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Preferential Attachment in Online Networks:  Measurement and Explanations
Preferential Attachment in Online Networks:  Measurement and Explanations
Preferential Attachment in Online Networks:  Measurement and Explanations
Preferential Attachment in Online Networks:  Measurement and Explanations
Preferential Attachment in Online Networks:  Measurement and Explanations
Preferential Attachment in Online Networks:  Measurement and Explanations
Preferential Attachment in Online Networks:  Measurement and Explanations
Preferential Attachment in Online Networks:  Measurement and Explanations
Preferential Attachment in Online Networks:  Measurement and Explanations
Preferential Attachment in Online Networks:  Measurement and Explanations
Preferential Attachment in Online Networks:  Measurement and Explanations
Preferential Attachment in Online Networks:  Measurement and Explanations
Preferential Attachment in Online Networks:  Measurement and Explanations
Preferential Attachment in Online Networks:  Measurement and Explanations
Preferential Attachment in Online Networks:  Measurement and Explanations
Preferential Attachment in Online Networks:  Measurement and Explanations
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Preferential Attachment in Online Networks: Measurement and Explanations

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We perform an empirical study of the preferential attachment phenomenon …

We perform an empirical study of the preferential attachment phenomenon
in temporal networks and show that on the Web, networks follow a
nonlinear preferential attachment model in which the exponent depends on
the type of network considered. The classical preferential attachment
model for networks by Barabási and Albert (1999) assumes a linear
relationship between the number of neighbors of a node in a network and
the probability of attachment. Although this assumption is widely made
in Web Science and related fields, the underlying linearity is rarely
measured. To fill this gap, this paper performs an empirical
longitudinal (time-based) study on forty-seven diverse Web network
datasets from seven network categories and including directed,
undirected and bipartite networks. We show that contrary to the usual
assumption, preferential attachment is nonlinear in the networks under
consideration. Furthermore, we observe that the deviation from
linearity is dependent on the type of network, giving sublinear
attachment in certain types of networks, and superlinear attachment in
others. Thus, we introduce the preferential attachment exponent $\beta$
as a novel numerical network measure that can be used to discriminate
different types of networks. We propose explanations for the behavior
of that network measure, based on the mechanisms that underly the growth
of the network in question.

