IEEE Workshop on Nonlinear Circuit Networks
December 3-4, 2021

Research on Synchronization Phenomena
in Complex Networks by Using Oscillators
Considering Euclidean Distance
Kiichi Yamashita, Tsuyoshi Isozaki, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {kiichi, isozaki, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—Complex networks have attracted a great deal ofattention in recent years. Because they have many properties
that are relevant to real-world networks. This has been studieda lot in terms of network topology and interactions between
nodes. However, most of these studies have analyzed networks
by coupling strength is kept constant between nodes for allconnections. In this study, we focus on the Euclidean distance
between nodes to weight the connections. Coupling strength isdetermined by it. Two networks, a random network and a scalefree network, were constructed by using 100 oscillators. Thesenetworks were analyzed in terms of synchronization phenomena.
As a result, it was confirmed that the synchronization phenomenain the two networks is significantly different when the couplingstrength is determined by Euclidean distance.

I. INTRODUCTION
Complex networks have the properties of real-world networks: scale-free, small-world, and clustered. They can be
represented in graph theory and modeled by nodes and edges.
Examples of these relationships are human relationships, transportation networks, neural networks and the Internet. Complex
networks have received much attention in various fields such associology, biology and engineering. Furthermore, in the fieldof engineering, complex networks using circuits have beenstudied, and interesting phenomena such as synchronization
between circuits have been observed [1]. Synchronization is
one of the most familiar phenomena that exist in nature.
It is a phenomena in which nonlinear systems that havebeen moving separately interact with each other and becomealigned. Synchronization phenomena in complex networks
have been studied in the past. These studies have focusedon the topological structure of the network and investigatedvarious synchronization phenomena caused by differences inthe topological structure [2]. From the results of these studies,
it was confirmed that the topological structure of the network
influences the synchronization. However, most of these studies
have examined synchronization phenomena in networks where
the coupling strength is kept constant for all the connections [3]. Therefore we investigate how the synchronization
phenomena differ when the coupling strength is considered as
the Euclidean distance of each coupled part depending on thenode arrangement.

In this study, two network models with different propertiesare constructed using 100 oscillators. For each network, thenode configuration is placed on the two-dimensional space andthe Euclidean distance is derived to set the coupling strength.
The synchronization between the nodes of the coupled part isinvestigated by the computer simulations.
II. N ETWORK MODELSIn this study, we use two network models, Barabási Albert
model (BA model) [4] and Erdős Rényi model (ER model) [5].
The number of nodes is 100 and the average degree of each
network model is set to be close to 4.0. Since the Euclidean
distance between nodes determines the coupling strength,
these two network models are constructed with fixed nodeconfigurations.
First, Fig. 1 shows the BA model, a model for scalefree networks used in this study. This network has properties
called network growth and preferential selection, which isthe property that hubs are created as new nodes are easily
connected to nodes with higher degree. Furthermore, since
the degree distribution follows a power law, a small numberof nodes are connected to many other nodes and have a large
degree, while many other nodes are connected to only a small
number of nodes.

Fig. 1. BA model.

Next, Fig. 2 shows the ER model, a model for random
networks used in this study. In this network, all pair nodes
are connected with the same probability. Furthermore, since
the degree distribution follows a binomial distribution, themajority of nodes have a degree close to the average degree.

- 80 -The normalized circuit equation described as follows:


100



∑

dxn

2

=αεx(1−x)−y−Eγ(x−x)
nnnknknk

n
 dτ

dyn

= xn

 dτFig. 2. ER model.

III. S YSTEM MODELFigure 3 shows a van der Pol oscillator. This oscillator isa simple circuit, consisting of only a capacitor, an inductor
and a nonlinear element. This circuit is considered as a single
node. By connecting these circuits with resistors, edges are
created to form each network.

nv

CiLiG

L

G

n,k=1

(n, k = 1, 2, · · ·, 100).

Here, The parameter of the van der Pol oscillator is setto ε = 0.1. Moreover α represents the small error of thecapacitor. The results of the synchronization rate in the twonetwork models are shown in the range of [0.95:1.05] inincrements of 0.001. Further, Enk represents the adjacency
matrix of the network. This is a matrix that indicates whether
node n and node k are connected or not. Enk = 1 if noden and node k are connected, and Enk = 0 if they are notconnected.
The coupling strength γnk is determined by using theparameter q as follows:√Lq1
γnk ==.
(4)
Rnk C
dnk 2

The parameter q is the weight of parameter that determines
the coupling strengths [6]. In this case, we set parameter q =0.01.
IV. R ESULTS
Fig. 3. van der Pol oscillator.

In order to investigate the synchronization phenomena in thecoupled parts, we define the synchronization between circuits
as follows:

The circuit equation is described as follows:
100
∑

dvn1


C
=−i−i
(vn − vk )−Lngn


dt
Rnk
n,k=1

din


= vnL


dt



|xn − xk | < 0.01.

(5)

(1)
(n, k = 1, 2, · · ·, 100).

Here, the parameter ig is the equation of the nonlinear element,
and described as follows:ig = −g1 v + g3 v 3 .

(2)

Where n and k are the number of circuits. Figure 4 shows
the differential voltage waveform observed in this study. The
two lines in Fig. 4 represent the threshold value shown inEq. (5). When the voltage difference meets this threshold, thecircuits are considered to be synchronized. This is the voltage
difference in the range of τ = 10000 to τ = 20000 where thesystem settles into a steady state, and the synchronization rateis calculated under these conditions.

