Preferential Attachment in Online Networks: Measurement and Explanations

Jérôme Kunegis, Institute for Web Science, University of Koblenz– LandauMarcel Blattner, Laboratory for Web Science, FFHS

Christine Moser, VU University Amsterdam

ACM Web Science 2013

With thanks to Hans Akkermans, Rena Bakhshi and Julie BirkholzFunded by the European Community's Seventh Framework Programme under grant agreement n° 257859, ROBUST

Jérôme Kunegiskunegis@uni-koblenz.de

Preferential Attachment in Online Networks: Measurement and Explanations 2

Communication network

Authorship network

Social network

c

Interaction network

Folksonomy

Rating network

Networks Are Everywhere

Jérôme Kunegiskunegis@uni-koblenz.de

Preferential Attachment in Online Networks: Measurement and Explanations 3

Power Laws – Scale Free Networks

C(d) ~ d − °

Degree (d)

Fre

quen

cy (C

(d))

Jérôme Kunegiskunegis@uni-koblenz.de

Preferential Attachment in Online Networks: Measurement and Explanations 4

Preferential Attachment Model

d = 3

d = 2

d = 2d = 4

d = 1

P({A, i}) ~ d(i)A

Jérôme Kunegiskunegis@uni-koblenz.de

Preferential Attachment in Online Networks: Measurement and Explanations 5

Linear vs Nonlinear Preferential Attachment

f(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. Math Debrecen 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-László Barabási & Réka Albert, Science 286, 5439 (1999), 509– 512.[4] Random Trees and General Branching Processes. Random Struct. Algorithms 31, 2 (2007), 186– 202.

Jérôme Kunegiskunegis@uni-koblenz.de

Preferential Attachment in Online Networks: Measurement and Explanations 6

Erdős–Rényi Model (1959)

P({i, j}) = p

● Every edge equiprobable● No structure

● Binomial degree distribution

C(d) ~ pd (1 − p)|V| − 1 − d[1] On Random Graphs I. Paul Erdős & Alfréd Rényi, Publ. Math Debrecen 6 (1959), 290– 197.

Jérôme Kunegiskunegis@uni-koblenz.de

Preferential Attachment in Online Networks: Measurement and Explanations 7

Barabási–Albert Model (1999)

P({A, i}) ~ d(i)

● Generative model

● Scale-free network

● Power law degree distribution

C(d) ~ d − °[1] Emergence of Scaling on Random Networks. Albert-László Barabási & Réka Albert, Science 286, 5439 (1999), 509– 512.

Jérôme Kunegiskunegis@uni-koblenz.de

Preferential Attachment in Online Networks: Measurement and Explanations 8

Sublinear Model

P({A, i}) ~ d(i)¯0 < ¯ < 1

● Stretched exponential degree distribution [1, Eq. 94]

[1] Evolution of Networks. Adv. Phys. 51 (2002), 1079– 1187.[2] Random Networks with Sublinear Preferential Attachment: Degree Evolutions. Electrical J. of Probability 14 (2009), 1222– 1267.

Jérôme Kunegiskunegis@uni-koblenz.de

Preferential Attachment in Online Networks: Measurement and Explanations 9

Superlinear Model

P({A, i}) ~ d(i)¯¯ > 1

● A single node attracts 100% of edges asymptotically● Power law degree distribution in the pre-asymptotic regime

[1] Random Trees and General Branching Processes. Random Struct. Algorithms 31, 2 (2007), 186– 202.

Jérôme Kunegiskunegis@uni-koblenz.de

Preferential Attachment in Online Networks: Measurement and Explanations 10

+ =

Network at time t1Degrees d1(u)

Network at time t2Degrees d1(u) + d2(u)

Added edgesDegrees d2(u)

Temporal Network Data

Hypothesis: d2 = ® d1¯

Jérôme Kunegiskunegis@uni-koblenz.de

Preferential Attachment in Online Networks: Measurement and Explanations 11

Empirical Measurement of β

d2 = 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 2V

pu 2V

Jérôme Kunegiskunegis@uni-koblenz.de

Preferential Attachment in Online Networks: Measurement and Explanations 12

Example Network: Facebook Wall Posts

Description: User– user wall postsFormat: Edges are directedEdge weights: Multiple edges are possible Metadata: Edges have timestamps Size: 63,891 verticesVolume: 876,993 edgesAverage degree: 27.45 edges / vertexMaximum degree: 2,696 edges

http://konect.uni-koblenz.de/networks/facebook-wosn-wall

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Preferential Attachment in Online Networks: Measurement and Explanations 13

Facebook Wall Post Preferential Attachment

Jérôme Kunegiskunegis@uni-koblenz.de

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Network Categories

Social network user– userRating network user – itemCommunication network user– userFolksonomy person – tag/itemWiki edit network editor– articleExplicit interaction network person– personImplicit interaction network person – item

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Preferential Attachment in Online Networks: Measurement and Explanations 15

Social network ¯ < 1Rating network ¯ < 1Communication network ¯ < 1Folksonomy ¯ < 1Wiki edit network Explicit interaction network ¯ > 1Implicit interaction network ¯ > 1

Comparison

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Preferential Attachment in Online Networks: Measurement and Explanations 16

Thank You

Datasets 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/