link connects different communities in the figure. Two diffusion processes are triggered by the maximum degree nodes, as in Figure 1a, and minimum degree nodes, as in Figure 1b, respectively. Theoretically, the requirement of simultaneous diffusion of source nodes is not necessary due to the independent infection process between each infected node and its susceptible neighbors. However, to ensure a clear, quick observation of a diffusion phenomenon, as in, say, letting source nodes (represented by two red nodes in Figure 1) initiate a diffusion process simultaneously. Based on the propagation rules defined in a typical “ two-state” diffusion model, 22 each infected node tries to infect all its susceptible neighbors with a certain probability at each time step, bringing uncertainty during the propagation process. For example, as shown in Figure 1a, “A” is infected by “E” at t = 3, rather than by its source node neighbor at t = 1 or t = 2. Meanwhile, at t = 8, “D” remains susceptible until infected by “C” at t = 10. In Figure 1a, information diffuses quickly at the initial stage. The speed of diffusion could be enhanced by increasing the initial number of source nodes. Note also two factors concerning the effect of network structure on the potential diffusion process:

Effective diffusion links. The effective
diffusion links represent the connec-
tions that make a key contribution to
the diffusion process; 22 for example,
the link between the two source nodes
in Figure 1a does not benefit subse-
quent diffusion. With the increment
of source nodes, there is a strong likeli-
hood that some might cluster together,
as outlined in Figure 1a, thus decreas-
ing the effective diffusion links. But
such a negative effect is unlikely to
show up in Figure 1b unless there are
more initial source nodes.

Community structure. Global diffusion in a network with community structure is restricted by intercommunity links; 16 that is, global diffusion is facilitated only when the nodes on the intercommunity links (also called “bridge nodes” by network architects) are infected. In Figure 1a, four nodes—“A,” “B,” “C,” and “D”—are bridge nodes. The global diffusion is suppressed temporarily because “D” remains susceptible at t = 8. Although “D” is not infected by “C” in Figure 1b, the other source node initiates new propagation in other communities, enhancing global diffusion.

Based on two factors—effective diffusion links and community structure—the two diffusion processes— maximum-degree-based and minimum-degree-based—in Figure 1a and Figure 1b might result in a crossover in terms of diffusion scale. The diffusion scale of Figure 1b would be greater than the diffusion scale of Figure 1a. Differing diffusion scales involve several questions: For example, do the most connected users always drive information diffusion in social networks? If not, what kind of influence would the community structure have on the diffusion process? To answer, we simulated information diffusion in both real-world and synthetic networks with community structure to investigate the potential crossover points of two diffusion processes.

Crossover in Terms of
Propagation Scale

Many real-world systems can be de-
scribed as networks; examples are
email, social, and technological. Here,
we select a benchmark university email
dataset and construct an interaction
network to demonstrate the influence
of two diffusion processes—maxi-
mum-degree-based and minimum-de-
gree-based—triggered by two kinds of
initial source nodes with greatest- and
least-degree centrality. The network
includes 1,133 nodes and 5,451 links. 9
The average degree and clustering coef-
ficient in the network are 9.62 and 0.25,
respectively. Specifically, the clustering
coefficient is used to denote the degree
to which the neighbors of a user know
potential communities, 21 model diffu-
sion dynamics, 6 and control informa-
tion dissemination and sharing19). In
particular, the influence of each node
in the diffusion process must be taken
into consideration. In simulation ex-
periments, the source nodes that trig-
ger diffusion are selected by researchers
at random from a network or based on
predefined measures of centrality.

In recent decades, multiple centrality measures have been proposed to statistically evaluate the importance or influence of a node (such as degree, 2 betweenness, 11 coreness, 14 and eigenvector3). Degree is used mainly for characterizing the partial influence of a node. 2 Betweenness reflects the potential power of a node in controlling information flow. 11 Coreness implies that if a node lies in the core part of a network, the node is more important. 14 And eigenvector accounts for two factors: a node’s connections and its neighbors’ influences. 3 State-of-the-art studies have looked into nodes with relatively greater centrality in information diffusion. However, the influence of nodes with relatively less centrality on the diffusion process has never been completely addressed. In this article, we aim to explain the importance of two kinds of nodes in the information-diffusion process in a community-based network. Our findings can help network administrators better understand the diversity of communities and associated complexity of the diffusion process.

Potential for a Crossover

Centrality characterizes the influence of a node or user in a network. Intuitively, nodes with relatively greater centrality should be more important than those with relatively less centrality, as they can lead to fast, large-scale diffusion. However, we often find diffusion breaks out from a group of nodes with relatively less centrality in the real world.

Figure 1 outlines two diffusion processes as triggered by different initial states in a community-based network. The dotted circles represent different communities in a network.

With a strong community structure,
the density of the intracommunity
links is much greater than that of the
intercommunity; for instance, only one
key insights

˽ Central users do not always contribute to information diffusion due to a crossover point of two diffusion processes triggered by source users with most and fewest connections, respectively.

˽ A strong community structure decreases the stability of the crossover point in terms of influence of two diffusion processes.

˽ Compared to the influence of community structure on the diffusion process, the increment of source nodes leads the diffusion scale to the appearance of an earlier crossover point.