cludes average time and standard devi-
ation of different crossover points with
respect to four kinds of centrality mea-
sures: degree,
2 betweenness,
11 k-core,
14
and eigenvector.
3 Figure 5 includes fur-
ther detail, in addition to the crossover
phenomenon:
Crossover points. Comparing the
statistical results in the phase II segment of the figure, although the increment of the mix parameter μ triggers
the crossover points slightly earlier,
it is still far less than the influence
resulting from increasing the initial
source nodes; and
Deviation. The deviation of different
crossover points tends to be stable in
the wake of a weaker community structure, or greater value for μ.
On the basis of the simulation results in synthetic networks, we found
two types of non-centrality-related network influence:
Strength of community structure.
The stability of crossover points is
inversely related to the strength of a
community structure, demonstrating
the strong (though indirect) influence
of community structure on the diffusion process; and
Increment of initial source nodes. The
increment of the initial source nodes is
the primary factor resulting in an earlier crossover phenomenon.
We likewise analyzed the influence of
community structure on two diffusion
processes—maximum-degree-based
and minimum-degree-based—to verify
our hypothesis, as proposed in Figure 1.
Influence of community structure.
Taking the synthetic network with μ
= 0.05 (Figure 5a) as an example, the
moment the crossover phenomenon
begins to emerge was visualized to
show the states of all nodes in two
propagation processes being initialized based on degree of centrality.
Figure 6 highlights the detailed states
of nodes in each community in various colors. Moreover, we extracted
five communities we labeled as “C0”,
“C1,” “C2,” “C3,” and “C4” that include
only two kinds of nodes.
Figure 6 outlines that a strong
community structure does not ben-
efit a subsequent propagation process.
When nodes with relatively greater
centrality are treated as sources, source
nodes tend to be clustered together,
decreasing (to some extent) the effec-
tive diffusion links. In a network with
a strong community structure, global
diffusion can be enhanced only when
the nodes on the intercommunity links
become infected. In the worst case, all
formed extensive simulations in two
such synthetic networks.
12 Compar-
ing the influence of the initial propor-
tion of source nodes and the strength
of community structure, Figure 5 in-
Figure 5. Average time of crossover points in synthetic networks with different community
structures.
The mix parameter μ controls the strength of community structure of the synthetic networks.
Each subgraph includes the average crossover point of four measures of centrality and the standard
deviation. The statistical results indicate the increment of the initial source nodes is the main factor
causing earlier crossover points, while the stronger a community structure a network has, the less
stable are the crossover points.
0.0 0.1 0.2 0.3 0.4 0.5
1E- 3
0.01
0.1
1
A
ve
ra
ge
t
i
me
o
fd
i
ffe
ren
t
c
r
o
sso
ve
rp
o
in
ts
A
ve
ra
ge
t
i
me
o
fd
i
ffe
ren
t
c
r
o
sso
ve
rp
o
in
ts
µ=0.05
Phase I
Phase II
0.0 0.1 0.2 0.3 0.4 0.5
1E- 3
0.01
0.1
1
µ=0.50
Phase I
Phase II
Initial proportion
of source nodes (i0)
(a) Synthetic network with µ = 0.05 (b) Synthetic network with µ = 0.50
Initial proportion
of source nodes (i0)
Figure 6. Visualization of two propagation processes—maximum-degree-based and
minimum-degree-based—in the synthetic network with μ = 0.05 when the crossover
phenomenon emerges; the susceptible nodes are marked in cyan.
The infected nodes, highlighted in red or blue, belong solely to the maximum-degree-based process or
the minimum-degree-based process, respectively. The black ones represent the infected nodes in both
processes. Five communities—“C0,” “C1,” “C2,” “C3,” and “C4”—include two kinds of nodes, demonstrating
that a strong community structure could hinder or even prevent global diffusion.
C1 C2
C3
C0
C4