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.

1. Preferential Attachment in Online Networks:
Measurement and Explanations
Jérôme Kunegis, Institute for Web Science, University of Koblenz– Landau
Marcel Blattner, Laboratory for Web Science, FFHS
Christine Moser, VU University Amsterdam
ACM Web Science 2013
With thanks to Hans Akkermans, Rena Bakhshi and Julie Birkholz
Funded by the European Community's Seventh Framework Programme under grant agreement n° 257859, ROBUST

2. Jérôme Kunegis
kunegis@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

3. Jérôme Kunegis
kunegis@uni-koblenz.de
Preferential Attachment in Online Networks: Measurement and Explanations
3
Power Laws – Scale Free Networks
C(d) ~ d− °
Degree (d)
Frequency(C(d))

4. Jérôme Kunegis
kunegis@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

5. Jérôme Kunegis
kunegis@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.

6. Jérôme Kunegis
kunegis@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.

7. Jérôme Kunegis
kunegis@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.

8. Jérôme Kunegis
kunegis@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.

9. Jérôme Kunegis
kunegis@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.

10. Jérôme Kunegis
kunegis@uni-koblenz.de
Preferential Attachment in Online Networks: Measurement and Explanations
10
+ =
Network at time t1
Degrees d1(u)
Network at time t2
Degrees d1(u) + d2(u)
Added edges
Degrees d2(u)
Temporal Network Data
Hypothesis: d2 = ® d1
¯

11. Jérôme Kunegis
kunegis@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
p
u 2V

12. Jérôme Kunegis
kunegis@uni-koblenz.de
Preferential Attachment in Online Networks: Measurement and Explanations
12
Example Network: Facebook Wall Posts
Description: User– user wall posts
Format: Edges are directed
Edge weights: Multiple edges are possible
Metadata: Edges have timestamps
Size: 63,891 vertices
Volume: 876,993 edges
Average degree: 27.45 edges / vertex
Maximum degree: 2,696 edges
http://konect.uni-koblenz.de/networks/facebook-wosn-wall

13. Jérôme Kunegis
kunegis@uni-koblenz.de
Preferential Attachment in Online Networks: Measurement and Explanations
13
Facebook Wall Post Preferential Attachment

14. Jérôme Kunegis
kunegis@uni-koblenz.de
Preferential Attachment in Online Networks: Measurement and Explanations
14
Network Categories
Social network user– user
Rating network user– item
Communication network user– user
Folksonomy person– tag/item
Wiki edit network editor– article
Explicit interaction network person– person
Implicit interaction network person– item

15. Jérôme Kunegis
kunegis@uni-koblenz.de
Preferential Attachment in Online Networks: Measurement and Explanations
15
Social network ¯ < 1
Rating network ¯ < 1
Communication network ¯ < 1
Folksonomy ¯ < 1
Wiki edit network
Explicit interaction network ¯ > 1
Implicit interaction network ¯ > 1
Comparison

16. Jérôme Kunegis
kunegis@uni-koblenz.de
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/