹ University of Koblenz–Landau, Germany ² University College Cork, Ireland HT 2012, Milwaukee, WI June 27, 2012
  2. 2. Everyone likes good things: Jérôme Kunegis et al. Diversity Dynamics in Online Networks 2
  3. 3. Or even better: Diversity! Jérôme Kunegis et al. Diversity Dynamics in Online Networks 3
  4. 4. Structural Diversity Jérôme Kunegis et al. Diversity Dynamics in Online Networks 4
  5. 5. (1) Length of paths Diversity No diversity“Large” world Small world Jérôme Kunegis et al. Diversity Dynamics in Online Networks 5
  6. 6. 90-percentile effective diameter ±0.9 Hop plot Jérôme Kunegis et al. Diversity Dynamics in Online Networks 6
  7. 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. 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. 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. 10. (2) Number of neighborsDiversity No diversityd(i) ¼ d(j) d(i) ¿ d(j) Jérôme Kunegis et al. Diversity Dynamics in Online Networks 10
  11. 11. Gini CoefficientJérôme Kunegis et al. Diversity Dynamics in Online Networks 11
  12. 12. “The Rich Get Richer”(Barabási & Albert 1999) Jérôme Kunegis et al. Diversity Dynamics in Online Networks 12
  13. 13. (3) Size of communitiesDiversity No diversity Jérôme Kunegis et al. Diversity Dynamics in Online Networks 13
  14. 14. Fractional Rank Spectrum of the graph = f¸1, ¸2, ¸3, . . .grankF = §k (¸k / ¸1) 2 = (kAkF / kAk2)2 Jérôme Kunegis et al. Diversity Dynamics in Online Networks 14
  15. 15. “Eigenvector Preferential Attachment” Ui1 Uj1 > ¸1 / 2jEjJérôme Kunegis et al. Diversity Dynamics in Online Networks 15
  16. 16. (4) Random walks Diversity No diversityPret(L) large Pret(L) small Jérôme Kunegis et al. Diversity Dynamics in Online Networks 16
  17. 17. Weighted Spectral DistributionPret(L) = §(i, j, . . . k) (d(i) d(j) . . . d(k)){1 = tr(NL) = § k ¸k Lwhere ¸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. 18. Eigenvalues of N Jérôme Kunegis et al. Diversity Dynamics in Online Networks 18
  19. 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. 20. (5) ControllabilityDiversity No diversity (Liu, Slotine & Barabási 2011) Jérôme Kunegis et al. Diversity Dynamics in Online Networks 20
  21. 21. Find a maximal directed 2-matching #Knobs needed = jVj { max jMj  Jérôme Kunegis et al. Diversity Dynamics in Online Networks 21
  22. 22. “Ne two rks Get E as ier to C ont rol”Jérôme Kunegis et al. Diversity Dynamics in Online Networks 22
  23. 23. (B) Experiments 20 networks from konect.uni-koblenz.de 9 authorship, 3 communication, 3 social, 3 interaction, 1 rating, 1 physicalMeasure 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. 24. Thank YouJérôme Kunegis @kunegisSergej SizovFelix SchwagereitDamien FayUniversity of Koblenz–Landau, GermanyUniversity College Cork, Irelandkonect.uni-koblenz.de Jérôme Kunegis et al. Diversity Dynamics in Online Networks 24
  25. 25. 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 normIsnt 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 WSDjustified? → By observing that all eigenvalues shrink Jérôme Kunegis et al. Diversity Dynamics in Online Networks 25
  26. 26. ReferencesJ. Kunegis, S. Sizov, F. Schwagereit, D. Fay. Diversity Dynamics inOnline Networks. Proc. Conf. on Hypertext and Social Media, 2012.J. Kunegis, J. Preusse. Fairness on the Web: Alternatives to the PowerLaw. Proc. Web Science Conf., 2012.Y.-Y. Liu, J.-J. Slotine, A.-L. Barabási. Controllability of ComplexNetworks. Nature, 473:167–173, May 2011.J. Leskovec, J. Kleinberg, C. Faloutsos. Graph Evolution: Densificationand 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. 27. Creditshttp://www.shewearsshortshorts.com/2012/01/downside.htmlhttps://twitter.com/#!/justinbieberhttp://www.iconspedia.com/icon/nerd-4255.htmlhttp://hk.digikey.com/1/3/index1227.html Jérôme Kunegis et al. Diversity Dynamics in Online Networks 27

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