we conducted more simulations, as we
explore in two real-world networks in
the next section.
Nonlinear Crossover Phenomenon
To obtain a deeper understanding of
such a phenomenon, we simulated
propagations in real-world networks:
Datasets. We included two real-world networks with a potential community structure—a U.S. political
weblog network (PolBlogs)
1 and a
scientific collaboration network (
Arxiv).
18 The PolBlogs network includes
1,490 nodes and 19,025 links. Its average degree and clustering coefficient
were 22. 44 and 0.36, respectively. Two
political communities represent liberal blogs and conservative blogs,
respectively.
1 Mark Newman of the
University of Michigan analyzed the
Arxiv network, with 56,276 nodes and
631,632 links,
18 looking to identify
community-based features on co-authorship patterns. The average degree
and clustering coefficient of the Arxiv
networks were 11. 23 and 0.69, respectively, and the overall Arxiv network
included 42 communities.
Experimental settings. The initial
proportion of source nodes we denote
as i0 varied from 0.01 to 0.5 and was divided into two parts. When the initial
proportion is between 0.01 and 0.05,
the rate of increase increases by 1%, after which the rate of increase increases
to 5%. We selected the initial source
nodes based on four kinds of centrality measures: degree,
2 betweenness,
11
k-core,
14 and eigenvector.
3
Experimental results. Under the
same experimental conditions as
outlined in the previous section, two
propagation processes are triggered by
source nodes with relatively greatest
and relatively least centrality. Our focus
is still on the critical crossover points.
Since they are relevant to the time steps
of each propagation, we recorded the
time each crossover point emerged
and normalized them based on tc/2000,
where 2,000 was the total time steps, as
is plotted in Figure 3. Despite different
centrality measures and networks, the
figure reveals several similarities:
Propagation scale. The crossover in
terms of propagation scale emerges
when the initial proportion of source
nodes is low (such as 1%). Experiments
on different kinds of networks show
that a stable state, when the crossover
phenomenon can be triggered, is indeed possible when i0 increases;
Crossover points. The time of different crossover points is generally
a decreasing function of the initial
proportion of source nodes i0; that
is, the crossover points come earlier with the increment of the initial
source nodes; and
Strength of community structure.
The different crossover points under
the same degree of centrality reveal
the strength of influence a community structure exerts on the propagation process.
Experimental results in real-world
networks demonstrate our assumption
that central users (or nodes with rela-
tively greatest centrality) do not always
drive information diffusion. Specifi-
cally, the crossover phenomenon pre-
tion process, triggered by minimum-
degree nodes, leads to a larger scale of
diffusion until the whole propagation
system is stable. The maximum differ-
ence is reached numerically at time t2,
even exceeding that of time t1. The time
tc is the exact crossover point of the two
propagation processes in Figure 2.
During the propagation process, the
most important period is between t1
and t2 when the two potential propagation processes undergo different transitions. The phenomenon in Figure 2
shows that, compared to nodes with
relatively greater centrality, those with
relatively less centrality could ensure
the stability of propagation, reflecting
its vital role in long-term diffusion.
Such an interesting phenomenon also
implies that in some cases, even central
users may not always drive information
diffusion. To validate this assumption,
Figure 2. Crossover of two propagation processes in terms of propagation scale in a
university email network.
Time tc represents the critical moment the crossover begins, indicating nodes with relatively greater
centrality do not always drive diffusion.
tc
t1 t2
(a)
tc
t1
t2
1
10
100 1000
2
4
6
8
10
12
Maximum degree
Minimum degree
P
ropag
a
t
ion
sc
ale
X102
1
10
100 1000
– 1
0
1
X102
Propagation step (t) Propagation step (t)
N
u
me
r
i
ca
l
d
if
fere
n
ce
o
f
t
wo
p
rop
aga
ti
on
s
ca
les
(b)
Figure 3. Nonlinear crossover phenomenon in networks with community structure.
Each point represents the potential crossover point in terms of propagation scale. The results indicate
both community structure and initial proportion of source nodes have influence on such phenomena.
0.0 0.1 0.2 0.3 0.4 0.5
1E- 3
0.01
0.1
1
Initial proportion
of source nodes (i0)
Initial proportion
of source nodes (i0)
0.0 0.1 0.2 0.3 0.4 0.5
1E- 3
0.01
0.1
1
T
ime
o
fd
i
ffe
ren
t
cro
s
so
ver
p
o
in
t
s
T
ime
o
fd
i
ffe
ren
t
cro
s
so
ver
p
o
in
t
s
Degree
Betweenness
K-core
Eigenvector
(a) PolBlogs (b) Arxiv