(June 2012), 70–75.
6. Gao, C. and Liu, J.M. Network-based modeling for
characterizing human collective behaviors during
extreme events. IEEE Transactions on System, Man,
and Cybernetics: Systems 47, 1 (Jan. 2017), 171–183.
7. Gao, C., and Liu, J. M. Modeling and restraining mobile
virus propagation. IEEE Transactions on Mobile
Computing 12, 3 (Mar. 2013), 529–541.
8. 8 Goel, S., Watts, D.J., and Goldstein, D.G. The
structure of online diffusion networks. In Proceedings
of the 13th ACM Conference on Electronic Commerce
(Valencia, Spain, June 4–8). ACM Press, New York,
9. Guimerá, R., Danon, L., Díaz-Guilera, A., Giralt, F., and
Arenas, A. Self-similar community structure in a
network of human interactions. Physical Review E 68,
6 (Dec. 2003), 065103.
10. Howard, B. Analyzing online social networks.
Commun. ACM 51, 11 (Nov. 2008), 14–16.
11. Kitsak, M., Gallos, L.K., Havlin, S., Liljeros, F., Muchnik,
L., Stanley, H.E., and Makse, H.A. Identification of
influential spreaders in complex networks. Nature
Physics 6, 11 (Aug. 2010), 888–893.
12. Lancichinetti, A., Fortunato, S., and Radicchi, F.
Benchmark graphs for testing community detection
algorithms. Physical Review E 78, 4 (Oct. 2008).
13. Leskovec, J., Kleinberg, J., and Faloutsos, C. Graph
evolution: Densification and shrinking diameters. ACM
Transactions on Knowledge Discovery from Data 1, 1
14. Liu, Y. Y., Slotine, J. J., and Barabási, A.-L.
Controllability of complex networks. Nature 473, 7346
(May 2011), 167–173.
15. McGoogan, C. What is WannaCry and how does
ransomware work? The Telegraph (May 18,
16. Nematzadeh, A., Ferrara, E., Flammini, A., and Ahn,
Y.-Y. Optimal network modularity for information
diffusion. Physical Review Letters 113, 8 (Aug. 2014),
17. Newman, M.E.J. Modularity and community structure
in networks. Proceedings of the National Academy of
Sciences 103, 23 (June 2006), 8577–8582.
18. Newman, M.E.J. Co-authorship networks and patterns
of scientific collaboration. Proceedings of the National
Academy of Sciences 101, Supplement 1 (Apr. 2004),
19. Ranjbar, A. and Maheswaran, M. Using community
structure to control information sharing in online
social networks. Computer Communications 41 (Jan.
20. Wang, W., Tang, M., Stanley, H.E., and Braunstein, L.A.
Unification of theoretical approaches for epidemic
spreading on complex networks. Reports on Progress
in Physics 80, 3 (Feb. 2017), 036603.
21. Xie, J.R., Kelley, S., and Szymanski, B.K. Overlapping
community detection in networks: The state-of-the-art and comparative study. ACM Computing Surveys
45, 4 (Aug. 2013), 43:1–43: 35.
22. Zou, C. C., Towsley D., and Gong W. Modeling and
simulation study of the propagation and defense
of Internet e-mail worms. IEEE Transactions on
Dependable and Secure Computing
4, 2 (Apr. 2007),
Chao Gao ( firstname.lastname@example.org) is a professor in the
College of Computer and Information Science, Southwest
University, Chongqing, China, and a visiting scholar in the
Humboldt University of Berlin, Germany.
Zhen Su ( email@example.com) is pursuing a master’s
degree in the College of Computer and Information
Science, Southwest University, Chongqing, China.
Jiming Liu ( firstname.lastname@example.org) (corresponding
author) is a professor of computer science and associate
vice president (research) at Hong Kong Baptist University,
Hong Kong, China.
