IEEE Workshop on Nonlinear Circuit Networks
December 4-5, 2020

Research on the Relationship between
Network Structure and Synchronization Phenomena
in Complex Networks by Using Oscillators
Tsuyoshi Isozaki, Takumi Nara, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {isozaki, nara, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—Complex networks have been the subject of great
attention. Some characteristics of complex networks are closelyrelated to the real-world network and have been studied in termsof network topology and interactions between nodes. However,many of these studies focus on local properties, making it difficult
to consider the properties of the network as a whole. In thisstudy, we focus on properties of the network as a whole andbuild three networks such as the Regular lattice network, random
network and scale free network, which are composed of 100 vander Pol oscillators. Further, we analyzed these networks fromthe perspective of synchronization phenomena in each network.
As a result, it is confirmed that synchronization conditions aredifferent in the three network models.

I. INTRODUCTION
Complex networks are modeled with nodes and edges basedon several properties of real-world networks. The first is theproperty of small world. Small world means that the distance
between nodes is short compared to the total number of nodes.
The second is the property of scale free. Scale-free means thata small number of nodes, called hubs, have large degree, whilealmost all nodes have relatively small degree. The third is theproperty of cluster. Cluster means that any node forms a small
network with links to neighboring nodes and represents a close
network relationship. For example, human relationships, transportation networks, neural networks, internet and so on. Thus
complex networks have attracted a lot of attention from various
fields such as sociology, biology and engineering [1]. Furtherin the field of engineering, research on complex networks
using circuits has been studied, and interesting phenomena
such as synchronization between circuits have been observed.
Synchronization is phenomena in which things that have been
moving differently start to move in the same way at thesame time due to interactions. The synchronization phenomena
in complex networks have been studied in the past [2]-[3].
These studies focused on the network topological structure
and investigated the various synchronizations created by thedifferent topological structures. It was confirmed that thenetwork topological structure influences on synchronization.
However, many of these studies focus on local properties,
making it difficult to consider the properties of the network as
a whole. The networks in the real-world are made up of many
nodes and edges, and more complex phenomena are expected

to be observed as the number of nodes and edges increases.
Therefore, we increase the number of nodes a little to makeit easier to characterize the network as a whole, and we also
use oscillators to make it easier to analyze.
In this study, we build three network models with different
properties by using 100 oscillators. We vary the couplingstrength of each network and investigate the synchronization
between nodes at that time by means of computer simulations.
II. N ETWORK MODELSIn this study, we use three network models, the Regularlattice network, Erdős Rényi model (ER random network) [4]and Barabási Albert model (BA scale free network) [5]. Eachof these network models has some properties.
First, Fig. 1 shows the Regular lattice network with 100
nodes. This network has properties such as regular coupling
and long distance between nodes.

Fig. 1. Regular lattice network.

Second, Fig. 2 shows the ER random network with 100
nodes. This network has properties such as the degree distribution follows the binomial distribution and short distance between nodes. Here, following the binomial distribution meansthat the number of nodes with degrees close to the average
degree is higher.

- 11 -III. S YSTEM MODEL
Figure 4 shows van der Pol oscillator. This oscillator
consists of a capacitor, an inductor and a nonlinear element.
Further, one circuit is considered to be a single node, anda network is formed by connecting circuits to each otherusing resistors. In this study, 100 van der Pol oscillators areconnected to form each network.

Fig. 2. ER random network.

Third, Fig. 3 shows the BA scale free network with 100nodes. This network has properties such as the degree distribution follows the power law, short distance between nodes
and growth and priority selection. Here, following the powerlaw distribution means that a few nodes are connected to manyother nodes and have a large degree, while most other nodes
are connected to only a few nodes and have a small degree.
In addition, priority selection means that newly added nodes
are more likely to be connected with nodes of higher degree
during formation of the network.

Fig. 4. van der Pol oscillator.

When the circuits are connected by using resistors, thecircuit equations are given as follows:
100∑
1 dvnC=−i−i−(vn − vk )
L(n)
g(n)
dtR
n,k=1(1)
din

L=vn
dt

(n, k = 1, 2, · · ·, 100).
where the parameter ig is the equation for the nonlinear
element and described as follows:ig = −g1 v + g3 v 3 .

(2)

By using the parameters and the variables:
√√
g1
g1 C
v=y, t = LCτ
x, i =g3
g3 L
√√
L1 L
ε = g1, γ=.
CR C

Fig. 3. BA scale free network.

√The conditions of the networks created in this case areshown in Table I.
TABLE IN ODE NUMBER N , AVERAGE NODE DEGREE davg , M AXIMUMNODE DEGREE dmax , M INIMUM NODE DEGREE dmin AND
AVERAGE DISTANCE Davg OF THREE NETWORK MODELS .
networkNdavg
dmaxdminDavg

Regular
100
4.0
4
4
12.87

ER random
100
4.08
10
1
3.37

The normalized circuit equations are described as follows:

100

∑
dxn
2

=αεx(1−x)−y−Eγ(x−x)
nnnknknk
n dτBA scale free100
3.92
3222.98dyn= xn dτIn this study, we set the average node degree to be close to4.0 each network model. Further as properties of the network,
the maximum node degree is the largest for the BA scale freenetwork, the average distance is the longest for the Regularlattice network.

n,k=1(n, k = 1, 2, · · ·, 100).

where Enk represents the connected state of node n and nodek. If node n and node k are connected, Enk = 1 and if they
are not connected, Enk = 0.