By using the normalization parameter and the variables asfollows:√√√
g1
g1 Cx, i =y, t = LCτv=g3g3 L√√L1 L
ε = g1, γ=.
CR C- 81 -(3)The number of pair nodes

xn-xk

0.01-0.01

Synchronization rates [%]τFig. 6. Relationship between the synchronization rate and the numberof pair nodes in the BA model and ER model determined byEuclidean distance.

Fig. 4. The differential voltage waveform.

TABLE IIAVERAGE SYNCHRONIZATION RATES OF TWO NETWORK MODELS .
NetworkAverage synchronization rates[%]

BA model50.811

ER model28.346

Finally, Fig. 7 and Fig. 8 show the results to compare
the case of coupling strength kept constant at the averagevalue obtained from Eq. (4) with the case of coupling strength
determined by Euclidean distance for the BA model and ERmodel. Here, the average coupling strength is γ ≃ 0.2816 forthe BA model and γ ≃ 0.3736 for the ER model. Table IIIand Table IV shows the average synchronization rate of eachnetwork under these conditions.

The number of pair nodes

The number of pair nodes

First, Fig. 5 shows the relationship between the synchronization rate and the number of pair nodes when the common
coupling strength is kept constant for the BA model and ER
model. Here, the coupling strength is set to γ ≃ 0.3276.
Table I shows the average synchronization rate for each
network under these conditions.

Synchronization rates [%]Fig. 5. Relationship between the synchronization rate and the numberof pair nodes in the BA model and ER model with the same couplingstrength.
Synchronization rates [%]Fig. 7. Relationship between the synchronization rate and the numberof pair nodes in the BA model with the coupling strength determinedby Euclidean distance and constant (average).
TABLE INetwork
Average synchronization rates[%]

BA model77.674

The number of pair nodes

AVERAGE SYNCHRONIZATION RATES OF TWO NETWORKMODELSWITH THE SAME COUPLING STRENGTH .
ER model81.074

Next, Fig. 6 shows the relationship between the synchronization rate and the number of pair nodes when the couplingstrength is determined by Euclidean distance between nodes
for the BA model and ER model using Eq. (4). Table II also
shows the average synchronization rate of each network underthese conditions.

Synchronization rates [%]Fig. 8. Relationship between the synchronization rate and the numberof pair nodes in the ER model with the coupling strength determinedby Euclidean distance and constant (average).

- 82 -TABLE III

AVERAGE SYNCHRONIZATION RATES FOR DIFFERENT METHODSOF DETERMINING THE COUPLING STRENGTH OF BA MODEL .
Coupling strength
Average synchronization rates[%]

Determindedby distance
50.811

Constant(average)
72.425

V. CONCLUSIONTABLE IVAVERAGE SYNCHRONIZATION RATES FOR DIFFERENT METHODSOF DETERMINING THE COUPLING STRENGTH OF ER MODEL .
Coupling strength
Average synchronization rates[%]

Determindedby distance
28.346the coupling strength is determined by Euclidean distance,
the synchronization rate is much lower than when it is kept
constant at the average value.

Constant(average)
84.425Figure 9 shows the relationship between the synchronization
rate and the number of pair nodes when the coupling strengthis determined by Euclidean distance for (a) BA model and (b)ER model.

In this study, 100 van der Pol oscillators were used toinvestigate the synchronization phenomena for two networks,
the BA model and the ER model. Where the coupling strength
was considered as the Euclidean distance between nodes. Theresults show that the synchronization rate of both networks
decreases when Euclidean distance is considered, comparedto the constant coupling strength. The decrease was smaller inthe BA model than in the ER model. This may be due to theinfluence of the hub in the BA model.
In the future, we would like to further investigate the effectof hubs and local node pairs such as indirectly coupled pairs.

The number of pair nodes

The number of pair nodes

R EFERENCES

Coupling strengh(a)BA modelγ

Coupling strenghγ(b)ER modelFig. 9. Coupling strength distribution.

First, we compare the network structure of the two networks
used in this study. If we keep the common coupling strength
constant, we can see from Fig. 5 and Table I. There is no
significant difference in the distribution of the synchronization
rate and the average synchronization rate.
As a result, we compare the networks when the couplingstrength is determined by Euclidean distance. Figure 6 showsthat the BA model has the largest number of node pairs witha synchronization rate of 90-100[%], while the ER model has
the largest number of node pairs with a synchronization rate of0-10[%]. Table II also shows that about half of the nodes in theBA model are synchronized, while only about a quarter of thenodes in the ER model are synchronized, indicating that there
are significant differences among the networks. In addition,
when the coupling strength is kept constant for each network,
the distribution of the synchronization rate is higher for both
BA model and ER model, as shown in Fig. 7 and Fig. 8. Table
III and IV shows that the average synchronization rate is also
higher when the coupling strength is constant. This trend ismore pronounced in the ER model. The synchronization rateof most pair nodes is 90-100[%] when the coupling strength
is constant, as shown in Fig. 8.
Further, Fig. 9 also shows that in both networks, there aremany pair nodes for which the distribution of the couplingstrength is lower than the average value. Accordingly, when[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,
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