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

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A
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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