Better synchronizability in generalized adaptive networks
Jun-Fang Zhu,
1
Ming Zhao,
1,2,
*
Wenwu Yu,
3,†
Changsong Zhou,
4,‡
and Bing-Hong Wang
1
1
Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
2
College of Physics and Technology, Guangxi Normal University, Guilin 541004, China
3
Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China
4
Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China
Received 17 September 2009; published 1 February 2010
In this paper, to study the interaction between network structure and dynamical property in the context of
synchronization, a previously proposed adaptive coupling method is generalized where the coupling strength of
a node from its neighbors not only develops adaptively according to the local synchronization property be-
tween the node and its neighbors dynamical part but also is modulated by its local structure, degree of the
node with the form 1 / k
i
topological part. We can show both numerically and analytically that the input
coupling strength of the network after adaptation displays a power-law dependence on the degree, k
-
, where
the exponent is controlled by as = 1+ / 2. Compared to the original adaptive coupling method, after the
addition of modulation, the distribution of the node’s intensity is tunable and can be more homogenous with
1, which results in better synchronizability. It is also found that the synchronization time can shrink greatly.
Our theoretical work in the context of synchronization provides not only a deeper understanding of the
interplay between structure and dynamics in real world systems, such as opinion formation and concensus, but
also potential approaches to manipulate the global collective dynamics through local adaptive control.
DOI: 10.1103/PhysRevE.81.026201 PACS numbers: 05.45.Xt, 87.18.Sn, 89.75.-k
Since the discovery and modeling of small world and
scale-free networks ten years ago 1,2, the dynamics in
complex networks have been a research topic of great atten-
tion. These dynamics include the epidemic and percolation
process, cascading behavior, traffic, opinion formation, syn-
chronization and consensues, etc. 3–7. Now it is well
known that not only the topology of the networks but also
the weights of nodes and strengths of links have great effects
on the dynamics in the networks 8–11. Most of these stud-
ies mainly considered the effects of structure on the dynam-
ics, while the dynamics have no effects on the network struc-
ture, i.e., no matter how the dynamics change, the topology
and the weights and strengths will keep unchanged. How-
ever, in most realistic dynamical networks, the dynamics can
sometimes induce changes in network structure. For ex-
ample, in airport networks of airports and airlines, the air-
lines affect the transportation and the demand of transporta-
tion between different airports may result in the increasing or
decreasing of the strengths of airlines or even the emergence
of new airlines. Another typical example is in the social net-
work where an epidemic spreads among the people. It is
obvious that the network structure will affect the direction
and speed of the epidemic spreading and meanwhile the sus-
ceptible will avoid contact with the infected by rewiring their
network connections, thus the network structure is changed
12. Similar interplay between structure and dynamics hap-
pens in some other realistic systems where synchronization
is relevant. For example, in neural systems, the synapses are
strengthened when two neural populations synchronize their
activity, which is known as the Hebbian learning rule and is
the mechanism underlying learning and memory of the brain.
It has been shown that applying such Hebbian-like rule cou-
pling strength increases faster when synchronization is stron-
ger to large network of oscillators will lead to the formation
of dynamical and structural clusters 13. In other systems,
global synchronization of the whole network is more desir-
able, for example, to achieve consensus on some opinion.
Base on our intuition and observation, we know that the
strategy often taken in such systems is “following the major-
ity.” The agents try to approach those individuals who have
different opinions from their own and persuade them to fol-
low their common opinion, while they do not have to put
much effort to enhance the interaction with the others al-
ready sharing similar opinions.
The impact of this type of adaptation on network structure
has been investigated recently 14 –16 in the context of syn-
chronization of complex networks. Mathematically, we can
represent this type of adaptation by a rule opposite to the
Hebbian one: the coupling strength increases proportional to
the synchronization difference in order to suppress the dif-
ference. In particular, we previously proposed an adaptive
coupling method where the input coupling strength of a node
from its neighbors develops adaptively according to the local
synchronization property between the node and its neighbors
14. In such adaptive networks, when complete synchroni-
zation is achieved, the coupling strength becomes stable and
weighted, which obeys a power-law form with respect to the
node degree. This makes the network synchronizability bet-
ter than the symmetrical coupling method 17,18 but not yet
in the optimal situation. In our previous studies on the syn-
chronization in weighted random networks, we had proved
that for random networks with a large minimum degree,
larger average degree and more homogenous distribution of
node’s intensity will ensure better network synchronizability,
where the intensity of a node is defined as its total input
coupling strengths 8,19. For a given average degree, the
network synchronizability will be optimized when all the
nodes’ intensities are equal. In the adaptive networks we pro-
*
zhaom17@gmail.com
†
wenwuyu@gmail.com
‡
cszhou@hkbu.edu.hk
PHYSICAL REVIEW E 81, 026201 2010
1539-3755/2010/812/0262016 ©2010 The American Physical Society 026201-1