pose of giving an intuitive demonstration of a crossover, as plotted in Figure 2a. We also investigated the dynamic changes of two propagation scales by calculating the numerical difference of two propagation processes at each time t, as plotted in Figure 2b. Three critical points are labeled t1, tc, and t2.

When t < tc, the difference between propagation scales is positive, as shown in Figure 2b. That difference corresponds to the stage (see t < tc in Figure 2a) when the propagation process, triggered by the maximum degree nodes, diffuses more quickly than the other process. The maximum difference is found the moment t = t1 in Figure 2b.

However, as the propagation contin-
ues (see t1 < t < t2), the numerical dif-
ference decreases sharply, as plotted
in Figure 2b. This unexpected change
implies the propagation process, trig-
gered by the minimum-degree nodes,
represents relatively greater propaga-
tion ability. The shift coincides with
the dynamic change of the propagation
scale in Figure 2a. When t > tc, the shift
is completely reversed. The propaga-
each other. 2 A greater value means a
network’s greater inherent tendency to
cluster of a network.

Simulation model. Each user is es-
sentially represented by two states
in the scenario of information diffu-
sion—“received” a message or “not
received” a message. We adopt a typi-
cal “two-state” diffusion model—the
interactive email model proposed by
Zou et al. 22 and implemented by Gao
et al. 7—as a testbed for characterizing
various kinds of information-diffusion
processes. 6, 7 Each node in the model
reflects one of two corresponding
states—“susceptible” or “infected”—
and the transition cannot be reversed;
that is, a user who receives a message is
denoted as an “infected” node, and oth-
ers are denoted as “susceptible.” In a
diffusion process, a basic step that ben-
efits the subsequent process is a user
must change state from “susceptible”
to “infected.” The diffusion process is
triggered by user behavior—the email-
checking time interval and the email-
clicking probability. The diffusion rate
is thus different for different users. By
assuming the behavior of each user
is independent, we used a Gaussian
distribution to depict the features of
two behaviors when the sample size is
large. 7 In this article, we use two nor-
mal distribution functions—N( 40, 202)
and N(0.5, 0.32)—to represent the fea-
tures of checking intervals and clicking
probability. 7, 22
Experimental settings. We set the per-
centage of initial source nodes at 20%.
We simulated two diffusion processes
triggered by maximum-degree nodes
and minimum-degree nodes in the
email network simultaneously and
independently. We averaged simula-
tion results by following 100 runs for
wiping off the computational fluctua-
tion. In each run, we terminated the
propagation process after 2,000 time
steps to ensure the whole system is
and would remain stable.

Experimental results. In general, we used the proportion or total number of infected nodes to evaluate a propagation process. Here, we adopt the total number of infected nodes at time t as the propagation scale for the pur-

Figure 1. Schematic of two diffusion processes: maximum-degree-based and minimum-degree-based.

Successful infection path Unsuccessful infection path Diffusion source Susceptible node at t = 8

Infected node at t = 8

(b) Diffusion process triggered by nodes with the minimum degree

(a) Diffusion process triggered by nodes
with the maximum degree
t= 2

t=0

t=0

t= 1

t= 2

t= 3

E
A
t= 7

t= 4
A
t= 14

t= 12 t= 9

t= 2

t=0 t= 3 t= 5

BC
t= 6
t= 6
t= 4

t= 7 t= 5

BC
t= 8
t=15D
t= 10
t= 11 t= 13

t= 14 t= 12 t= 14

 

D t=0

t= 3 t= 6

t= 5

t= 8

t= 12 t= 9

Figure 1(a)

Figure 1(b)

A crossover

02468101214 2

4 6 8 10 12 14 16 18

The propagation step (t)
Maximum degree
Minimum degree T h
e
nu
m
b
er
of
in
fe
c
te
d
no
d
es

(c) Comparison of total number
of infected nodes between (a) and (b)

During diffusion, all nodes in a network are divided into three categories: source nodes, infected nodes, and susceptible nodes. The source nodes receive information first and trigger the overall diffusion process. The infected and susceptible nodes represent nodes that have or have not received information.

At each time step, each infected node tries to infect all susceptible neighbors with a certain probability. The final infected time of each node is labeled; for example, at time step t = 8, two snapshots are used to present two diffusion processes: (a) the process is triggered by two highly connected source nodes; and (b) the process starts from the relatively least-connected source nodes. In particular, solid and dashed arrows associated with links denote successful and unsuccessful infection paths. Section (c) reports the dynamic changes of infected nodes in (a) and (b) at each time step. The crossover of the two propagation scales in (a) and (b) is plotted in (c).