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  • 1. Preferential Attachment in Online Networks:Measurement and ExplanationsJérôme Kunegis, Institute for Web Science, University of Koblenz– LandauMarcel Blattner, Laboratory for Web Science, FFHSChristine Moser, VU University AmsterdamACM Web Science 2013With thanks to Hans Akkermans, Rena Bakhshi and Julie BirkholzFunded by the European Communitys Seventh Framework Programme under grant agreement n° 257859, ROBUST
  • 2. Jérôme Kunegiskunegis@uni-koblenz.dePreferential Attachment in Online Networks: Measurement and Explanations2Communication networkAuthorship networkSocial networkcInteraction networkFolksonomyRating networkNetworks Are Everywhere
  • 3. Jérôme Kunegiskunegis@uni-koblenz.dePreferential Attachment in Online Networks: Measurement and Explanations3Power Laws – Scale Free NetworksC(d) ~ d− °Degree (d)Frequency(C(d))
  • 4. Jérôme Kunegiskunegis@uni-koblenz.dePreferential Attachment in Online Networks: Measurement and Explanations4Preferential Attachment Modeld = 3d = 2d = 2d = 4d = 1P({A, i}) ~ d(i)A
  • 5. Jérôme Kunegiskunegis@uni-koblenz.dePreferential Attachment in Online Networks: Measurement and Explanations5Linear vs Nonlinear Preferential Attachmentf(d) ~ 1 Erdős–Rényi model [1]f(d) ~ d¯, 0 < ¯ < 1 Sublinear model [2]f(d) ~ d Barabási–Albert model [3]f(d) ~ d¯, ¯ > 1 Superlinear model [4][1] On Random Graphs I. Paul Erdős & Alfréd Rényi, Publ. MathDebrecen 6 (1959), 290– 197.[2] Random Networks with Sublinear Preferential Attachment:Degree Evolutions. Electrical J. of Probability 14 (2009), 1222– 1267.[3] Emergence of Scaling on Random Networks. Albert-LászlóBarabási & Réka Albert, Science 286, 5439 (1999), 509– 512.[4] Random Trees and General Branching Processes. Random Struct.Algorithms 31, 2 (2007), 186– 202.
  • 6. Jérôme Kunegiskunegis@uni-koblenz.dePreferential Attachment in Online Networks: Measurement and Explanations6Erdős–Rényi Model (1959)P({i, j}) = p●Every edgeequiprobable●No structure●Binomial degreedistributionC(d) ~ pd(1 − p)|V| − 1 − d[1] On Random Graphs I. Paul Erdős & Alfréd Rényi, Publ.Math Debrecen 6 (1959), 290– 197.
  • 7. Jérôme Kunegiskunegis@uni-koblenz.dePreferential Attachment in Online Networks: Measurement and Explanations7Barabási–Albert Model (1999)P({A, i}) ~ d(i)●Generative model●Scale-freenetwork●Power law degreedistributionC(d) ~ d− °[1] Emergence of Scaling on Random Networks. Albert-LászlóBarabási & Réka Albert, Science 286, 5439 (1999), 509– 512.
  • 8. Jérôme Kunegiskunegis@uni-koblenz.dePreferential Attachment in Online Networks: Measurement and Explanations8Sublinear ModelP({A, i}) ~ d(i)¯0 < ¯ < 1●Stretchedexponential degreedistribution[1, Eq. 94][1] Evolution of Networks. Adv. Phys. 51 (2002), 1079– 1187.[2] Random Networks with Sublinear Preferential Attachment: Degree Evolutions. Electrical J. ofProbability 14 (2009), 1222– 1267.
  • 9. Jérôme Kunegiskunegis@uni-koblenz.dePreferential Attachment in Online Networks: Measurement and Explanations9Superlinear ModelP({A, i}) ~ d(i)¯¯ > 1●A single nodeattracts 100% ofedgesasymptotically●Power law degreedistribution in thepre-asymptoticregime[1] Random Trees and General Branching Processes. Random Struct. Algorithms 31, 2 (2007), 186– 202.
  • 10. Jérôme Kunegiskunegis@uni-koblenz.dePreferential Attachment in Online Networks: Measurement and Explanations10+ =Network at time t1Degrees d1(u)Network at time t2Degrees d1(u) + d2(u)Added edgesDegrees d2(u)Temporal Network DataHypothesis: d2 = ® d1¯
  • 11. Jérôme Kunegiskunegis@uni-koblenz.dePreferential Attachment in Online Networks: Measurement and Explanations11Empirical Measurement of βd2 = e®(1 + d1)¯− ¸Find (®, ¯) using least squares:min Σ (® + ¯ ln[1 + d1(u)] { ln[¸ + d2(u)])2" = exp{ 1 / |V| Σ (® + ¯ ln[1 + d1(u)] { ln[¸ + d2(u)])2}®, ¯ u 2Vpu 2V
  • 12. Jérôme Kunegiskunegis@uni-koblenz.dePreferential Attachment in Online Networks: Measurement and Explanations12Example Network: Facebook Wall PostsDescription: User– user wall postsFormat: Edges are directedEdge weights: Multiple edges are possibleMetadata: Edges have timestampsSize: 63,891 verticesVolume: 876,993 edgesAverage degree: 27.45 edges / vertexMaximum degree: 2,696 edgeshttp://konect.uni-koblenz.de/networks/facebook-wosn-wall
  • 13. Jérôme Kunegiskunegis@uni-koblenz.dePreferential Attachment in Online Networks: Measurement and Explanations13Facebook Wall Post Preferential Attachment
  • 14. Jérôme Kunegiskunegis@uni-koblenz.dePreferential Attachment in Online Networks: Measurement and Explanations14Network CategoriesSocial network user– userRating network user– itemCommunication network user– userFolksonomy person– tag/itemWiki edit network editor– articleExplicit interaction network person– personImplicit interaction network person– item
  • 15. Jérôme Kunegiskunegis@uni-koblenz.dePreferential Attachment in Online Networks: Measurement and Explanations15Social network ¯ < 1Rating network ¯ < 1Communication network ¯ < 1Folksonomy ¯ < 1Wiki edit networkExplicit interaction network ¯ > 1Implicit interaction network ¯ > 1Comparison
  • 16. Jérôme Kunegiskunegis@uni-koblenz.dePreferential Attachment in Online Networks: Measurement and Explanations16Thank YouDatasets available at:http://konect.uni-koblenz.de/Read our blog:https://blog.west.uni-koblenz.de/2013-04-29/the-linear-preferential-attachment-assumption-and-its-generalizations/

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