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.
Preferential Attachment in Online Networks: Measurement and Explanations
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
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