Jürgen Kurths (Juergen. Kurths@pik-potsdam.de)
is a professor of nonlinear dynamics in the Humboldt
University of Berlin, Germany, and Chair of the Research
Domain Transdisciplinary Concepts in the Potsdam
Institute for Climate Impact Research, Potsdam,
© 2019 ACM 0001-0782/19/2 $15.00
source nodes are distributed over only
one community, thereby suppressing
However, diffusion is quite different
when nodes characterized by relatively
least centrality are viewed as sources.
Source nodes under such conditions
are distributed over more communities and more likely to facilitate global
diffusion. Moreover, the worst case
is unlikely to appear due to the relatively greater proportion of low-degree
nodes in a network. That is why there
are fewer red nodes in “C0,” “C1,” “C2,”
“C3,” and “C4” than blue nodes. As the
two propagation processes in Figure
6—maximum-degree-based and minimum-degree-based—proceed, such
phenomenon will intensify. Finally, the
various diffusion scenarios we have addressed also increase the fluctuation of
For networks with weak community
structures, the increasing proportion
of intracommunity links makes global
diffusion more likely, making crossover points relatively stable.
We have explored the nonlinear crossover of two diffusion processes—
cen-tral-user-based and boundary-user-based—triggered by two opposite
initial states in networks with community structure. We first considered the
universality of the crossover phenomenon, then offered a detailed comparison with respect to the influence of
community structure and initial proportion of source nodes on the diffusion process. The results were twofold:
Networks with weak community structure could increase the stability of
crossover points; and compared to the
influence of community structure, the
increment of the initial source nodes
is the primary factor leading to an earlier crossover phenomenon.
The crossover phenomenon shows
the topology of a network is a major
factor affecting the diffusion process.
A deep understanding of diffusion
dynamics requires consideration of
both network topology and dynamical
correlations. Many popular theoreti-
cal approaches (such as mean field,
dynamical message passing, and pair-
wise approximation) are used to study
the dynamics of different kinds of in-
formation diffusion, but the difficulty
of capturing both network topology
and dynamical correlations remains
an open topic.
20 Even with the con-
tinuous-time Markov approach, the
complicated master equations lead
to yet another challenge—that the
approach is unlikely to directly yield
analytical or numerical results for
large-scale networks. Studies inves-
tigating the balance between poten-
tial diffusion dynamics and solving
computational complexity are still
This article has offered insight into
the dynamics of information diffusion in community-based networks.
For instance, compared with the ability of nodes with relatively greater
centrality to dramatically enhance
diffusion speed at the initial stage,
nodes with relatively least centrality
could in fact have a greater propagation effect in the long term, especially
when a network includes more initial
source nodes. However, we are not
saying nodes with relatively least centrality are critically important. It is the
topological structure that establishes
an explicit and complex connection
between the two kinds of nodes. In
some cases, such connections suggest
users with relatively least centrality
should be taken into consideration, as
they could still significantly influence
This work was supported by the National Natural Science Foundation
of China (grant No. 61402379), Hong
Kong Research Grants Council (No.
HKBU12202415), CQ CSTC (grant No.
cstc2018jcyjAX0274), the Fundamental Research Funds for the Central Universities (grant No. XDJK2016A008),
and Chongqing Graduate Student Research Innovation Project (grant No.
1. Adamic, L.A. and Glance, N. The political blogosphere
and the 2004 U. S. election: Divided they blog. In
Proceedings of the Third International Workshop on
Link Discovery (Chicago, IL, Aug. 21–25). ACM Press,
New York, 2005, 36–43.
2. Albert, R. and Barabási, A.-L. Statistical mechanics of
complex networks. Reviews of Modern Physics 74, 1
(Jan. 2002), 47–97.
3. Borgatti, S.P. Centrality and network flow. Social
Networks 27, 1 (Jan. 2005), 55–71.
4. De Meo, P., Ferrara, E., Fiumara, G., and Provetti, A. On
Facebook, most ties are weak. Commun. ACM 57, 11
(Oct. 2014), 78–84.
5. Doerr, B., Fouz, M., and Friedrich, T. Why rumors spread
so quickly in social networks. Commun. ACM 55, 6