the diffusion network. Specifically, although the initial source nodes in two propagation processes share the same proportion, the potential processes can be different in light of the diversity of the underlying diffusion network.

Since both the initial proportion of source nodes and the strength of community structure influence potential crossover points, we explored more simulations in synthetic networks to identify the influence of these factors.

Influence Comparison

To help us understand the influence of the strength of community structure on the diffusion process, we adopted a community-network generator12 with tunable parameters:

Datasets. We built two synthetic networks by varying the mix parameter μ = 0.05 and 0.5. This parameter controls the strength of a community structure, indicating that with a smaller μ, the community structure of a synthetic network is stronger. The generator includes two kinds of parameters— specified and default settings. We assigned the specified settings as follows: total number of nodes = 1,000; average degree = 15; maximum degree in the network = 50; and maximum and minimum community sizes = 50 and 20, respectively. We kept the default settings, with the exponent for the degree distribution at 2; the exponent for the community-size distribution at 1; and the number of overlapping nodes and number of memberships of the overlapping nodes both at 0.

Experimental results. Following the
same experimental scenario, we per-
vails and will intensify when the initial
proportion of source nodes increases.
We also investigated the influence of
different initial states on effective dif-
fusion links to verify our hypothesis, as
proposed in Figure 1.

Diffusion links analysis. Taking the email network as an example, we evaluated two opposite initial states under four kinds of centrality measures by calculating the average distance of source nodes. This distance can reveal the degree to which source nodes are close to each other. A shorter average distance refers to a relatively greater probability of being clustered together. Diffusion links between source nodes could thus be decreased. As outlined in Figure 4, under the condition of nodes with relatively greatest centrality functioning as source nodes, the average distance of these sources is much shorter than the distance under nodes with relatively least centrality being treated as source nodes. The reason for the shorter distance is that nodes with relatively least centrality are located at the boundary of a network, and vice versa. Hence, when nodes with relatively greatest centrality are selected as sources, the increasing proportion of source nodes can lead to a relative decrease in effective diffusion links. Moreover, the subsequent propagation process would be suppressed. How nodes with relatively greatest centrality might enhance information diffusion depends on the number of initial source nodes clustering together. In particular, when there are few initial source nodes (such as less than 1%), the propagation ability of nodes with relatively greatest centrality can take full effect.

Behind the crossover phenomenon, this shift is derived by taking into account two propagation processes— great-est-centrality-based and least-centrality-based—as triggered by different initial states. In the domain of social networks, analysis of a diffusion process is associated with a selected propagation model and the topology of an underlying network. In our experiments, we simulated two propagation processes simultaneously based on the same model, indicating the crossover phenomenon is independent of the selected simulation models. The only factor that should be relevant to this observed phenomenon is thus the structure of

Figure 4. Average distance of source nodes in the email network. Statistical results indicate source nodes with relatively greater centrality tend to be clustered together.

0.0 0.1 0.2 0.3 0.4 0.5

0

2

4

6

8

Minimum degree Minimum betweenness Minimum k-core Minimum eigenvector Maximum degree Maximum betweenness Maximum k-core Maximum eigenvector

A
ve
r
a
ge
di
st
an
ce
of
s
ou
rc
e
n
od
e
s
Initial proportion
of source nodes (i0)

The underlying
attack reflects a
malicious diffusion
in the presence
of communities;
that is, the
homogeneous
feature of
individuals leads
to the community’s
vulnerability.