- 12 -(3)xn-xkwhere n and k are number of circuits. Figure 5 shows thedifferential voltage waveform observed in this study. The twolines in Fig. 5 represent the thresholds shown in Eq. (4). If thevoltage difference falls within these thresholds, it is consideredsynchronous. When calculating the synchronization rates, weuse the voltage differences in the range of 10000 to 20000 in
τ, taking into account the time it takes to settle to a steadystate.

0.01
-0.01

The number of pair of nodes

The number of pair of nodes(c)γ = 2.0(d)γ = 2.6

Fig. 6. Regular.

The number of pair of nodes(4)Synchronization rates [%]Synchronization rates [%]Synchronization rates [%](a)γ = 0.4(b)γ = 0.8
The number of pair of nodes|xn − xk | < 0.01

Synchronization rates [%]The number of pair of nodes

We set the parameters of van der Pol oscillator as ε =0.1 and α = 1.0. Here, α represents the small error on thecapacitor, given randomly in the [0.95:1.05] range for eachcircuit. In this study, we vary the coupling strength γ in each
network model and show results on the synchronization rates
in the three network models.
In order to analyze synchronization, we define the synchronization as following equation.

The number of pair of nodes

IV. R ESULTS

Synchronization rates [%]Synchronization rates [%](c)γ = 1.0(d)γ = 1.2

Fig. 5. The differential voltage waveform.

Figures 6-8 show the results of the Regular lattice, the ER
random network and the BA scale free networks, respectively.
In the Figs. 6-8, the vertical axes are the number of pairs ofnodes and horizontal axes are synchronization rates.

The number of pair of nodes

τThe number of pair of nodes

Fig. 7. ER random.

Synchronization rates [%]Synchronization rates [%]Synchronization rates [%](a)γ = 1.0

Synchronization rates [%]Synchronization rates [%](c)γ = 1.0(b)γ = 1.6- 13 -(b)γ = 0.8
The number of pair of nodes

The number of pair of nodes

The number of pair of nodes

The number of pair of nodes(a)γ = 0.4

Synchronization rates [%](d)γ = 1.2

Fig. 8. BA scale free.

Synchronization rates [%]As a result, we can see from Fig. 6 that the number ofpairs of nodes with synchronization rates of 0-10[%] is quite
high when the coupling strength γ=1.0. When the couplingstrength was increased, the number of pairs of nodes withsynchronization rates of 0-10[%] decreases and the numberof pairs of nodes with synchronization rates of 90-100[%]
increases. However, it was confirmed that the coupling strength
needs to be considerably increased in this network. Further, wecan see from Figs. 7 and 8 that when the coupling strength isincreased, the number of pairs of nodes with a synchronization
rate of 0-10[%] decreases and the number of pairs of nodes
with a synchronization rate of 90-100[%] increases. It wasalso confirmed that these networks varied when the couplingstrength is smaller compared to the Regular lattice network.
Figure 9 shows the relation between synchronization rates
and coupling strength. In the Fig. 9, the vertical axes aresynchronization rates and horizontal axes are coupling strength
γ. The red line is the results of the Regular lattice network,
the yellow line is the results of the ER random network andthe green line is the results of the BA scale free network. We
can see the change in the synchronization rates with respectto the coupling strength in each network from this figure.

phenomena between nodes in each networks. We confirmedthat in the Regular lattice network, the node synchronization
rates are slightly higher when the coupling strength is quite
high, but the nodes are less synchronized than in other networks. Further, it was confirmed that the synchronization rates
of nodes in the ER random network increase as the couplingstrength increases, but the oscillators in the BA scale free do
not work well when the coupling strength is too high. It is
confirmed that when the average distance between nodes, theregular lattice network has the longest distance of 12.87, theER random network has 3.37 and the BA scale free has 2.98,
so ER random network and BA scale free are close to eachother.
In the future, we will continue to investigate the relationshipbetween the average distance between nodes and in this results.
R EFERENCES[1] Jiajing Wu, Chi K. Tse, Francis C. M. Lau and Ivan W. H. Ho, “Analysisof communication network performance from a complex network perspective”, IEEE Transactions on Circuits and Systems-I: Regular Papers,
vol. 60, No. 12, December 2013.
[2] Kenta Ago, Yoko Uwate and Yoshifumi nishio, “Influence of local
bridge on a complex network of coupled chaotic circuits , Proceedingsof International Symposium on Nonlinear Theory and its Applications(NOLTA’14), pp. 731-734, September. 2014.
[3] Shuhei Hashimoto, Yoko Uwate and Yoshifumi Nishio, “Synchronization
phenomena of chaotic circuit networks with distributed hub including
positive and negative coupling”, Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS’18), May 2018.
[4] D. Erdős and D. Rényi, “On the evolution of random graphs,” Publ. Math.
Inst. Hung. Acad. Sci, vol.5, pp. 17-60, 1960.
[5] R. Albert and A.-L. Barabasi, “Topology of evolving networks: Local
events and university,” Phys. Rev. Lett., vol.85, pp. 5234-5237, 2000.

RegularER random
BA scale freeCoupling strength γFig. 9.

The relation between synchronization rates and couplingstrength.

It was confirmed that the synchronization rates of the nodes
in the Regular lattice network are higher slightly as thecoupling strength is increased, but the synchronization rates
are lower than that of the ER random network and the BA scale
free network. In the ER random network, it was confirmedthat the synchronization rates of nodes are considerably higher
when the coupling strength is increased. In the BA scale
free network, it was confirmed that the synchronization rates
increased when the coupling strength was small, as in the ER
random network, but the oscillator did not work well aroundthe coupling strength more than the value of 1.2.
V. CONCLUSION
In this study, we have built three network models by using
100 van der Pol oscillators and investigated synchronization- 14 -